Dan Dissertation

THE UNIVERSITY OF CHICAGO

DISPLACED LOGNORMAL AND DISPLACED HESTON VOLATILITY SKEWS:

ANALYSIS AND APPLICATIONS TO STOCHASTIC VOLATILITY SIMULATIONS

A DISSERTATION SUBMITTED TO

THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCE

IN CANDIDACY FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

DEPARTMENT OF STATISTICS

BY

DAN WANG

CHICAGO, ILLINOIS

JUNE 2010

To my parents

TABLE OF CONTENTS

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

PUBLICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 DISPLACED LOGNORMAL PROCESS . . . . . . . . . . . . . . . . . . . . . . . 42.1 Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Displaced Lognormal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Implied volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.2 Global behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.3 At-the-money behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.4 Short-expiry behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Displaced anti-Lognormal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.1 Implied volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2 Global behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.3 At-the-money behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.4 Short-expiry behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 DISPLACED HESTON PROCESS . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1 Displaced Heston . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.1 Implied volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1.2 At-the-money behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.3 Short-expiry behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Displaced anti-Heston . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.1 Implied volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.2 Short-expiry behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.3 Generalization of short-expiry behavior . . . . . . . . . . . . . . . . . 20

iii

4 CALIBRATION OF DL AND DH PROCESS . . . . . . . . . . . . . . . . . . . . 224.1 Calibration of DL and DH Process . . . . . . . . . . . . . . . . . . . . . . . 22

4.1.1 Calibration of DL process . . . . . . . . . . . . . . . . . . . . . . . . 224.1.2 Calibration of DH process . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2 Calibration DL and DH to CEV/SABR . . . . . . . . . . . . . . . . . . . . . 234.2.1 CEV and SABR stochastic volatility models . . . . . . . . . . . . . . 234.2.2 Calibration of DL to CEV/SABR . . . . . . . . . . . . . . . . . . . . 254.2.3 Calibration of DH to SABR . . . . . . . . . . . . . . . . . . . . . . . 26

5 VARIANCE REDUCTION IN MONTE CARLO SIMULATION . . . . . . . . . . 275.1 Variance Reduction Using Control Variate . . . . . . . . . . . . . . . . . . . 27

5.1.1 DL or DH as a control variate . . . . . . . . . . . . . . . . . . . . . . 285.1.2 Example I: Discretely sampled barrier option under CEV/SABR dy-

namics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.1.3 Numerical results I: Discretely sampled barrier option under CEV/SABR

dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.1.4 Example II: Discretely sampled arithmetic Asian option under SABR

dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.1.5 Numerical results II: Discretely sampled arithmetic Asian option under

SABR dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2 Variance Reduction Combining Control Variate and Importance Sampling . . 35

5.2.1 Importance sampling on options pricing . . . . . . . . . . . . . . . . 355.2.2 Drifted DH/DL process . . . . . . . . . . . . . . . . . . . . . . . . . . 375.2.3 Combine control variate with importance sampling . . . . . . . . . . 375.2.4 Example: Discretely sampled barrier option under SABR dynamics . 385.2.5 Numerical results: Discretely sampled barrier option . . . . . . . . . 40

6 DISCRETISATION SCHEME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.1 Partial Strong Convergency of Stochastic Volatility Process . . . . . . . . . . 446.2 Strong Convergence of Mean-reverting CEV Process . . . . . . . . . . . . . . 466.3 Discretisation Schemes Used in the Monte Carlo Simulation . . . . . . . . . 48

7 LARGE-EXPIRY IMPLIED VOLATILITY OF DISPLACED LOGNORMAL . . 507.1 Large-strike and Large-expiry Behavior . . . . . . . . . . . . . . . . . . . . . 50

7.1.1 Case one: K = S0exT , x ∈ R/[−1

2σ2, 1

2σ2] . . . . . . . . . . . . . . . 50

7.1.2 Case two: K = S0exT , x ∈ (−σ2/2, σ2/2) . . . . . . . . . . . . . . . 52

7.1.3 Case three: K = S0exTα . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.2 Fixed-strike Large-expiry Implied Volatility . . . . . . . . . . . . . . . . . . . 56

8 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Appendix

iv

A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63A.1 Appendix: Proof of Theorem 1–Global Behavior of Displaced Lognormal . . 63A.2 Appendix: Proof of Theorem 2–At-the-money Behavior of Displaced Lognormal 69A.3 Appendix: Proof of Theorems 3 and 6–Short-expiry Behavior of DL . . . . . 70A.4 Appendix: Proof of Theorem 4–Global Behavior of Displaced anti -Lognormal 71A.5 Appendix: Proof of Theorem 5–At-the-money Behavior of Displaced anti -

Lognormal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77B.1 Appendix: Proof of Theorem 7–At-the-money Behavior of Displaced Indepen-

dent Stochastic Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77B.2 Appendix: Proof of Theorems 8 and 9– Short-expiry Behavior of DH . . . . 78B.3 Appendix: Proof of Propositions 3.1.2 and 3.2.1–Level, Slope and Convexity

of DH Short-expiry Implied Volatility . . . . . . . . . . . . . . . . . . . . . . 80

C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85C.1 Appendix: Proof of Theorem 12–Partial Strong Convergency of Stochastic

Volatility Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85C.2 Appendix: Coefficients of SABR Satisfy the Local Lipschtiz Condition (*) . . 88C.3 Appendix: Proof of Proposition 6.2.2 and Theorem 13–Strong Convergence

of Mean-reverting CEV Process . . . . . . . . . . . . . . . . . . . . . . . . . 89

D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94D.1 Appendix: Proof of Theorem 14 –Large-strike and Large-expiry Asymptotic

of Displaced Lognormal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94D.2 Appendix: Proof of Theorem 15–First Order Approximation of Large-strike

Large-expiry Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . 98D.3 Appendix: Proof of Theorem 16–Asymptotic Formula of Black-Scholes Call

Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99D.4 Appendix: Proof of Theorems 17 and 18– Second and Third Order Approxi-

mation of Large-strike Large-expiry Implied Volatility . . . . . . . . . . . . 101D.5 Appendix: Proof of Theorem 19–Large-strike Large-expiry At-the-money Im-

plied Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102D.6 Appendix: Proof of Theorem 20–Approximation of Implied Volatility when

K = S0 exp(xTα) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102D.7 Appendix: Proof of Theorem 21–Fixed-strike Large-expiry Implied Volatility 103

D.7.1 Asymptotic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 103D.7.2 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

D.8 Appendix: Proof of Theorem 22–Approximation of Fixed-Strike Large-expiryImplied Volatility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

v

LIST OF FIGURES

2.2.1 Theorem 3 formula and Exact σimp for T = 1.0. . . . . . . . . . . . . . . . 82.3.1 Theorem 5.1 bounds of σatm for T = 3.0. . . . . . . . . . . . . . . . . . . 11

3.2.1 Theorem 9 formula and exact σimp. . . . . . . . . . . . . . . . . . . . . . . 19

7.1.1 Theorem 14 formula and Exact σimp for T = 10. . . . . . . . . . . . . . . 517.1.2 Theorem 17 formula and Exact σimp. . . . . . . . . . . . . . . . . . . . . 537.1.3 Theorems 17 and 18 formula and Exact σimp. . . . . . . . . . . . . . . . . 587.1.4 Theorems 17 and 18 formula and Exact σimp. . . . . . . . . . . . . . . . . 587.1.5 Theorem 20 formula and Exact σimp for T = 10. . . . . . . . . . . . . . . 597.1.6 Theorem 20 formula and Exact σimp for T = 7. . . . . . . . . . . . . . . . 597.2.1 Theorem 22 formula and Exact σimp for T = 20. . . . . . . . . . . . . . . 60

vi

LIST OF TABLES

5.1.1 Percentage reduction of variance, using DL for down-and-out call on CEV 325.1.2 Percentage reduction of variance, using DL for down-and-out call on SABR 325.1.3 Percentage reduction of variance, using DH for down-and-out call on SABR 325.1.4 Percentage reduction of variance, using DL as control for Asian Option on

SABR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.1.5 Percentage reduction of variance, using LN as control for Asian Option on

SABR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2.1 Percentage reduction of variance, using importance sampling for down-and-

out call on SABR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2.2 Percentage reduction of variance, using ISDL and ISDH for down-and-out

call on SABR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2.3 Percentage reduction of variance, using importance sampling for down-and-

in call on SABR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2.4 Percentage reduction of variance, using ISDL and ISDH for down-and-in call

on SABR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

vii

ABSTRACT

We analyze the displaced (anti-)lognormal (DL) and displaced (anti-)Heston (DH) volatility

skew. In particular, for the displaced lognormal, we prove the global monotonicity of the

implied volatility, and an at-the-money bound on the steepness of the downward volatility

skews, which therefore cannot reproduce some features observed in the equity market. A

variant, the displaced anti-lognormal, overcomes this steepness constraint, but its state space

is bounded above and unbounded below. We prove the global monotonicity of its implied

volatility too. For the displaced Heston dynamics, we show that the at-the-money slope has

the same sign as the displacement. What’s more, we give an explicit formula for the DL and

DH’s short-expiry limiting volatility skew, which allows direct calibration of their parameters

to volatility skews implied by market data or by other models. In the end, we analyze the

large-expiry limiting volatility of the displaced lognormal and give an asymptotic formula of

it in the region of large-strike and fixed-strike respectively.

We propose using the DL/DH dynamics as a control variate, to reduce variance in Monte

Carlo simulations of the CEV and SABR local/stochastic volatility models. We give simula-

tion results to show that a carefully constructed control variate can significantly reduce the

variance in the Monte Carlo simulations. We further propose a combination of the impor-

tance sampling and the control variate to reduce the variance. Numerical simulations show

that significant variance reduction can be achieved.

Finally we discuss the convergency of the discretisation schemes of the stochastic pro-

cesses encountered in the Monte Carlo simulations. Under some regularity conditions, we

give a partial strong convergency result for the stochastic volatility process. Moreover, we

give a strong convergency result for the mean-reverting CEV process.

viii

ACKNOWLEDGEMENTS

First and foremost, I am deeply grateful to my advisor, Professor Roger Lee, for sharing

with me his insights and wisdom, for his instructive guidance and invaluable advice, for his

constant support and encouragement over the past few years. Without him, this dissertation

would not be possible.

I am grateful to Professor Steven Lalley and Professor Per Mykland for being on my

committee and for their valuable suggestions and comments. In addition, I thank Professor

Per Mykland for many inspiring discussion on topics of application of Statistics to Finance.

Also, I would like to express my sincere thanks to all the faculty, stuff and students

of the Department of Statistics at the University of Chicago. Because of them, I have a

wonderful education here. I feel fortunate to study and pursue my doctoral degree in such

an intellectually stimulating and friendly environment.

I would also like to thank my friends who made my time so enjoyable that I will al-

ways cherish my time here. They are my source of advice and discussion. Particularly, I

would like to thank Xiaohui, Han, Changgee, Wenlong, William, Han, Yibi, Yingying, Oli,

Omar, Baoguan, Zuoheng, Christina, Dale, Winfried, Mike, Minsun and Darongsae. I thank

Nathaniel for proof reading my thesis.

Finally, to my parents, to whom this thesis is dedicated, goes my deepest appreciation.

I thank them for their love, care and support over the years. Words cannot express the

gratitude I owe them. I owe a special thanks to my dearest husband, Zhuo, for his love and

care, for his understanding, support, and for accompanying me through this work.

ix

PUBLICATION

Part of the thesis has been published in Lee and Wang (2009), which include the following

materials: Chapter 2.1, 2.2, 2.3.4, Chapter 4.1.1, 4.2.1, 4.2.2, and the parts related to the

DL process in Chapter 5.

x

CHAPTER 1

INTRODUCTION

Given an empirically-observed or model-generated price of a call or put, the implied volatility

is by definition the volatility parameter for which the Black-Scholes formula recovers the

given option price. Regardless of what process actually underlies the given option price, the

implied volatility provides a canonical language or scale by which option prices are commonly

quoted and compared. At any expiry, the volatility skew – meaning the implied volatility as

a function of all strikes – captures the full risk-neutral underlying distribution at that expiry,

and hence constitutes a natural framework to understand and to compare distributions.

In particular, comparison can occur between an empirically-observed volatility skew and

a model-generated volatility skew, for the purpose of calibrating the model’s parameters, or

for the purpose of understanding what empirical features can or cannot be reproduced by

the model. Comparison can also occur, between volatility skews generated by two different

models, for the purpose of approximating the features of a more complex model, using a

simpler model.

The lognormal (Black-Scholes 1973) model generates a flat implied volatility skew, which

does not agree with the sloping skews observed empirically in equity, FX, and interest

rate markets. Displacing the lognormal (Rubinstein 1983) does generate a sloping implied

volatility skew. Marris (1999), Brigo and Mercurio (2002), Joshi and Rebonato (2003),

and Svoboda-Greenwood (2009), have investigated the displaced lognormal (and extensions

thereof), as a pricing model or as a analytical approximation to other models, motivated

largely by applications to interest rate derivatives. In contrast, we draw motivation mainly

from problems arising in equity markets, such as how to calibrate to volatility skews that

slope downward more steeply than all displaced lognormal skews; and we intend to use the

calibrated process less for its analytical pricing than for its applicability to Monte Carlo

pricing.

First, we bound the level and slope of the implied volatility skews generated by dis-

placed lognormal diffusions in various regimes (global, or at-the-money, or short-expiry).

1

We prove, among other results, the global monotonicity of implied volatility, and an at-the-

money upper bound on the absolute slope of downward volatility skews, under displaced

lognormal dynamics, which therefore cannot model some features (non-monotonicity and a

steep downward slope) observed in equity market volatility skews. A variant, the displaced

anti -lognormal, overcomes the steepness constraint, but its state space is bounded above

and unbounded below, unlike stock prices. For the displaced anti -lognormal, we prove the

global monotonicity of implied volatility, and an at-the-money bound for the level and slope

of the implied volatility.

The Heston model (Heston 1993) generates the volatility smile and has an analytical

solution for call and put options. We propose a displaced (anti-)Heston diffusion and analyze

its implied volatility. We show that the slope of its at-the-money implied volatility has the

same sign as the displacement and its short-expiry implied volatility lacks the skewness

observed in the equity market; while the displaced anti -Heston diffusion overcomes the

skewness problem, its state space is bounded above and unbounded below.

For both the displaced (anti-)lognormal (DL) and the displaced (anti-)Heston (DH) dif-

fusions, we find an explicit formula for their short-expiry limiting volatility skew. This helps

to calibrate the model to the empirical data or other models. Moreover, for any general

process which is a positive martingale, and its corresponding (anti-)displaced process, we

find an explicit formula between the short-expiry implied volatilities of the two processes.

In light of these restrictions on what features the DL and the DH can model, we then

exploit the DL or DH, not as a model, but as a control variate, to reduce variance in Monte

Carlo simulation of other models, such as the CEV and SABR local/stochastic volatility

models. Fisher and Tataru (2010) state that “During the past few years, practitioners have

settled on a consensus of using a mixed stochastic/local volatility (SLV) model as the market

standard for pricing barrier options.” CEV/SABR belong to the SLV family of models so

pricing options under them is of practical importance. We therefore consider pricing barrier

option and Asian option under the CEV/SABR model. We give numerical examples which

illustrate significant variance reductions, when the control variate on DL/DH is carefully

constructed. What’s more, we explore the combination of the control variate and other

variance reduction technique, particularly importance sampling. We give numerical examples

to show the significance of the variance reduction.

In the Monte Carlo simulation, most of the stochastic processes can not be exactly

sampled. Thus we refer to discretisation schemes to conduct the Monte Carlo simulation.

2

We discuss the conditions which guarantee the convergence of the discretized process to the

underlying continuous process. These convergence results serve as theoretical foundations

for the Monte Carlo simulation.

Finally, we analyze the large-expiry implied volatility of displaced lognormal dynamics.

We give explicit approximation formula of the implied volatility in the large-strike large-

expiry case and the fixed-strike large-expiry case.

Chapter 2 discusses the features of the implied volatility of the DL diffusion. Chapter 3

discusses the features of the implied volatility of the DH diffusion. Chapter 4 illustrates how

to calibrate the parameters of the DL and DH processes. Chapter 5 discusses how to use DL

or DH processes to construct control variate to reduce variance in Monte Carlo simulations

of option pricing; it also discusses how to combine the control variate with the importance

sampling to achieve variance reduction. Chapter 6 discusses the discretisation schemes used

in the Monte Carlo simulations. Chapter 7 discusses the large-expiry behavior of the implied

volatility of displaced lognormal dynamics. Appendix contains most of the proofs.

3

CHAPTER 2

DISPLACED LOGNORMAL PROCESS

2.1 Implied Volatility

We work under martingale measure, and we either assume zero interest rates, or stipulate

that all prices are quoted as forward prices.

Our definition of the implied volatility skew will refer to the function CBS , specified as

follows.

Define CBS : R3∗ × R+ → R and PBS : R3

∗ × R+ → R, where R∗ := R\0 and

R+ := (0,∞), by

CBS(s, k,Σ, T ) := sN(d+)− kN(d−) (2.1)

PBS(s, k,Σ, T ) := kN(−d−)− sN(−d+) (2.2)

d± :=log(s/k)

Σ√T± Σ√T

2(2.3)

where N denotes the standard normal CDF. We may suppress the last argument (T ) of CBS .

Definition 1 (Implied volatility). Let X be a process with X0 > 0. For all positive K,T

such that

(X0 −K)+ < E(XT −K)+ < X0, (2.4)

define the implied volatility of X at (K,T ) to be the σimp > 0 such that

CBS(X0, K, σimp, T ) = E(XT −K)+. (2.5)

Write σXimp(K,T ) for this implied volatility, which is well-defined, because CBS(X0, K, ·, T )

is strictly increasing on R+, and has range ((X0 −K)+, X0).

We refer to the function σXimp(·, T ) as the implied volatility skew of X at expiry T .

4

2.2 Displaced Lognormal

Definition 2. A process S follows displaced lognormal dynamics, with displacement θ ∈ R,

if

dSt = σ(St − θ)dWt, S0 > θ, σ > 0, (2.6)

where W is Brownian motion.

Thus S − θ is a driftless geometric Brownian motion with volatility σ, and the interval

of points attainable by S is (θ,∞). If modeling a nonnegative underlying such as a stock

price, this model for θ < 0 will misprice deep-out-of-the-money puts, due to the possibility

of ST < 0. This model has further limitations, even for at-the-money contracts, as we will

see later.

For K > θ, a K-strike T -expiry European call option on a displaced lognormal S has

price

E(ST −K)+ = E((ST − θ)− (K − θ))+ = CBS(S0 − θ,K − θ, σ, T ). (2.7)

In general, payoffs invariant to parallel shifts of the S path and the contract parameters can

be priced using Black-Scholes model valuation methods, but applied to displaced arguments.

2.2.1 Implied volatility

Let us apply Definition 1 to the case that X = S, a displaced lognormal. If θ ≥ 0 then

(2.4) holds for all K,T positive. If θ < 0, then the first inequality in (2.4) holds for all K,T

positive, but the second inequality E(ST −K)+ < S0 may fail for small positive K, due to

the nonzero probability that ST < 0. We therefore take care to define the strike interval on

which implied volatility exists. For displaced lognormal S, let

KS(T ) := K > θ+ : CBS(S0 − θ,K − θ, σ, T ) < S0 (2.8)

which is a semi-infinite interval, by monotonicity of CBS in K. For each T > 0 and each

K ∈ KS(T ), equation (2.5) defines the implied volatility of S to be the σimp such that

CBS(S0, K, σimp, T ) = CBS(S0 − θ,K − θ, σ, T ). (2.9)

5

To abbreviate the σSimp(K,T ) and KS(T ) notations for displaced lognormal S, we will sup-

press the S superscript, and possibly also the T argument.

2.2.2 Global behavior

Theorem 1 establishes the following global properties of σimp: If θ < 0 then σimp is every-

where strictly decreasing in K, and bounded below by σ. If θ > 0 then σimp is everywhere

strictly increasing in K, and bounded above by σ. In both cases, the global bounds are also

asymptotes.

Theorem 1 (Global behavior). Implied volatilities in the displaced lognormal model (2.6)

have the following global properties.

1. (Monotonicity in strike). For all T > 0 and K ∈ K(T ),

sgn∂σimp

∂K(K) = sgn θ. (2.10)

2. (Upper/lower bound). For all T > 0 and K ∈ K(T ):

If θ > 0 then σimp(K) < σ. If θ < 0 then σimp(K) > σ. (2.11)

3. (Sharpness of bound). For all T > 0, we have σimp(K)→ σ as K →∞.

Hence supK∈K(T ) σimp(K) = σ if θ > 0; and infK∈K(T ) σimp(K) = σ if θ < 0.

Proof. Appendix A.1.

Brigo-Mercurio (2001) proves the K = S0 case of (2.10, 2.11). Theorem 1 extends to all

K ∈ K(T ).

Remark 2.2.1. Empirical volatility skews are typically not monotonic over the entire range of

strikes; a volatility skew which slopes downward in the central portion of the strike range will

usually still turn upward at sufficiently large strikes. Theorem 1 proves that the displaced

lognormal cannot reproduce this empirical feature.

6

2.2.3 At-the-money behavior

Theorems 2 and 3 focus on two different subsets of the (K,T ) domain.

Theorem 2 examines the at-the-money strike K = S0. Specifically, if T > 0 and S0 ∈K(T ), then define the at-the-money implied volatility σatm(T ) := σimp(S0, T ), which may

be abbreviated as σatm. We bound the level σatm and also the slope of σimp at-the-money.

By “slope” we always mean ∂ log σimp/∂ logK, the strike-elasticity of implied volatility.

Theorem 2 (At-the-money behavior). At-the-money implied volatilities in the displaced

lognormal model (2.6) have the following properties.

1. (At-the-money level). If T > 0 and S0 ∈ K(T ) then

σatm ≥(

1 +|θ|S0

)σ if θ ≤ 0;

σatm ≤(

1− θ

S0

)σ if θ ≥ 0.

(2.12)

2. (At-the-money slope). If θ < 0 and T > 0 and S0 ∈ K(T ) then

1

2

|θ||θ|+ S0

≤∣∣∣∣∂ log σimp

∂ logK

∣∣∣∣K=S0

=N(σatm

√T/2)−N(σ

√T/2)

φ(σatm√T/2)

√Tσatm

<1

2eσ

2atmT/8. (2.13)


Remark 2.2.2. If T ≤ 1 and σatm ≤ 100%, then (1/2)eσ2atmT/8 < 0.57. Empirically, however,

equity volatility skews typically slope downward more steeply than−0.57. Indeed, in S&P500

daily data from all dates (1996–2008) in the OptionMetrics database, the approximate1 3-

month-expiry at-the-money slope ∂ log σimp/∂ logK is more negative than −1.29 on 90% of

the days in the sample. Theorem 2 proves that the displaced lognormal cannot reproduce

steepness of this magnitude.

1. Our source is the “Volatility Surface” data set, which contains volatility skews in-terpolated by OptionMetrics using kernel smoothing. We approximate the slope aslog(σimp(K1)/σimp(K0))/ log(K1/K0), where K1 is the strike of a 0.50-delta call, and K0 is thestrike of a 0.55-delta call, as computed by OptionMetrics.

7

2.2.4 Short-expiry behavior

Theorem 3 takes the short-expiry T ↓ 0 limit of the implied volatility skew, and expresses the

solution explicitly. The K = S0 case is known (indeed Rebonato (2004) refines the K = S0

formula, to address the case of T large). The contribution of Theorem 3 is to find and prove

a short-expiry σimp formula valid for every strike K > θ+.

Figure 2.2.1: Theorem 3 formula and Exact σimp for T = 1.0.

60 70 80 90 100 110 120 130 140 150 16017.5

18

18.5

19

19.5

20

20.5

21

21.5

22

K

Per

cent

age

poin

ts

S0=100, σ(1−θ/S

0)=20%

θ=25

θ=−50

Exact σimp

Theorem 3 formula

Theorem 3. For all K > θ+, in the displaced lognormal model (2.6),

limT→0

σimp(K,T ) =

σ log(S0/K)

log((S0 − θ)/(K − θ))if K 6= S0

σ(1− θ/S0) if K = S0.

(2.14)


Remark 2.2.3. The formula in the right-hand side of (2.14) provides, moreover, a remarkably

accurate approximation to σimp(K,T ) even for some T not close to 0. Figure 2.2.1 compares

the Theorem 3 formula and the exact σimp(K,T ), at expiry T = 1.0.

8

For some expirations of moderate length, therefore, the σ log(S0/K)/ log((S0−θ)/(K−θ))formula may still facilitate calibration of the displaced lognormal parameters (σ, θ) to an

empirically observed volatility skew, or to a model-generated volatility skew.

2.3 Displaced anti-Lognormal

The previous section’s results show that with θ < 0, the displaced lognormal produces

downward-sloping implied volatility, but not of steepness commensurate with typical equity

options data – regardless of how large a negative value θ takes.

A related process, however, does generate arbitrarily large downward slopes.

Definition 3. A process S follows displaced anti -lognormal dynamics if

dSt = σ(St − θ)dWt, 0 < S0 < θ, σ < 0, (2.15)

where W is a Brownian motion.

Thus θ − S is a driftless geometric Brownian motion with volatility −σ > 0, and the

interval of points attainable by S is (−∞, θ).Displaced anti-lognormal pricing calculations have the tractability of the displaced log-

normal. For instance, to price a T -expiry call struck at K < θ, on a displaced anti-lognormal

S,

E(ST −K)+ = E(θ −K − (θ − ST ))+

= PBS(θ − S0, θ −K,−σ, T ) = CBS(S0 − θ,K − θ, σ, T ).(2.16)

So we have formally the same CBS formula as in the displaced lognormal case. Here its

first three arguments are negative, which presents no problem; the CBS function is still

well-defined by (2.1). An equivalent way to express the result, without negative arguments,

is CBS(θ −K, θ − S0,−σ, T ).

Recognizing the similarities between the displaced lognormal and anti-lognormal, the

following terminology groups them together:

Definition 4 (DL). A process S which satisfies either the displaced lognormal (2.6) or the

displaced anti-lognormal (2.15) specification is said to be a DL process.

9


Implied volatility for displaced anti -lognormal S is defined on the strike interval

KADL(S, T ) := K ∈ (0, θ) : CBS(S0 − θ,K − θ, σ, T ) < S0. (2.17)

For K ∈ KADL(S, T ), we have

S0 > CBS(S0−θ,K−θ, σ) = PBS(θ−S0, θ−K,−σ) > ((θ−K)− (θ−S0))+ = (S0−K)+,

so (2.4) holds and σSimp(K,T ) is thereby well-defined. To abbreviate the σSimp(K,T ) and

KADL(S, T ) notations for displaced anti-lognormal S, we will suppress the S superscript,

and possibly also the T argument. Note that we have

PBS(S0, K, σimp) = E(K − S0)+ = CBS(θ − S0, θ −K, |σ|). (2.18)

Consequently,

CBS(S0, K, σimp) = E(ST −K) + PBS(S0, K, σimp) = (S0 −K) +CBS(θ− S0, θ−K, |σ|).(2.19)

2.3.2 Global behavior

For the implied volatility under the displaced anti -lognormal, we have the global monotonic-

ity behavior similar to the behavior under the displaced lognormal. This is given by Theorem

4.

Theorem 4 (Global behavior). Implied volatilities in the anti-lognormal model (2.15) de-

creases monotonically in strike. That is, for all T > 0 and K ∈ KADL(T ),

sgn∂σimp

∂K(K) = − sgn θ. (2.20)


10


Theorem 5 examines the at-the-money strike K = S0. σatm(T ) is abbreviated as σatm. We

bound the level σatm and also the slope of σimp at-the-money.

100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

θ

σ atm

σatm

(θ/S0−1)|σ|

|σ|

Figure 2.3.1: Theorem 5.1 bounds of σatm for T = 3.0. Process parameters are S0 = 100, K =

100, σ = −0.3. The solid blue line is σatm, the black dashed line is (θ/S0 − 1)|σ| and the red

horizontal line is at the level of |σ|. As we can see from the figure, when θ ≥ 2S0, we have

σatm > (θ/S0 − 1)|σ|; though when S0 < θ ≤ 2S0, it looks like the two lines coincide with each

other, if we zoom in, we could see that σatm < (θ/S0 − 1)|σ|.

Theorem 5 (At-the-money behavior). At-the-money implied volatilities in the anti-lognormal

model (2.15) have the following properties.

1. (At-the-money level). If θ > S0, T > 0 and S0 ∈ KADL(T ) then

σatm ≤( θS0− 1)|σ| if S0 < θ ≤ 2S0;

σatm ≥( θS0− 1)|σ| if θ ≥ 2S0.

(2.21)

2. (At-the-money slope). If θ > S0, T > 0 and S0 ∈ KADL(T ) then

1

2

θ

θ − S0≤∣∣∣∣∂ log σimp

∂ logK

∣∣∣∣K=S0

≤ θ

2(θ − S0)eσ

2atmT/8. (2.22)

11


Remark 2.3.1. θ = 2S0 is the critical case. When θ < 2S0, we have σatm <(θS0− 1)|σ|;

when θ > 2S0, we have σatm >(θS0−1)|σ|; the equality happens when θ = 2S0. This could

be explained by using the put-call parity as follows. When K = S0 and θ = 2S0, we have

(ST −K)+ = (K − (θ − ST ))+. (2.23)

If St follows the displaced anti -lognormal dynamics with parameters (θ = 2S0, σ), then using

K = S0 = θ−K, the right-hand side of (2.23) is PBS(θ−S0, θ−K, |σ|) = PBS(S0, K, |σ|) =

CBS(S0, K, |σ|). Therefore the implied volatility of the process S is exactly |σ|.

Figure 2.3.1 illustrates the Theorem 5.1 bounds of σatm. The figure also suggests that

we will not have |σ| as the upper or lower bound for σimp as in the displaced lognormal case.


Theorem 3 extends to all DL processes:

Theorem 6 (Short-expiry behavior). For all K > θ+ in the displaced lognormal model (2.6),

as well as for all K ∈ (0, θ) in the displaced anti-lognormal model (2.15), the conclusion

(2.14) holds.


The Theorem 6 conclusion, and its derivative with respect to logK, yield the short-expiry

limiting volatility skew’s level and slope

σimp

∣∣∣∣T↓0, K=S0

=σ(S0 − θ)

S0,

∂ log σimp

∂ logK

∣∣∣∣T↓0, K=S0

=θ

2(S0 − θ). (2.24)

This holds under all DL dynamics. The distinction is that under displaced lognormal dynam-

ics, we have θ < S0, hence the short-expiry slope cannot be more negative than −1/2. Under

displaced anti -lognormal dynamics, we have S0 < θ, hence (2.24) can produce arbitrarily

steep negative slopes.

12

CHAPTER 3

DISPLACED HESTON PROCESS

3.1 Displaced Heston

Definition 5. A process S follows displaced Heston dynamics, with displacement θ ∈ R, if

dSt = σt(St − θ)dWt, S0 > θ,

dσ2t = κ(µ− σ2

t )dt+ εσtdBt,

dBt = ρdWt +

√1− ρ2dW ∗t .

(3.1)

where Wt, W∗t are i.i.d. Brownian motions and σt is non-negative for all t > 0.

Thus S−θ follows the Heston model, and the interval of points attainable by S is (θ,∞).

As with the displaced lognormal model, if modeling a nonnegative underlying process such

as a stock price, for θ < 0 this model will misprice deep-out-of-the-money puts, due to the

possibility of ST < 0.

We would like to characterize the implied volatility of displaced Heston dynamics. We will

have similar at-the-money behavior and short-expiry behavior as in the DL case. However,

we will not have the global behavior of implied volatility.


Implied volatility for the displaced Heston S is defined on the following strike interval:

KDH(T ) := K > θ+ : (S0 −K)+ < E(ST −K)+ < S0 (3.2)

For each T > 0 and eachK ∈ KDH(T ), (2.5) defines the implied volatility of the displaced

Heston to be σDHimp (K,T ) > 0 such that

CBS(S0, K, σDHimp (K,T ), T ) = E(ST −K)+. (3.3)

We will suppress the T argument of σimp(K,T ).

13


Although for the displaced Heston process, we will not have the global behavior that the

slope of implied volatility is determined by the sign of the displacement θ, we are able to

show that when ρ = 0, this feature still holds. We will prove this for a more general DISV

process, defined as follows.

Definition 6. (DISV) A process S follows displaced independent stochastic volatility dy-

namics, with displacement θ ∈ R, if


dσt = f(σt)dt+ g(σt)dW∗t ,

(3.4)

where Wt, W∗t are i.i.d. Brownian Motion.

Implied volatility for the DISV S is defined on the following strike interval:

KDISV (T ) := K > θ+ : (S0 −K)+ < E(ST −K)+ < S0. (3.5)

Using (2.5), for each T > 0 and each K ∈ KDISV (T ), the implied volatility of the DISV

model σDISV (K,T ) is defined such that

CBS(S0, K, σDISVimp (K,T ), T ) = E(ST −K)+. (3.6)

We can suppress the T in σDISVimp (K,T ).

Theorem 7. If T > 0 and S0 ∈ KDISV (T ), the slope of the at-the-money implied volatility

in the DISV model (3.4) has the following property:

sgn∂σDISVimp

∂K(K)

∣∣∣∣K=S0

= sgn θ. (3.7)

Proof. Appendix B.1.

Remark 3.1.1. The slope of at-the-money implied volatility depends on the assumption that

Wt, W∗t are independent. It may not hold when this condition is violated.

14

Corollary 3.1.1. If T > 0, S0 ∈ KDH(T ) and ρ = 0, the slope of the at-the-money implied

volatility in the displaced Heston model (3.1) has the following property:

sgn∂σimp

∂K(K)

∣∣∣∣K=S0

= sgn θ. (3.8)

Proof. With ρ = 0, the displaced Heston model belongs to the DISV family of models.


Durrleman (2004) gives the short-expiry implied volatility of the Heston model as a function

of the strike. We will derive the short-expiry implied volatility of the displaced Heston

model using his results. Particularly, we connect the short-expiry implied volatility of the

displaced Heston model with that of the Heston model, and we give an approximation of the

short-expiry implied volatility level, slope and convexity.

Short-expiry behavior of Heston process

Recall that the Heston process proposed by Heston (1993) as follows.

dSt = σtStdWt,


t )dt+ εσtdBt,

dBt = ρdWt +

√1− ρ2dW ∗t ,

(3.9)


For St under the displaced Heston dynamics (3.1), denote St = St − θ, then St follows

the Heston dynamics (3.9).

The implied volatility of the Heston process St is defined on the following strike interval:

KH(T ) := K > θ+ : (S0 − (K − θ))+ < E(ST − (K − θ))+ < S0. (3.10)

For each T > 0 and each K ∈ KH(T ), (2.5) defines the implied volatility of the displaced

Heston to be σS,Himp (K, T ) > 0 such that

CBS(S0, K, σS,Himp (K, T ), T ) = E(ST − K)+, (3.11)

15

where K = K − θ. We will suppress the S and T argument of σS,Himp (K, T ) as σHimp(K).

A call option on the displaced Heston process starting at S0 with strike K can be con-

sidered as a call option on a Heston process starting at S0 with strike K:

E(ST −K)+ = E((ST − θ)− (K − θ))+ = E(ST − K)+. (3.12)

We take care here to define the domain of K to make sure that the implied volatilities

of both the Heston model and the displaced Heston model exist. Combine (3.2) and (3.10)

and define

KDH∩H(T ) = KDH(T )∩KH(T ) := K > θ+ : (S0−K)+ < E(ST−K)+ < max(S0, (S0−θ))

to be the domain of K.

The notation A(T ) ∼ B(T ) means that A(T )/B(T ) → 1 as T ↓ 0. Durrleman (2004)

shows that, for the Heston model, the near-money short-expiry implied volatility of the

vanilla call with K-strike, T -maturity and initial value S0 can be approximated as

σHimp(K, T ) ∼√σ2

0 + a0 log(S0/K) + b0T

2+c02

log2(S0/K), (3.13)

where

a0 = −ερ2, b0 = κ(µ− σ2

0) +ερ

2σ2

0 −ε2

6(1− ρ2/4), c0 =

ε2

6σ20

(1− 7ρ2

4).

Short-expiry behavior of displaced Heston process

When near the expiry, the σimp is well-defined for both the Heston process and the displaced

Heston process, so K ∈ KDH∩H(T ). Theorem 8 says that the short-expiry implied volatility

of the displaced Heston model can be expressed in terms of the short-expiry implied volatility

of the Heston model. Proposition 3.1.2 gives the short-expiry implied volatility’s level, slope

and convexity at-the-money.

Theorem 8. For K > θ+, the short-expiry relationship between the implied volatilities of

16

the displaced Heston model (3.1) and the Heston model (3.9) is given as follows.

limT→0

σDHimp (K,T ) =

limT→0

σHimp(K − θ, T )× log(S0/K)

log((S0 − θ)/(K − θ))if K 6= S0

limT→0

σHimp(K − θ, T )× (1− θ/S0) if K = S0.(3.14)


Proposition 3.1.2. For the displaced Heston model (3.1) with ρ = 0, for K > θ+, the

at-the-money implied volatility level, slope and convexity when T ↓ 0 are given as follows.

level: (σDHimp (K,T ))2∣∣∣∣K=S0

=

(σ2

0 + κ(µ− σ20)T

2− ε2T

12

)× (1− θ/S0)2 +O(T 2),

slope:∂ log σDHimp (K,T )

∂ logK

∣∣∣∣K=S0

=θ

2(S0 − θ)+O(T ),

convexity:∂2σDHimp (K,T )

∂K2

∣∣∣∣K=S0

=

c0

(σHimp(S0 − θ, T )

)−1

2(S0 − θ)S0+ σHimp(S0 − θ, T )H(S0, θ) +O(T ),

(3.15)

where c0 = ε2

6σ20

and H(S0, θ) =4θ2−5θS06S3

0(S0−θ).


Remark 3.1.2. The short-expiry at-the-money slope is θ2(S0−θ)

, suggesting that when θ > 0,

the at-the-money slope is positive; when θ < 0, the at-the-money slope is negative, and

is bounded by 12 . We have mentioned in Remark 2.2.2 that the empirical equity volatility

skews typically slope downward more steeply than −0.57. Theorem 3.1.2 suggests that the

short-expiry displaced Heston cannot reproduce steepness of this magnitude.

3.2 Displaced anti-Heston

Remark 3.1.2 says that with θ < 0, the steepness of the short-expiry implied volatility from

the displaced Heston has an upper bound of 12 at-the-money. This means that the displaced

Heston is not a suitable model for the equity data. A related process, however, which is

analogous to the displaced anti -lognormal process, will generate arbitrary large downward

slopes.

17

Definition 7. A process S follows displaced anti -Heston if:

dSt = (−σt)(St − θ)dWt, 0 < S0 < θ,


t )dt+ εσtdBt,

dBt = ρdWt +

√1− ρ2dW ∗t ,

(3.16)


Thus θ − S follows the Heston dynamics with volatility process σt, and the interval of

points attainable by S is (−∞, θ).Recognizing the similarities between the displaced Heston and anti-Heston, the following

terminology groups them together:

Definition 8. (DH) A process S which satisfies either the displaced Heston (3.1) or the

displaced anti-Heston (3.16) specification is said to be a DH process.


Implied volatility for the displaced anti -Heston S is defined on the following strike interval:

KADH(T ) := 0 < K < θ : (S0 −K)+ < E(ST −K)+ < S0. (3.17)

For each T > 0 and each K ∈ KADH(T ), (2.5) defines the implied volatility of the displaced

anti -Heston to be σADHimp (K,T ) > 0 such that

CBS(S0, K, σADHimp (K,T ), T ) = E(ST −K)+. (3.18)

Denote S0 = θ − S0, K = θ −K, then S is the Heston process (3.9). Implied volatility for

the Heston S is defined on the following strike interval

KAH(T ) := 0 < K < θ : (S0 − K)+ < E(ST − K)+ < S0. (3.19)

For each T > 0 and each K ∈ KAH(T ), (2.5) defines the implied volatility of the Heston to

be σHimp(K,T ) > 0 such that

CBS(S0, K, σHimp(K, T ), T ) = E(ST − K)+. (3.20)

18

Again, to make sure the implied volatilities of both the Heston model and the displaced

anti -Heston model exist, we define a domain of K as

KADH∩AH(T ) = KADH(T )∩KAH(T ) = K : 0 < K < θ, σADHimp (K,T ) ∃, σHimp(K, T ) ∃.

We can suppress the T in σADHimp (K,T ) as σADHimp (K), and σHimp(K, T ) as σHimp(K).


When near the expiry, the σimp is well-defined for both the Heston process and the displaced

anti -Heston process, so K ∈ KADH∩AH(T ). Theorem 9 gives a relationship between the

short-expiry implied volatilities of the displaced anti -Heston model and the Heston model.

Proposition 3.2.1 gives the short-expiry at-the-money implied volatility’s level, slope and

convexity.

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.20.18

0.185

0.19

0.195

0.2

0.205

0.21

0.215

0.22

0.225

0.23

Impl

ied

Vol

atili

ty

log(K/S)

T=1/12

S=100, σ0=0.04

Th−ImpVolImpVol

−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.20.18

0.185

0.19

0.195

0.2

0.205

0.21

0.215

0.22

0.225

0.23

Impl

ied

Vol

atili

ty

log(K/S)

T=4/12

S=100, σ0=0.04

Th−ImpVolImpVol

Figure 3.2.1: Theorem 9 formula and exact σimp. Parameters of the displaced anti -Heston dy-

namics are S0 = 100, σ0 = 0.04, κ = 0.2, µ = 0.002, ε = 0.0175, θ = 600, ρ = 0. In the left plot,

T = 1 month and in the right plot, T = 4 month. The DH model parameters are chosen to be close

to the ones calibrated from the SABR model using S = 100, β = 0.2, αSβ0 = 0.2, ρ = −0.4, ν =

0.2, T = 1 year. The details of the calibration are in section 4.1. The dashed black line is the exact

σimp by (3.18). The solid green line is Theorem 9 formula, where σHimp(θ −K,T ) is approximated

using (3.13).

19

Theorem 9. For K ∈ (0, θ), the short-expiry relationship between the implied volatilities of

the displaced anti-Heston model (3.16) and the Heston model (3.9) is given as follows.

limT→0

σDHimp (K,T ) =

limT→0

−σHimp(θ −K,T )× log(S0/K)

log((S0 − θ)/(K − θ))if K 6= S0

limT→0

−σHimp(θ −K,T )× (1− θ/S0) if K = S0.(3.21)

Proof. Appendix B.2

Remark 3.2.1. Figure 3.2.1 compares the Theorem 9 formula and the exact σimp(K,T ), at

expiry T = 1/12 and T = 4/12. It suggests that the formula in the right-hand side of (3.21)

provides, a remarkably accurate approximation to σimp(K,T ) even for T not close to 0.

Proposition 3.2.1. For the displaced anti-Heston (3.16) with ρ = 0, for K ∈ (0, θ), the

at-the-money implied volatility level, slope and convexity when T ↓ 0 are given in the follows.

level: (σADHimp (K,T ))2∣∣∣∣K=S0

=

(σ2

0 + κ(µ− σ20)T

2− ε2T

12

)× (1− θ/S0)2 +O(T 2),

slope:∂ log σADHimp (K,T )

∂ logK

∣∣∣∣K=S0

=θ

2(S0 − θ)+O(T ),

convexity:∂2σADHimp (K,T )

∂K2

∣∣∣∣K=S0

= −c0

(σHimp(θ − S0, T )

)−1

2(S0 − θ)S0− σHimp(θ − S0, T )H(S0, θ) +O(T ),

(3.22)

where c0 = ε2

6σ20

and H(S0, θ) =4θ2−5θS06S3

0(S0−θ).


3.2.3 Generalization of short-expiry behavior

The relationship between the short-expiry σimp of the Heston process and the DH process

given in Theorems 8 and 9 can be generalized to a relationship between the short-expiry

σimp of any process which is a positive martingale and its corresponding (anti-)displaced

process.

Denote the process X to be a positive martingale.

20

Definition 9. A process S is said to be a displaced-X process, with displacement θ ∈ R, if

St − θ = Xt, for any t ≥ 0.

Definition 10. A process S is said to be an displaced-anti-X process, with displacement

θ ∈ R, if θ − St = Xt, for any t ≥ 0.

Denote σXimp(K,T ) to be the implied volatility of the process X by (2.5). Denote

σDXimp (K,T ) and σADXimp (K,T ) to be the implied volatilities of the displaced-X process and

the displaced-anti-X process. Denote KX(T ) to be the domain of K where σXimp(K,T ) and

σDXimp (K,T ) are well-defined as T ↓ 0. Denote KADX(T ) to be the domain of K where

σXimp(K,T ) and σADXimp (K,T ) are well-defined as T ↓ 0. Theorems 10 and 11 give the

short-expiry relationship between σXimp(K,T ) and σDXimp (K,T ), as well as σXimp(K,T ) and

σADXimp (K,T ).

Theorem 10. For a process X which is a positive martingale, for K > θ+ and K ∈KDX(T ), the short-expiry relationship between the implied volatilities of X and its corre-

sponding displaced-X process is given as follows.

limT→0

σDXimp (K,T ) =

limT→0

σXimp(K − θ, T )× log(S0/K)

log((S0 − θ)/(K − θ))if K 6= S0

limT→0

σXimp(K − θ, T )× (1− θ/S0) if K = S0.(3.23)

Proof. Similar to the proof of Theorem 8.

Theorem 11. For a process X which is a positive martingale, for K ∈ (0, θ) and K ∈KADX(T ), the short-expiry relationship between the implied volatilities of X and its corre-

sponding displaced-anti-X process is given as follows.

limT→0

σADXimp (K,T ) =

limT→0

−σXimp(θ −K,T )× log(S0/K)

log((S0 − θ)/(K − θ))if K 6= S0

limT→0

−σXimp(θ −K,T )× (1− θ/S0) if K = S0.(3.24)

Proof. Similar to the proof of Theorem 9.

21

CHAPTER 4

CALIBRATION OF DL AND DH PROCESS

4.1 Calibration of DL and DH Process

Whether one chooses to use the DL/DH as a model, or as an approximation of another

model, or (as we will) as a control variate for another model, in any case the DL parameters

(σ, θ) and DH parameters (σ0, κ, ν, ρ, ε, θ) require estimation/calibration. We use Theorem

6 implications (2.24) to fit the DL parameters to a given implied volatility level and slope.

We use Proposition 3.1.2 and Proposition 3.2.1 to fit the DH parameters to a given implied

volatility’s level, slope and curvature.

4.1.1 Calibration of DL process

Given a short-expiry at-the-money skew level a and slope b (either from some model, or from

direct empirical measurement), and given an underlying level S0, there exists a DL process,

with S0 = S0 and parameters (θ, σ), such that the DL skew’s short-expiry level and slope

(2.24) match the given level a and slope b, provided that b 6= −1/2. Explicitly, we find

σ = a(1 + 2b), θ =2b

1 + 2bS0. (4.1)

In the case of slope b > −1/2, the calibrated DL process is a displaced lognormal. In the case

of slope b < −1/2, the calibrated DL process is a displaced anti -lognormal. The singular

case of slope b = −1/2 can be matched by normal or “Bachelier” dynamics dSt = aS0dWt.

The DL and Bachelier models belong to the family dSt = (σSt+A)dWt, where σ 6= 0 in the

case of DL, and σ = 0 in the singular case of Bachelier.

Remark 4.1.1. Although (2.24) is a short-expiry limit, Remark 2.2.3 indicates its accuracy

at T of moderate length. Therefore (4.1) may still facilitate calibration of (σ, θ) to volatility

skews (a, b) even at moderately long expiries.

22

4.1.2 Calibration of DH process

Given a short-expiry at-the-money skew level a, slope b and convexity c (either from some

model, or from direct empirical measurement), and given an underlying level S0, there exists

a DH process, with S0 = S0 and parameters (σ0, κ, ν, ρ, ε, θ), such that the DH skew’s

short-expiry level, slope and convexity (3.15) or (3.22) match the given level a, slope b and

convexity c, provided that b 6= −1/2. Recall that we have restricted ρ = 0 in (3.15) and

(3.22). In application, we will also prefix κ. With (ρ, κ) prefixed, we find explicitly,

θ =2b

1 + 2bS0, σ0 = |a(2b+ 1)|, µ =

ε2

6κ+ σ2

0

ε =

√6σ2

0 × 2(S0 − θ)S0

(c− a(2b+ 1)H(S0, θ)

)(|a(2b+ 1)|

),

(4.2)

If b > −12 , we calibrate the parameters to the displaced Heston model; if b < −1

2 , we calibrate

the parameters to the displaced anti -Heston model. For the singular case b = −12 , we take

the same approach as in the DL case.

Compared with the calibration of the DL parameters given in (4.1), we see that the θ is

the same in both the DL and the DH case. In the DL case, we have σ = a(2b+ 1); while in

the DH case, we have σ0 = |a(2b+ 1)|.

Remark 4.1.2. Although (3.14) and (3.21) are short-expiry limit, Remark 3.2.1 indicates

their accuracy at T of moderate length. Therefore (4.2) may still facilitate calibration of

(σ0, κ, ν, ρ, ε, θ) to volatility skews (a, b, c) even at moderately long expiries.

4.2 Calibration DL and DH to CEV/SABR

4.2.1 CEV and SABR stochastic volatility models

For many local or stochastic volatility models, there exist explicit short maturity approxi-

mations of implied volatility, such as in Lewis (2000) and Berestycki et al. (2004), making it

easy to calculate the implied volatility level a, slope b and convexity c, and to calibrate DL

parameters σ and θ via (4.1), or to calibrate DH parameters (σ0, κ, ν, ρ, ε, θ) via (4.2).

Two such models capable of generating realistically steep at-the-money implied volatility

skews are the Constant Elasticity of Variance (CEV) model and the SABR model. In the

23

CEV model (Cox 1996),

dSt = αSβt dWt, S0 > 0, (4.3)

where β ≤ 1, and absorption is imposed at S = 0.

Remark 4.2.1. Boundary Conditions for the CEV:

• β < 1/2: 0 is attainable. The origin is a regular boundary point and is specified as a

killing boundary by adjoining a killing boundary condition.

• β = 1/2: 0 is attainable and strong reflecting.

• β > 1/2: 0 is not attainable. 0 is exist boundary.

The CEV model can generate a steep downward implied volatility skew at-the-money.

Indeed, by Berestycki et al. (2002) and Roper (2009), for all K > 0,

limT→0

σCEVimp (K,T ) =

α(1− β) log(S0/K)

S1−β0 −K1−β

if K 6= S0

αSβ−10 if K = S0.

(4.4)

Differentiating with respect to logK, we have

∂ log σCEVimp

∂ logK

∣∣∣∣T↓0, K=S0

=β − 1

2, (4.5)

which can take arbitrarily large negative values.

The widely-used SABR model (Hagan et al. 2002) generalizes the CEV, by making

the coefficient α stochastic, with volatility-of-volatility ν ≥ 0, and correlation ρ ∈ [−1, 1]

between S and α:

dSt = αtSβt dWt, S0 > 0

dαt = ναtdBt, α0 > 0

dBt = ρdWt +

√1− ρ2dW∗t

(4.6)

where W and W∗ are independent Brownian motions, and absorption is imposed at S = 0.

Taking ν = 0 in the SABR model reduces to the CEV case.

24

According to a short-expiry approximation in Hagan et al. (2002, eq. 3.1a),

σSABRimp (K,T ) ≈ α0S

β−10

(1 +

(β − 1

2+

ρν

2αSβ−10

)log(K/S0)

). (4.7)

The approximation’s slope at K = S0 is

∂ log σSABRimp

∂ logK

∣∣∣∣T↓0, K=S0

≈ β − 1

2+

ρν

2αSβ−10

, (4.8)

reflecting the contributions to the SABR volatility skew, not just from the functional rela-

tionship between price levels and volatility, as expressed by β, but also from the correlation

between price increments and volatility, as expressed by ρ.

4.2.2 Calibration of DL to CEV/SABR

For the SABR process, a = α0Sβ−10 , and b is given by (4.8), so we have

σ = α0βSβ−10 + ρν, θ = S0 −

α0Sβ0

α0βSβ−10 + ρν

. (4.9)

The ν = 0 special case of (4.9) gives the DL parameters that match the CEV level and slope:

σ = αβSβ−10 , θ = S0(β − 1)/β. (4.10)

In the CEV case, Marris (1999) and Svoboda-Greenwood (2009) have already investigated

displaced lognormal approximation, by an approach which chooses parameters such that the

displaced lognormal instantaneous volatility approximates the CEV instantaneous volatility

function S 7→ αSβ , in contrast to our approach which matches the implied volatility functions.

Their approach arrived at the same result (4.10) as our approach, in the CEV case.

A distinction is that our implied volatility approach is intended to apply moreover to

models, such as SABR, where instantaneous volatility varies not just as a function of S, but

also other stochastic factors. The implied volatility skew reflects the dependence of volatility

on the S level together with the other stochastic factors in the model, such as α in the SABR

case.

25

4.2.3 Calibration of DH to SABR

Durrleman (2004) shows that (3.13) can also be used as an approximation for the short-expiry

implied volatility of SABR model (4.6) with

σ0 = α0Sβ−10 , a0 = −(νρ+ (β − 1)σ0)σ0,

c0 =ν2

6(4− 3ρ2) + νρ(β − 1)σ0 +

5

6(β − 1)2σ2

0,

b0 =σ2

0

6(ν2(2− 3ρ2) + 6νρβσ0 + (β − 1)2σ2

0).

(4.11)

From this, we derive the approximation of the short-expiry SABR implied volatility level,

slope and convexity as follows.

a(T ) =

√σ2

0 + b0T

2,

b(T ) = −(σatm)−2a0

2,

c(T ) =(σatm)−2a0

2K

∂σatm

∂K+a0 + c0

2K2(σatm)−1,

(4.12)

where σatm is approximated by (3.13) with K = S0. The parameters of DH are chosen by

plugging (4.12) into (4.2).

Remark 4.2.2. In (4.12), limT→0 a(T ) = αSβ−10 , limT→0 b(T ) = β−1

2 + ρν

2αSβ−10

, the same

as the level and slope given in (4.7) and (4.8).

26

CHAPTER 5

VARIANCE REDUCTION IN MONTE CARLO SIMULATION

5.1 Variance Reduction Using Control Variate

Theorems 1 and 2 imply that the displaced lognormal is inconsistent with the steep downward

slopes (Remark 2.2.2) and non-monotonicity (Remark 2.2.1) typical of stock market volatility

skews. The displaced anti -lognormal, by (2.24), overcomes the steepness constraint, but

introduces other drawbacks: its paths, which take values in (−∞, θ), are bounded above and

unbounded below – the opposite of the behavior desirable in a model of stock prices. Such

paradox applies to the DH process too. Proposition 3.1.2 implies that the displaced Heston

is inconsistent with the steep downward slopes (Remark 3.1.2). The displaced anti -Heston

overcomes the steepness constraint but its paths are unbounded from below.

For these reasons, we do not generally advocate the DL/DH to model stock price pro-

cesses. Rather, we propose the DL/DH to generate control variates to reduce variance in

the Monte Carlo pricing of derivative contracts under commonly-used dynamics which do

match the empirical at-the-money volatility skew.

Indeed, suppose the underlying S dynamics follow some specification that a modeler

deems appropriate, such as the CEV or the SABR stochastic volatility model. Suppose the

modeler intends to price a derivative contract for which the desired model lacks analytical

pricing formulas, such as a discretely-monitored barrier option on the CEV/SABR process

S. Let the contract’s payoff Y be given by a specified function of the S path. In the absence

of analytical solutions, consider the use of Monte Carlo simulation to estimate the price EY.

The basic Monte Carlo estimator is the sample average

C :=1

M

∑m

Ym (5.1)

where the simulations Y1, . . .YM are iid as Y. To improve accuracy, in the sense of reducing

variance, let us apply the control variate technique, where the control comes from a DL

process calibrated by (4.1), or from a DH process calibrated by (4.2).

27

There exist, of course, other variance reduction methods, combinable with a DL/DH

control variate. We do not investigate them in this section; rather we maintain focus on

the DL/DH control, with the intent of illustrating how much variance reduction the DL/DH

control brings by itself. In section 5.2 we will discuss combining the DL/DH control with

other techniques.

5.1.1 DL or DH as a control variate

To make explanation clear, we first focuses on using DL as control variate. Using DH as

control variate follows the same spirit.

The control variate estimator of EY, using a control Y , where Y has a known expectation

C := EY and a known simulation methodology, is defined by

Ccv :=1

M

∑m

(Ym − βYm + βC

), (5.2)

where the simulated pairs (Y1, Y1), . . . , (YM , YM ) are iid as (Y, Y ). Good choices of Y have

large correlation ρY,Y with Y, because increasing |ρY,Y | decreases the estimator’s variance.

Specifically,

Var Ccv = (1− ρ2Y,Y ) Var C (5.3)

for the optimal choice of the β coefficient, namely β = Cov(Y, Y )/Var(Y ), which may also

be estimated by simulation. For further details see, for instance, Boyle et al. (1997).

Because the payoff Y is a specified function of the S path, we choose Y to be that same

payoff function applied to the S path, where S follows a DL process driven by a Brownian

motion that also drives S. Aiming to produce high correlation ρY,Y , we choose the S process

parameters by taking S0 = S0 and applying (4.1) to find (σ, θ) such that the short-expiry

at-the-money volatility skews implied by S and by S agree in both level and slope.

The suitability of the DL process S to serve in this role stems from a confluence of

flexibility and tractability; the DL is potentially flexible enough to generate significant cor-

relation between Y and Y (by linking the parameters of S and S, as discussed above), and

yet potentially tractable enough to allow analytic evaluation of EY and unbiased simulation

of Y , as discussed below.

For shift-invariant contracts (including barriers and lookbacks), exact evaluation of C =

EY under DL dynamics is just as easy as under Black-Scholes dynamics; more precisely,

28

if the contract’s payoff is invariant to parallel shifts of the underlying price path and the

contract parameters (such as strike and barrier level), then Black-Scholes model valuation

methods, applied to shifted arguments, produce the contract’s DL valuation. In the DL case,

if, moreover, we can simulate the exact distribution of Y – which is often the case, because

S is a transformed Gaussian – then Y can serve as a control variate that reduces variance

without introducing any bias.

When using DH as control variate, we have one more source of flexibility, namely the

stochastic volatility process. If Y depends on the path S where S depends on a stochastic

volatility process Σ, S is driven by a Brownian motion W and Σ is driven by W and W∗,

where W and W∗ are independent, we choose Y to be the same payoff function applied to

the S path, where S follows a DH process driven by W, and the volatility process of S is

driven by W∗. We choose the S process parameters by taking S0 = S0 and applying (4.2)

to find (σ0, κ, ν, ρ, ε, θ) such that the short-expiry at-the-money volatility skews implied by

S and by S agree in level, slope and convexity.

The DH control variate has an advantage over the DL control variate in that the DH

model is more flexible and the S path tracks the S path more closely. Consequently, Y will

have higher correlation with Y so we can achieve greater variance reduction. On the other

hand, the option pricing formula on the DH model is more complicated than on the DL

model.

5.1.2 Example I: Discretely sampled barrier option under CEV/SABR

dynamics

To take a concrete example, consider a discretely sampled barrier option on S, which follows

CEV (4.3) or SABR (4.6) dynamics. In particular, let the contract be a down-and-out call

with expiry T , barrier H, strike K, sampling dates t1 < t2 < · · · < TN = T , and payoff

Y := (ST −K)+1(minn

Stn > H). (5.4)

Analytical solutions exist for continuous barriers in the CEV model (Davydov-Linetsky

2001), but not for discrete barriers, nor for the SABR model, so we turn to Monte Carlo

simulation.

29

DL as control variate

To generate a control variate on DL, we apply the same payoff function to a DL process,

driven by the same Brownian motion W = W. More precisely,

Y := (ST −K)+1(minnStn > H)

dSt = σ(St − θ)dWt,(5.5)

where S0 = S0, and (σ, θ) are calibrated by (4.9) in the SABR case, or (4.10) in the CEV

case.

In the DL case, This Y is easily simulated without bias, and the value of C = EY can

be computed by shifting any of the fast and exact (up to numerical truncation/quadrature

error) solutions for discrete barrier option prices in the Gaussian framework, such as Broadie

and Yamamoto (2005), or in the Levy framework, such as Petrella and Kou (2004) or Feng

and Linetsky (2008).

An alternative to (5.5) is to choose instead a continuously-monitored control

Y ∗ := (ST −K)+1( mint∈[0,T ]

St > H). (5.6)

The control expectation EY ∗ has a simple exact formula, and the control Y ∗ can be simulated

without bias, using Brownian bridge techniques of Beaglehole et al. (1997).

DH as control variate

To generate a control variate on DH, we apply the same payoff function to a DH process,

driven by the same Brownian motion W = W, W ∗ = W∗. The DH process parameters

(σ0, κ, ν, ρ, ε, θ) are calibrated by (4.2) where a, b, c are determined by (4.12). More precisely,

if b > −12 , the control variate is based on the displaced Heston model,



dσ2t = κ(ν − σ2

t )dt+ εσtdW∗t , dWtdW

∗t = 0

(5.7)

30

If b < −12 , the control variate is based on displaced anti -Heston model:


dSt = (−σt)(St − θ)dWt, 0 < S0 < θ,


t )dt+ εσtdW∗t , dWtdW

∗t = 0

(5.8)

In both (5.7) and (5.8), S0 = S0 and σt is non-negative for all t > 0.

In the DH case, the Y can be simulated from an efficient discretisation scheme which

guarantees strong convergence, such as Lord et, al (2006). We discuss this more in detail in

Chapter 6. The value of C = EY can be computed by shifting any of the fast solutions for

discrete barrier option price under the Heston model, such as Griebsch and Wystup (2008).

5.1.3 Numerical results I: Discretely sampled barrier option under

CEV/SABR dynamics

DL as Control Variate

Our experiments simulate the payoff (5.4), where

K = S0 = 100, H = 95, T = 4/12, N = 84. (5.9)

In the CEV case we take β ∈ −0.5,−1.0,−1.5, with α such that αSβ−10 ∈ 0.15, 0.20, 0.25,

based on Hirsa-Courtadon-Madan’s (2003) estimates of S&P500 CEV parameters; our β is

what they denote as β + 1. We use the control (5.5), where (σ, θ) are tuned to the CEV by

(4.10).

In the SABR case we take β = 0.2, with α0 such that α0Sβ−10 = 0.2, with an array of

choices for (ν, ρ). We use the control (5.5), where (θ, σ) are tuned to the SABR by (4.9).

We run 1000 paths. Ten equal spaced points are sampled each day. Tables 5.1.1 and

5.1.2 report the estimated “percentage reduction of variance”

100%− Var(Ccv)

Var(C), (5.10)

31

Table 5.1.1: Percentage reduction of variance, using DL for down-and-out call on CEVT = 4 months

β αSβ−10

0.15 0.20 0.25-0.50 99.99% 99.99% 99.99%-1.00 99.99% 99.98% 99.93%-1.50 99.98% 99.89% 98.67%

Payoff: (5.4) with (5.9). Control: (5.5) with (4.10).

Table 5.1.2: Percentage reduction of variance, using DL for down-and-out call on SABRT = 4 months

ρ ν0.20 0.40 0.60

-0.4 98.74% 96.07% 92.42%-0.6 98.83% 97.17% 93.70%-0.8 99.64% 98.71% 96.76%

Payoff: (5.4) with (5.9). Control: (5.5) with (4.9).

where each Var is the scaled sample variance of the summands in (5.1) and (5.2) respec-

tively. Equivalently, the percentage reduction of variance equals the “R-squared” of an OLS

regression of the CEV/SABR barrier-option payoff Y on the DL control payoff Y .

DH as Control Variate

Our next experiments simulate the payoff (5.4) with parameters given by (5.9), using DH

as control variate for the down-and-out discrete barrier under the SABR process. The

parameters of the SABR process are β = 0.2, with α0 such that α0Sβ−10 = 0.2, with an

Table 5.1.3: Percentage reduction of variance, using DH for down-and-out call on SABRT = 4 months

ρ ν0.20 0.40 0.60

-0.4 99.76% 99.03% 99.41%-0.6 99.73% 98.95% 99.35%-0.8 99.00% 98.60% 98.73%

Payoff: (5.4) with (5.9). Control: (5.7) with (4.2) or (5.8) with (4.2).In (4.2), a, b and c are calculated from (4.12).

32

array of choices for (ν, ρ). The results are shown in Table 5.1.3.

Comparing the results in Table 5.1.2 and 5.1.3, we can see that using DH as control

variate achieves more variance reduction. As we noted before, the reason is that the DH

process is a two-factor stochastic dynamics which enable the process S to better track the

original process, S.

Remark 5.1.1. In the reported Table 5.1.3 and any other reported results related to DH

process in this paper, we arbitrarily chose κ = 0.2. Numerical simulations suggest that other

choices of κ give relatively same results.

5.1.4 Example II: Discretely sampled arithmetic Asian option under

SABR dynamics

The next example we consider is the discretely sampled arithmetic Asian option on S, which

follows the SABR dynamics (4.6). Let the contract be an Asian call option with expiry T ,

strike K, sampling dates t1 < t2 < · · · < tN = T , and payoff

Y := (1

N

N∑i=1

Si −K)+ (5.11)

There is no analytical solution for the arithmetic Asian option price under the SABR process,

so we turn to Monte Carlo simulation.

DL as control variate

For the control variate, we could apply the same payoff to the DL process. Although there

is no exact analytical solution for the value of the discrete arithmetic Asian option under

the DL process, it could be computed by shifting the numerical solution for the discrete

arithmetic Asian option under Geometric Brownian motion, such as Milevsky and Posner

(1998), Fusai and Meucci (2008).

Here we consider a slightly different payoff which we know how to compute exactly under

the DL process. We call the new payoff discrete exponential Asian option. The DL process

is driven by the same Brownian motion which drive the SABR process, W = W. More

33

precisely, the control is,

Y ∗ :=

(θ + exp( 1N

∑Ni=1 log(Si − θ))−K)+ if displaced lognormal is used

(θ − exp( 1N

∑Ni=1 log(θ − Si))−K)+ if displaced anti -lognormal is used

dSt = σ(St − θ)dWt

(5.12)

The value C∗ = EY ∗ can be calculated analytically under the DL process.

Lognormal as control variate

To make a comparison, we also use the control variate under the lognormal(LN) process.

The control variate is constructed by applying the discrete exponential Asian option to the

lognormal process, driven by the same Brownian motion W = W. More precisely,

Y ∗ := (exp(1

N

N∑i=1

log(Si))−K)+

dSt = σLNStdWt

(5.13)

The value C∗ = EY ∗ can be calculated analytically. In (5.13), we choose σLN to be the

short-expiry at-the-money implied volatility of the SABR process, which is

σLN = α0Sβ−10 . (5.14)

5.1.5 Numerical results II: Discretely sampled arithmetic Asian option

under SABR dynamics

Our experiments simulate the payoff (5.11), where

K = S0 = 100, T = 4/12, N = 84, β = 0.2, α0Sβ−10 = 0.2 (5.15)

and an array of choices for (ν, ρ). We use control (5.12) where (θ, σ) are tuned to SABR by

(4.9). We also report variance reduction results using control (5.13) with parameters (5.14).

We run 1000 paths. Ten equal spaced points are sampled each day. Table 5.1.4 and 5.1.5

report the estimated “percentage reduction of variance”. The results suggest that the DL

control achieves significantly larger variance reduction than the LN control.

34

Table 5.1.4: Percentage reduction of variance, using DL as control for Asian Option onSABR

T = 4 monthsρ ν

0.20 0.40 0.60-0.4 99.57% 98.43% 96.19%-0.6 99.69% 98.74% 97.36%-0.8 99.83% 99.24% 98.21%

Payoff: (5.11) with (5.15). Exponential Asian option on DL control: (5.12) with (4.9).

Table 5.1.5: Percentage reduction of variance, using LN as control for Asian Option on SABRT = 4 months

ρ ν0.20 0.40 0.60

-0.4 75.34% 78.61% 76.06%-0.6 76.61% 77.97% 78.95%-0.8 78.96% 80.76% 79.18%

Payoff: (5.11) with (5.15). Exponential Asian option on LN control: (5.13) with (5.14).

5.2 Variance Reduction Combining Control Variate and

Importance Sampling

In this section, we study the issue of combining the importance sampling with the control

variate. Importance sampling itself is a popular variance reduction technique, see Bolye,

Broadie and Glasserman (1997) for more details.

5.2.1 Importance sampling on options pricing

Recall the SABR process in (4.6), if we define

dWt := dWt − atdt,

then by the Girsanov Theorem, there is a measure P such that under which Wt, W∗t are

independent standard Brownian motions, and Wt is a drifted Brownian motion. Under P,

35

the original process S becomes a drifted SABR process:

dSt = αtSβt (dWt + atdt), S0 > 0

dαt = ναtdBt, α0 > 0

dBt = ρdWt +

√1− ρ2dW∗t

(5.16)

Again by the Girsanov Theorem, the likelihood ratio or Radon-Nikodym derivative be-

tween the original measure P and the new measure P is

r(S) :=dPdP

= exp

(−∫ T

0asdWs +

1

2

∫ T

0a2sds

). (5.17)

For any function G(.) : C→ R, where C is the domain of S, the Girsanov Theorem says

EG(S) = Er(S)G(S), (5.18)

where E is the expectation which is taken under P. Therefore, the importance sampling

estimator (IS-estimator) is:

CIS :=1

M

∑m

r(Sm)G(Sm). (5.19)

r(Sm)G(Sm) can be considered as a weighted payoff.

The drift at is chosen in a heuristic way similar to Bolye et al (1997). Denote

a0 =

(| log(S/K)|+ | log(K/H)|

)/T + 1

2α20S

2β−20

α0Sβ−10

. (5.20)

We would like to control the variance of the likelihood ratio, which is exp(∫ T

0 a2sds), so that

it does not explode in real applications. We will arbitrarily set a constant Ca < ∞ and

set a0 = min(Ca, a0). Next, for the down-and-out call options, we choose at = a0; for the

down-and-in call options, we choose at = −a0 before the barrier is reached and at = a0 after

the barrier is reached. The intuition behind this is to make the path less(more) likely to

reach the barrier in the down-and-out(down-and-in) case.

36

5.2.2 Drifted DH/DL process

The drifted DH/DL dynamics is derived similarly as the drifted SABR dynamics. Recall that

if we define dWt := dWt− atdt, then under the new measure P, Wt and W∗t are independent

standard Brownian motions. Consequently, the DL dynamics defined in (2.6) and (2.15),

DH dynamics defined in (3.1) and (3.16) become drifted DL/DH processes under the new

measure P. These drifted processes are listed below.

Under P, the displaced lognormal dynamics becomes the drifted displaced lognormal dy-

namics:

dSt = σ(St − θ)(dWt + atdt), S0 > θ, σ > 0. (5.21)

The drifted displaced anti-lognormal dynamics is

dSt = σ(St − θ)(dWt + atdt), 0 < S0 < θ, σ < 0. (5.22)

The drifted displaced Heston dynamics is

dSt = σt(St − θ)(dW1t + atdt), S0 > θ


t )dt+ εσtdBt,

dBt = ρdWt +

√1− ρ2dW ∗t .

(5.23)

The drifted displaced anti-Heston dynamics is

dSt = (−σt)(St − θ)(dW1t + atdt), 0 < S0 < θ


t )dt+ εσtdBt,

dBt = ρdWt +

√1− ρ2dW ∗t .

(5.24)

In both the drifted displaced Heston model and the drifted displaced anti -Heston model,

σt is non-negative for all t > 0.

5.2.3 Combine control variate with importance sampling

Since the IS-estimator CIS has smaller variance than the naive estimator C, we can construct

a control variate for the IS-estimator. We call the new estimator ISCV-estimator, and denote

it as CISCV. CISCV will reduce variance further than CIS.

37

The detail of the ISCV-estimator is as follows. First, considering the weighted payoff in

(5.19) as one payoff function on S:

Y = H(S) := r(S)G(S), (5.25)

where S is sampled from (5.16). As in section 5.1.2, we want to choose a control variate Y

that is highly correlated with Y. Y is constructed by applying the new payoff function H(.)

onto the drifted DL/DH process S:

Y := H(S) = r(S)G(S). (5.26)

It turns out that H(S) = r(S)G(S) can also be considered as an IS-estimator for the non-

drifted DL/DH process S, because by the Girsanov Theorem,

EG(S) = Er(S)G(S). (5.27)

Finally, we propose the combined estimator of the importance sampling and the control

variate as follows.

CISCV =1

M

∑m

r(Sm)G(Sm)− β(

1

M

∑m

r(Sm)G(Sm)− EG(S)

). (5.28)

where Si is sampled from (5.16), and Si is sampled from any of the drifted DL/DH process

(5.21)-(5.24).

Remark 5.2.1. This estimator coincides with the estimator proposed by Hesterberg (1996),

but comes from a different angle. In Hesterberg’s approach, both the importance sampling

estimator and the control variate estimator are regarded as weighted sums and are com-

bined together as a double weighted sum. Our approach is to find a control variate for the

importance sampling estimator.

5.2.4 Example: Discretely sampled barrier option under SABR dynamics

We consider a discretely sampled barrier option on S, which follows the SABR (4.6) dynamics.

In particular, let the contract be a down-and-in call with expiry T , barrier H, strike K,

38

sampling dates t1 < t2 < · · · < tN = T , and payoff:

G(S) := (ST −K)+1( min1≤n≤N

Stn < H). (5.29)

To construct the CISCV, we apply the payoff function G(.) to a drifted DL/DH process S,

where S is driven by the Brownian motion W = W. If using DH, the volatility of S is driven

by W ∗ = W∗. in the DL case, (σ, θ) are tuned by (4.9); in the DH case, (σ0, κ, ν, ρ, ε, θ) are

tuned by (4.2) where a, b, c are determined by (4.12).

In practice, there is no way of exactly sampling from the continuous SABR process and

the correlated DH process, so discretisation schemes are used. We discretize the SABR

process into I subintervals and sample discretely. The discretisation of the SABR process is:

log Si+1 = log Si −α2i

2S

2β−2i ∆t+ αiS

β−1i Wi

log σi+1 = log σi −ν2

2∆t+ ν(ρ(Wi − E(Wi)) +

√1− ρ2W∗i )

(5.30)

where Wi, W∗i ∼ i.i.d. N(0,∆t), i = 1, · · · , I. To sample from the drifted SABR process,

we take Wi ∼ N(ai∆t,∆t) in (5.30). ai is chosen by the rule at the end of section 5.2.1.

The likelihood-ratio is calculated as

r(S) =I∏i=1

f(Wi)

fα(Wi)= exp(−

I∑i=1

aiWi +1

2

I∑i=1

a2i∆t). (5.31)

where f(.) is the density function ofN(0,∆t) and fα(.) is the density function ofN(ai∆t,∆t).

To construct ISCV with DL control, we sample from the drifted DL process and apply

the payoff G(.) to it:

G(S) := (ST −K)+1( min1≤n≤N

Stn < H),

log(Si+1 − θ) = log(Si − θ)−σ2

2∆t+ σWi, if b > −1

2

log(θ − Si+1) = log(θ − Si)−σ2

2∆t+ σWi, if b < −1

2

(5.32)

where S0 = S0, W = W ∼ N(ai∆t,∆t). We are not showing the degenerate case where

b = 12 and Si is sampled from a geometric Brownian motion.

39

Similarly, to construct ISCV with DH control, we sample from the drifted DH process

and apply the same payoff G(.) to it:

G(S) := (ST −K)+1( min1≤n≤N

Stn < H),

log(Si+1 − θ) = log(Si − θ)−σ2i

2∆t+ |σi|Wi, if b > −1

2

log(θ − Si+1) = log(θ − Si)−σ2i

2∆t− |σi|Wi, if b < −1

2

σ2i+1 = σ2

i + κ(µ− σ2i )∆t+ ε

√|σ2i |(ρ(Wi − E(Wi)) +

√1− ρ2W ∗i ),

(5.33)

where S0 = S0, W = W ∼ N(ai∆t,∆t), W∗ = W∗ ∼ N(0,∆t), and σt is non-negative for

all t > 0.

Remark 5.2.2. The volatility process of DH is a mean-reverting square process. There are

various discretisation schemes, see Lord et al (2008) for more detail. We choose the first

order Euler discretisation scheme with reflection principal proposed by Higham and Mao

(2005), where the partial strong convergence of Si is ensured. More discussion of this is in

Chapter 6.

The likelihood ratio in both (5.32) and (5.33) is the same as in (5.31):

r(S) = exp(−I∑i=1

aiWi +1

2

I∑i=1

a2i∆t) (5.34)

Finally, the ISCV-estimator is constructed as

CISCV =1

M

∑m

(G(Sm)r(Sm)− βG(Sm)r(Sm) + βEG(Sm)

). (5.35)

5.2.5 Numerical results: Discretely sampled barrier option

In this section, we compare the variance reduction of the importance sampling, with that of

combining importance sampling with the control variate. Denote ISDL to be the estimator

which combines the importance sampling and DL control variate from (5.32) as in (5.35).

Denote ISDH to be the estimator which combines the importance sampling and the DH

control variate from (5.33) as in (5.35).

40

The experiments simulate the payoffs (5.4) and (5.29), where

K = S0 = 100, H = 95, T = 4/12, N = 84, β = 0.2, α0Sβ−10 = 0.2. (5.36)

We simulate 1000 pathes. For the DH parameters in ISDH, prefix ρ = 0, κ = 0.2. Table 5.2.1

and 5.2.2 show the variance reduction of the down-and-out call options using importance

sampling, ISDL and ISDH. Table 5.2.3 and 5.2.4 show the variance reduction of the down-

and-in call options using the three estimators. For both options, we see that ISDL and ISDH

achieve more variance reduction than importance sampling.

Table 5.2.1: Percentage reduction of variance, using importance sampling for down-and-outcall on SABR

T = 4 months, Importance Samplingρ ν

0.20 0.40 0.60-0.4 69.46% 68.32% 67.00%-0.6 69.07% 67.70% 66.26%-0.8 68.77% 67.31% 65.69%

Payoff: (5.4) with (5.36).

Table 5.2.2: Percentage reduction of variance, using ISDL and ISDH for down-and-out callon SABR

T = 4 months, ISDLρ ν

0.20 0.40 0.60-0.4 97.02% 89.99% 83.31%-0.6 97.32% 90.91% 85.71%-0.8 97.48% 94.55% 89.30%

T = 4 months, ISDHρ ν

0.20 0.40 0.60-0.4 98.16% 98.87% 97.20%-0.6 97.91% 97.34% 96.87%-0.8 97.69% 95.55% 96.45%

Payoff: (5.4) with (5.36).ISDL: Combined importance sampling and DL control variate: (5.32) and (5.35). Parametersof DL: (4.9).ISDH: Combined importance sampling and DH control variate: (5.33) and (5.35). Parame-ters of DH: (4.2).

41

Table 5.2.3: Percentage reduction of variance, using importance sampling for down-and-incall on SABR

T = 4 months, Importance Samplingρ ν

0.20 0.40 0.60-0.4 33.71% 32.35% 32.13%-0.6 35.85% 33.09% 30.97%-0.8 35.95% 33.25% 32.10%

Payoff: (5.29) with (5.36).

Table 5.2.4: Percentage reduction of variance, using ISDL and ISDH for down-and-in call onSABR

T = 4 months, ISDLρ ν

0.20 0.40 0.60-0.4 96.06% 87.59% 79.68%-0.6 96.58% 88.80% 82.50%-0.8 96.77% 93.42% 87.49%

T = 4 months, ISDHρ ν

0.20 0.40 0.60-0.4 97.89% 98.75% 95.96%-0.6 97.36% 97.00% 94.46%-0.8 97.06% 94.01% 94.63%

Payoff: (5.29) with (5.36).ISDL: Combined importance sampling and DL control variate: (5.32) and (5.35). Parametersof DL: (4.9).ISDH: Combined importance sampling and DH control variate: (5.33) and (5.35). Parame-ters of DH: (4.2).

We summarize the discussion of this Chapter here. As mentioned by Fisher and Tataru

(2010), the stochastic/local volatility (SLV) model has become market standard for barrier

option pricing. The CEV/SABR model belongs to the family of SLV models, so pricing

barrier options under them is of practical interest. This section uses DL and DH as control

variate to reduce variance in the Monte Carlo simulation of the barrier options. More

specifically, we give numerical examples to demonstrate that the DL and DH controls can

provide significant variance reduction for barrier option pricing under the CEV/SABR model,

as shown in Tables 5.1.1, 5.1.2 and 5.1.3. Moreover, we show that DL/DH in concert with

importance sampling is superior to the importance sampling alone, by comparing Table 5.2.1

with 5.2.2, as well as Table 5.2.3 with 5.2.4. Finally, we show that the DL controls also yield

significant variance reduction for Asian options.

42

CHAPTER 6

DISCRETISATION SCHEME

For most general stochastic processes, exact simulation methods do not exist and we have

to refer to discretisation schemes to simulate them. There are three basic criteria in the

discretisation schemes: rate of convergence, stability and positivity. These criteria have been

widely and intensively discussed in the literature. For more detail, see Lord et al (2008),

Kahl (2004), Zhang et al (2004) and Higham and Mao (2005). We use some discretisation

schemes in section 5.2.5 and this chapter we will validate their usage.

There are many numerical schemes. Euler-Maruyama (Euler for short) scheme is a

straightforward discretisation scheme, wherein one discretises the time interval of interest,

and simulates the process at the discretisation points. Under certain conditions, it can be

shown that the Euler scheme converges to the true process as the time intervals are made

finer and finer. Sometimes, in order to reserve the positivity of the underlying process, the

Log-Euler scheme is used, where one simulates from the discretisation of the logarithm of

the original process.

Consider a stochastic process:

dSt = A(t, St)dt+B(t, St)dWt, (6.1)

where Wt is Brownian motion. Denote the discrete approximation of the process as S∆tt .

Most existing proofs of the convergence of S∆tt to St rely on the linear growth and global

Lipschitz conditions (Kloeden and Platen (1992)):

(linear growth) there exists a positive constant L1 such that

|B(t, x)|2 ≤ L1(1 + x2); (6.2)

(global Lipschitz ) there exists a positive constant L2 such that

|B(t, x)−B(t, y)|2 ≤ L2|x− y|2. (6.3)

43

However, for the CEV/SABR processes, the strong convergence rule shown in Kloeden

and Platen (1992) can not be applied since Sβt does not satisfy the global Lipschitz condition.

Zhang et al (2004) relax the condition of the convergence to local Lipschitz. For the process

(6.1), they show that its Euler discretisation converges to the underlying continuous process

in L2 sense before a stopping time, if St satisfies a local Lipschitz condition before the

stopping time. Since the convergence they prove is up to a stopping time, we refer to it

as partial strong convergence. The result can be extended to the Log-Euler discretisation:

if the logarithm of the underlying process satisfies the local Lipschitz condition before the

stopping time, then the Log-Euler discretisation converges to the logarithm of the underlying

process. We extend the result further to a stochastic volatility case, where A(.), B(.) could

depend on a stochastic volatility process σt. This result serves as the theoretical foundation

of the Monte Carlo simulation of the CEV/SABR process.

6.1 Partial Strong Convergency of Stochastic Volatility Process

Consider the stochastic process

dSt = A(t, St, σt)dt+B(t, St, σt)dW1t, (6.4)

dσt = C(σt)dW2t, dW1tdW2t = ρdt (6.5)

Approximate St, σt using Euler scheme on time points tn = n∆t, where n = 1, ...N, and

N∆t = T . The discretisation scheme is given as

S∆tt+∆t = S∆t

t + A(t, S∆tt , σ∆t

t )∆t+B(t, S∆tt , σ∆t

t )∆W1,t,

σ∆tt+∆t = f(σ∆t

t ,∆t,∆W2,t),(6.6)

where ∆Wi,t = Wi,t+∆t −Wi,t, i = 1, 2.

Remark 6.1.1. We do not specify the discretisation scheme of σt here. It could be any

reasonable discretisation scheme, for example, the Euler scheme as:

σ∆tt+∆t = σ∆t

t + C(σ∆tt )∆W2,t; (6.7)

44

or the Log-Euler scheme as:

σ∆tt+∆t = σ∆t

t exp

(C(σ∆t

t )

σ∆tt

∆W2,t −1

2

(C(σ∆t

t )

σ∆tt

)2

∆t

). (6.8)

For the SABR process, the σt can be simulated exactly using the Log-Euler scheme.

Define the local Lipschitz condition (*) as follows: let Ω1, Ω2 be compact sets, there exist

positive constants K1(Ω1×Ω2) and K2(Ω1×Ω2), such that for any x1, x2 ∈ Ω1, z1, z2 ∈ Ω2,

|A(t, x1, z1)−A(t, x2, z2)|2∨|B(t, x1, z1)−B(t, x2, z2)|2 ≤ K1(Ω1×Ω2)|x1−x2|2+K2(Ω1×Ω2)|z1−z2|2.(6.9)

If local Lipschitz condition (*) holds, then for the bounded Ω1, Ω2, there exists a positive

constant K3(Ω1 × Ω2), such that for all x ∈ Ω1, z ∈ Ω2,

|A(t, x, z)|2 ∨ |B(t, x, z)|2 ≤ K3(Ω1 × Ω2). (6.10)

Theorem 12 says that under the local Lipschitz condition (*) and some other regularity

conditions, the discretized process S∆tt will have L2 convergency to the true process S before

a stopping time.

Theorem 12. Let Ω1, Ω2 be bounded regions and denote the stopping time

τ = inft ≥ 0 : S∆tt /∈ Ω1 or St /∈ Ω1 or σt /∈ Ω2.

For discretisation scheme in (6.6), if the following conditions are satisfied:

i. A(t, St, σt) and B(t, St, σt) satisfy the local Lipschitz condition (*);

ii. for ∆t small enough and t ∈ [tn, tn+1), there is a constant D(Ω2) such that

E(σt − σ∆ttn )2 ≤ D(Ω2)∆t; (6.11)

then for ∆tT < 1, ∃ constant C(Ω1 × Ω2), such that

E(

sup0≤t≤τ∧T

|S∆tt − St|2

)≤ C(Ω1 × Ω2)∆t. (6.12)

45

In other words, as long as S∆tt and St remain in the domain Ω1 and σt remains in the

domain Ω2, the Euler scheme S∆tt converges to the true process St as ∆t→ 0.

Proof. Appendix C.1.

Remark 6.1.2. If σt follows the geometric Brownian motion such that σt = νσtdWt, then

σ∆ttn

can be simulated exactly as σtn . Since σt = σtne∆sνX−1

2∆sν2where ∆s = t − tn and

X ∼ N(0, 1), substitute σ∆tt with σtn , and we have

E(σt − σ∆ttn )2 = E(σtn)2(e∆sν2

− 1) = etnν2(e∆sν2

− 1).

Since ∆s < ∆t, for any tn < T and small enough ∆t, there is a constant K such that

etnν2(e∆sν2 − 1) < K∆t. Therefore σt, σ

∆tt satisfy the condition (ii) in Theorem 12.

Remark 6.1.3. In the SABR process, we have A(t, St, σt) = 0 and B(t, St, σt) = σtSβt . In the

logarithm of the SABR process, we have A(t, St, σt) = 12σ

2t S

2β−2t and B(t, St, σt) = σtS

β−1t .

In both cases, A(.) and B(.) satisfy the local Lipschitz condition (*). (see appendix C.2

for detail.) So, Theorem 12 indicates that the Euler (Log-Euler) discretisation scheme of

the SABR process (with σt simulated exactly) will converge to the true process on the time

interval τ ∨ T , where τ = inft ≥ 0 : S∆tt /∈ [a1, a2] or St /∈ [a1, a2] or σt /∈ [b1, b2]; 0 < a1 <

a2, 0 < b1 < b2. This holds for the CEV process too since the CEV process is a special

case of the SABR process. In terms of our Monte Carlo simulation of the CEV/SABR

process, if St starts not close to 0, we could construct an interval [a1, a2] such that with

large probability, the simulated path will be within the range, and we could consider the

discretized path as a good approximation of the true continuous path.

6.2 Strong Convergence of Mean-reverting CEV Process

Kahl and Jackel (2006) consider a stochastic volatility process which they call it mean-

reverting CEV process. It is defined as follows:

dVt = λ(µ− Vt)dt+ σVβt dWt. (6.13)

They point out that the boundary behavior of the process as follows.

1. 0 is an attainable boundary for 0 < β < 12 and for β = 1

2 if λµ < σ2

2 .

46

2. 0 is unattainable for β > 12 .

3. ∞ is unattainable for all β > 0.

They also develop a numerical scheme to simulate the stock process whose volatility process

is the mean-reverting CEV process.

There is a lot literature on the particular case of the mean-reverting CEV process, when

β = 12 , which is called the mean-reverting square-root process. In the Heston model, the

volatility process is the mean-reverting square-root process. For the mean-reverting square-

root process, Higham and Mao (2005) propose a discretisation scheme called the reflection

Euler scheme:

V ∆tt+∆t = V ∆t

t + λ(µ− V ∆tt )∆t+ σ

√|V ∆tt |∆Wt, ∆Wt = Wt+∆t −Wt. (6.14)

They prove that this discretisation scheme has the strong convergency property. They also

propose a discretisation scheme for the Heston model as follows:

S∆tt+∆t =

√|V ∆tt |S

∆tt ∆W1,t,


t + λ(µ− V ∆tt )∆t+ σ

√|V ∆tt |∆W2,t, corr(∆W1,t,∆W2,t) = ρ.

(6.15)

They prove that such discretized process S∆t will converge to the true process S. This result

serves as the theoretical foundation for our Monte Carlo simulation of the DH process.

We extend the strong convergence to the general mean-reverting CEV process under the

following conditions:

Condition A: 0 < λ∆t < 2 and |(1− λ∆t)2 + βσ2∆t| < 1.

Condition B : 12 ≤ β < 1.

We consider the discretisation scheme with reflection principle on the time points tn =

n∆t, with n = 1, ..., N and N∆t = T as follows:


t + λ(µ− V ∆tt )∆t+ σ|V ∆t

t |β∆Wt, ∆Wt = Wt+∆t −Wt. (6.16)

Proposition 6.2.1 gives the first moment of the discretized process (6.16). Proposition 6.2.2

says that the second moment of the discretized process (6.16) is bounded. Theorem 13 says

that the discretisation scheme (6.16) converges to the true process (6.13) as ∆t → 0 in L1

sense.

47

Proposition 6.2.1. For the discretized process (6.16),

EV ∆tn = (1− λ∆t)n(EV ∆t

0 − µ) + µ, (6.17)

so that when 0 < λ∆t < 2,

limt→∞

EV ∆tt → µ. (6.18)

Proof. For SDE in (6.16), take the expectation of both sides,

EV ∆tn+1 = EV ∆t

n (1− λ∆t) + λ∆tµ (6.19)

⇒ EV ∆tn+1 − µ = (1− λ∆t)(EV ∆t

n − µ) (6.20)

⇒ EV ∆tn+1 = (1− λ∆t)n(EV ∆t

0 − µ) + µ. (6.21)

So when 0 < λ∆t < 2, limt→∞ EV ∆tt → µ; otherwise, limt→∞ EV ∆t

t →∞.

Proposition 6.2.2. For the discretized process (6.16), under conditions A and B, E(V ∆tn )2

and E(V ∆tn )2β is bounded for all n.


Theorem 13. For the mean-reverting CEV process in (6.13), under conditions A and B,

the discretisation process in (6.16) will have the L1 convergence:

lim∆t→0

sup0≤t≤T

E|Vt − V ∆tt | = 0. (6.22)


6.3 Discretisation Schemes Used in the Monte Carlo Simulation

In the Monte Carlo simulation conducted in Chapter 5, we use discretisation schemes to

simulate the processes whose exact simulation is difficult. Based on the discussion in section

6.1 and 6.2, the first order LogEuler scheme is applied to simulate the DL process and

the CEV/SABR process. The first order Euler scheme with reflection principle is applied to

simulate the DH process. The discretisation schemes for each process are summarized below.

48

First order LogEuler scheme:

Denote z ∼ N(0,∆t), w ∼ N(0,∆t), w⊥z.

Discretisation of the displaced lognormal process (exact simulation):

∆ log(S∆tt − θ) = −σ

2

2∆t+ σz

Discretisation of the displaced anti -lognormal process (exact simulation):

∆ log(θ − S∆tt ) = −σ

2

2∆t+ σz.

Discretisation of the CEV/SABR process:

∆ logS∆tt = αt(S

∆tt )β−1∆t− 1

2α2t (S

∆tt )2β−2z,

∆ logαt = −ν2

2∆t+ ν(ρz +

√1− ρ2w).

First order LogEuler scheme with reflection principle:

Denote z ∼ N(0,∆t), w ∼ N(0,∆t), w⊥z.

Discretisation of the displaced Heston process:

∆ log(S∆tt − θ) = −

V ∆tt

2∆t+

√|V ∆tt |z;

∆V ∆tt = κ(µ− V ∆t

t )∆t+ ε√|V ∆tt |w.

Discretisation of the displaced anti -Heston process:

∆ log(θ − S∆tt ) = −

V ∆tt

2∆t−

√|V ∆tt |z;

∆V ∆tt = κ(µ− V ∆t

t )∆t+ ε√|V ∆tt |w.

49

CHAPTER 7

LARGE-EXPIRY IMPLIED VOLATILITY OF DISPLACED

LOGNORMAL

7.1 Large-strike and Large-expiry Behavior

In this section, we analyze the implied volatility of the displaced lognormal dynamics (2.6)

in the large-strike and large-expiry region. Our goal is to give a large-expiry asymptotes of

σimp(K,T ).

7.1.1 Case one: K = S0exT , x ∈ R/[−1

2σ2, 1

2σ2]

First, consider the strike to be K = S0exT , where x is constant. Denote x(T ) = 1

T log((K −θ)/(S0 − θ)). Define the domain of T as follows:

KT := T > 0 : S0exT > θ+ and (S0 −K)+ < CBS(S0 − θ, S0e

xT − θ, σ, T ) < S0. (7.1)

The implied volatility σimp(x, T ) is defined on KT such that

CBS(S0, K, σimp(x, T ), T ) = CBS(S0 − θ,K − θ, σ, T ). (7.2)

We can suppress the x in σimp(x, T ) to be σimp(T ) when no ambiguity arises.

Theorems 14, 15 and 17 give the large-expiry asymptotic approximation of σimp(x, T ),

where Theorem 14 focuses on the region where x ∈ R/[−12σ

2, 12σ

2], Theorems 15 and 17 on

the region where x ∈ (−σ2/2, σ2/2).

Denote

A(T ) =(−x(T ) + 1

2σ2)2

2σ2(7.3)

and define the domain

Kx := T ∈ R+ : A(T ) + x ≥ 0 (7.4)

50

Theorem 14. For the displaced lognormal mordel (2.6), if x ∈ R/[−12σ

2, 12σ

2] and T ∈KT ∩Kx, then the following asymptotic limit of σ2

imp(x, T ) holds.

σ2imp(x, T ) = σ2

∞(x, T ) + a1(x, T )/T + o(1/T ) as T →∞, (7.5)

where

σ2∞(x, T ) := 2

(2A(T ) + x− 2

√A2(T ) + A(T )x

), if x ∈ R/[−1

2σ2,

1

2σ2] (7.6)

ABS(x, σ, a1) := exp

(1

8a1

(4x2

σ4− 1

))σ3

x2 − σ4/41x 6=±σ2/2 (7.7)

a1(x, T ) = 2(x2

σ4∞(x)− 1

4)−1 log

(S0

S0

ABS(x, σ, 0)

ABS(x, σ∞, 0)

). (7.8)

Proof. Appendix D.1.

We suppress the T in x(T ), A(T ), σ2∞(x, T ), a1(x, T ) to be x, A, σ2

∞(x), a1(x), with

the understanding that these variables all depend on T .

Remark 7.1.1. Figure 7.1.1 shows the exact σimp and Theorem 14 formula. We can see that

the approximation is reasonably accurate.

Figure 7.1.1: Theorem 14 formula and Exact σimp for T = 10.

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.05

0.1

0.15

0.2

0.25

0.3

x

σ imp

S0=100, σ=0.2, θ=−20, T=10

S0=100, σ=0.2, θ=60, T=10

σimp

Theorem 14 formula

Remark 7.1.2. We can write A + x =(−x+σ2/2)2

σ2 + (x − x) and limT→∞ x(T ) = x, so if

x ∈ R/[−12σ

2, 12σ

2], the condition A+ x > 0 always holds for large T .

51

Remark 7.1.3. The derivation of Theorem 14 depends on whether x ∈ R/[−12σ

2, 12σ

2] or

not. However, since x will be in a close neighborhood of x when T is large, we express the

condition in terms of x.

Remark 7.1.4. If θ > 0 and x < 0, the strike K will eventually become less than θ as T

increases. Thus ST > θ > K, the option price will always be S0 − K for large T . (7.5)

should be regarded as an approximation of the implied volatility on the domain of T where

S0exT > θ+.

7.1.2 Case two: K = S0exT , x ∈ (−σ2/2, σ2/2)

Theorem 15 gives the large-expiry asymptotic approximation of σimp(x, T ) in the region

which is complement to that in Theorem 14. Theorems 17 and 18 refine the approximation

to higher orders.

Theorem 15. For the displaced lognormal process (2.6), when x ∈ (−σ2/2, σ2/2),

1. If θ < 0, then σimp(x, T ) is not defined for large T .

2. If θ > 0, for T ∈ KT and x ∈ [0, σ2/2), then

limT→∞

σimp(x, T ) = 2x. (7.9)


Before stating Theorem 17, we give an asymptotic approximation of the Black-Scholes

call option in the large-strike, large-expiry case.

Theorem 16. For x > 0, for ∀B ∈ R, limT→∞ a(T ) = const, we have the asymptotic

behavior for the Black-Scholes call option formula in the large-strike, large-expiry case as

follows.

1

S0CBS(S0, S0e

xT ,

√2x+ 2B

√2x

T+a(T )

T, T ) = N(B)+

e−12B

2(a(T )/2−B2 − 1)√

2πT√

2x(1+O(1/

√T )).

(7.10)


52

Remark 7.1.5. Forde et al (2009) give the asymptotic behavior of the Black-Scholes call

option when the volatility is expressed as√σ2 + a/T . Theorem 16 is different from theirs

in that we have order of 1/√T and we restrict the leading term σ2 to be 2x.

Theorem 17 refines the asymptotic behavior given in Theorem 15. It elaborates to higher

order of approximation.

Figure 7.1.2: Theorem 17 formula and Exact σimp.

2 3 4 5 6 7 8 9 10

x 104

0.0762

0.0763

0.0764

0.0765

0.0766

0.0767

0.0768

0.0769

0.077

Time

σ imp

S0=100, σ=0.2, θ=60, x=0.003

σimp

approx σimp

Theorem 17. For the displaced lognormal process (2.6), if θ > 0, x ∈ (0, σ2/2) and T ∈ KT ,

the second order approximation of σimp(x, T ) when T →∞ is given as:

σ2imp(x, T ) = 2x+ 2B

√2x

T+a(T )

T+ o(

1

T), (7.11)

where B = N−1(S0−θS0

), a(T ) = 2(B2+1+C(T )e12B

2√2x), C(T ) =

S0S0e−1

2 ( x2

σ2−x+σ2

4 )T( σ3

x2−σ4/4).

Proof. See appendix D.4.

Remark 7.1.6. Figure 7.1.2 shows the approximation in Theorem 17 is highly accurate when

T is large. In the plot, we have T ∈ [2× 104, 105].

53

Theorem 18 refines Theorem 17’s approximation to the order of 1T√T

.

Theorem 18. For the displaced lognormal process (2.6), if θ > 0, x ∈ (0, σ2/2) and T ∈ KT ,

the third order approximation of σimp(x, T ) as T →∞ is given as:

σ2imp(x, T ) = 2x+ 2B

√2x

T+a

T+d(T )

T√T

+ o(1

T√T

) (7.12)

where B = N−1(S0−θS0

), a = 2(B2 + 1), d(T ) =5B3+3B+C(T )eB

2/24x√T√

2x,

C(T ) =S0S0e−1

2 ( x2

σ2−x+σ2

4 )T( σ3

x2−σ4/4).


Figures 7.1.3 and 7.1.4 compare Theorem 17 and Theorem 18’s approximations of σimp.

In both figures, the left plot is the Theorem 17 formula and the exact σimp, and the right

plot is the Theorem 18 formula and the exact σimp. In Figure 7.1.3, T ∈ [2000, 10000] and

in Figure 7.1.4, T ∈ [200, 1000]. All the other parameters are the same. We can see that for

T ∈ [2000, 10000], Theorem 18 significantly improves the results of Theorem 17. However,

neither of them are a good approximation for T ∈ [200, 1000] . To sum up, Theorem 18 gives

a good approximation of σimp when T is large.

Theorem 19 gives an explicit formula for the at-the-money implied volatility.

Theorem 19. For the displaced lognormal process (2.6), the at-the-money implied volatility

on T ∈ KT is

σimp(0, T ) =2√TN−1

(S0 − θS0

N(σ√T/2) +

θ

2S0

). (7.13)

Thus

limT→∞

σimp(0, T )√T = 2N−1

(1− θ

2S0

)(7.14)


Remark 7.1.7. For σimp(0, T ) to exist, we needS0−θS0

N(σ√T/2) + θ

2S0∈ [0, 1]. So, when

θ < 0, σimp(0, T ) becomes undefined for large T .

54

7.1.3 Case three: K = S0exTα

Theorem 14 can be extended to the region where K approaches ∞ such that K = S0exTα

for x 6= 0 and α > 1. This is given in Theorem 20.

Denote x = 1Tα log(

S0exTα−θS0−θ

), M = xTα−1, M = xTα−1, A =

(−M+σ2/2

σ

)2

. Define

the domain of T as

KTα := T > 0 : S0exTα > θ+ and (S0−S0e

xTα)+ < CBS(S0−θ, S0exTα−θ, σ, T ) < S0,

(7.15)

the implied volatility σimp(x, T ) is well defined by (2.5) when T ∈ KTα .

Theorem 20. For displaced lognormal dynamics (2.6), for x > 0, T ∈ KTα, we have

σ2imp(x, T ) = σ2

∞ + a1(T )/T + o(1/T ), (7.16)

where

σ2∞ = 2(M + A−

√2MA+ A2), (7.17)

a1 = 2(M2

σ4∞− 1

4)−1 log

(S0

S0

ABS(M, σ, 0)

ABS(M,σ∞, 0)

), (7.18)

and ABS(.) is defined by (7.7).


Figure 7.1.5 and 7.1.6 compare the Theorem 20 formula and the exact σimp. We can see

the approximation is accurate with moderate T . In Figure 7.1.5, the smallest K is about 700;

in Figure 7.1.6, the smallest K is about 200. Letting K approaches infinity in the fashion of

K = S0exTα , we could have a highly accurate approximation of σimp with reasonable expiry

time T .

Remark 7.1.8. In both Theorems 14 and 20, we have to make the strike K large to have

reasonably accurate approximation. Theorem 14 uses a large x (e.g. x = 0.2, T = 10, α = 1)

to obtain a good approximation; Theorem 20 uses a small x but a large α (e.g. x =

0.003, T = 10, α = 3).

55

7.2 Fixed-strike Large-expiry Implied Volatility

In this section, we fix the strike, and analyze the implied volatility of the displaced lognormal

dynamics when time to expiry is large. We show the asymptotic behavior and the mono-

tonicity of the implied volatility. In this section, denote K = S0ex, where x is a constant

and define the domain of T as

KT := T > 0, (S0 −K)+ < CBS(S0 − θ,K − θ, σ, T ) < S0. (7.19)

σimp(x, T ) is well-defined when T ∈ KT by (2.5).

Theorem 21. For the displaced lognormal process (2.6), the fixed-strike large-expiry implied

volatility has the following properties.

1. (Asymptotic behavior).

(a) For all K > θ+, T ∈ KT and 2S0 > θ > 0,

limT→∞

σimp(x, T )√T = σblsimpv(S0, K, S0 − θ), (7.20)

where σblsimpv(S0, K, P ) is the implied volatility of the one-year call option with

price P defined as follows:

CBS(S0, K, σblsimpv(S0, K, P ), 1) = P. (7.21)

(b) For θ < 0, the σimp(x, T ) is not defined when T →∞.

2. (Monotonicity in time) For all θ > 0, K > θ+ and T ∈ KT

limT→∞

sgn∂σimp

∂T(x, T ) = − sgn θ (7.22)


Theorem 22 refines the results of Theorem 21.1. The notation A(T ) ∼ B(T ) means

A(T )/B(T )→ 1 as T ↑ ∞.

56

Theorem 22. For the displaced lognormal dynamics with K = S0ex, the following holds for

the fixed-strike, large-time implied volatility:

σimp(x, T ) ∼ 1√T

(σblsimpv(S0, K, S0 − θ) + z

), (7.23)

where z =−1+

√1+4M(−1

8+ x2

8a2 )

−14+ x2

4a2

, a =σ2blsimpv(S0,K,S0−θ)

2 ,

M =√

2π exp(12(−x+a√

2a)2)

(1S0CBS(S0 − θ,K − θ, σ, T )− S0−θ

S0

).


Figure 7.2.1 shows Theorem 22 formula and exact σimp. We can see that for θ close to

S0 the approximation is quite accurate even for moderate large T .

57

Figure 7.1.3: Theorems 17 and 18 formula and Exact σimp.

2000 3000 4000 5000 6000 7000 8000 9000 100000.076

0.0765

0.077

0.0775

0.078

0.0785

0.079

0.0795

Time

σ imp

S0=100, σ=0.2, θ=60, x=0.003

Exact σimp

approximated σimp

2000 3000 4000 5000 6000 7000 8000 9000 100000.0762

0.0764

0.0766

0.0768

0.077

0.0772

0.0774

0.0776

0.0778

0.078

Time

σ imp

S0=100, σ=0.2, θ=60, x=0.003

Exact σimp

approximated σimp

Figure 7.1.4: Theorems 17 and 18 formula and Exact σimp.

200 300 400 500 600 700 800 900 10000.08

0.085

0.09

0.095

0.1

0.105

Time

σ imp

S0=100, σ=0.2, θ=60, x=0.003

Exact σimp

approximated σimp

200 300 400 500 600 700 800 900 10000.076

0.077

0.078

0.079

0.08

0.081

0.082

0.083

0.084

0.085

0.086

Time

σ imp

S0=100, σ=0.2, θ=60, x=0.003

Exact σimp

approximated σimp

58


3 4 5 6 7 8 9 10

x 10−3

0.15

0.155

0.16

0.165

0.17

0.175

0.18

0.185

x

σ imp

S0=100, σ=0.2, θ=60 ,T=10, α=3

Exact σimp

approximated σimp


3 4 5 6 7 8 9 10

x 10−3

0.105

0.11

0.115

0.12

0.125

0.13

0.135

0.14

0.145

0.15

x

σ imp

S0=100, σ=0.2, θ=60, α=2.8 ,T=7

Exact σimp

approximated σimp

59


−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.30.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

x

σ imp

S0=100, σ=0.4, θ=60, T=20

σimp

approx σimp

60

CHAPTER 8

CONCLUSION

By establishing properties shared by all displaced lognormal volatility skews, Theorems 1 and

2, in effect, exhibit limitations on what phenomena the displaced lognormal can faithfully

model. In particular, the displaced lognormal’s everywhere-monotonic skews (Theorem 1)

and its state space (θ,∞) 6= R+ may be drawbacks, when pricing contracts having sensitivity

to the tail behavior of an underlying process whose true state space is R+ or whose true

volatility skew is non-monotonic. The slope-constrained skew (Theorem 2) of the displaced

lognormal may be a drawback, when modeling markets, such as typical equity markets,

which exhibit downward skews of steepness greater than the Theorem 2 upper bound. The

displaced anti -lognormal overcomes the slope constraint, but it imposes bounds from above

on S, while allowing unbounded negative S. Furthermore, the displaced anti -lognormal has

the everywhere-monotonic skews (Theorem 4).

We therefore do not endorse the DL as a model of equity markets, but we do propose the

DL as a control variate to improve the accuracy of Monte Carlo pricing under alternative

dynamics (such as CEV and SABR) that do model equity prices. The DL is effective as a

control variate, because it is simple enough to admit both unbiased simulation and explicit

pricing formulas for many contracts, yet flexible enough to generate high correlations and

hence significant variance reduction with respect to the CEV/SABR dynamics. We give

two examples. The first is the down-and-out call option under the CEV/SABR dynamics.

The DL control generates large variance reduction as shown in Table 5.1.1 and 5.1.2. The

second example is the discrete arithmetic Asian option. We compare the variance reduction

of the DL control and the lognormal control. The DL control generates significant more

satisfactory variance reduction than LN control as shown in Table 5.1.4 and 5.1.5.

Another displaced process, the displaced (anti -)Heston process, will generate more vari-

ance reduction when used as control variate. We are able to have explicit pricing formulas

for the calls/puts and barrier options on the DH process. Though no exact simulations of

the DH process are available, we have a discretisation scheme which guarantees convergence

61

to the true process. We show that DH control improves the variance reduction of DL control

when applied to the SABR dynamics in Table 5.1.3.

Moreover, we analyze the variance reduction by combining the DL/DH control variate

with importance sampling. The combined technique is superior to the pure importance

sampling when applied to options under the SABR dynamics, as shown in Table 5.2.1 and

5.2.2, Table 5.2.3 and 5.2.4.

Toward either purpose – as a model, or as a computational device on behalf of another

model – the DL calibrates easily to a given volatility skew level and slope, via the short-

expiry limiting implied volatility formula of Theorems 3 and 6; and the DH calibrates easily

to a given volatility skew level, slope and convexity, via the short-expiry limiting implied

volatility formula of Theorems 8 and 9.

Furthermore, we give an explicit relationship between the short-expiry implied volatilities

of any process which is a positive martingale with its corresponding (anti-)displaced process

(Theorems 10 and 11). Combined with Durrleman’s (2004) approximation of the short-expiry

near-the-money implied volatility, we are able to give an approximation of the short-expiry

near-the-money implied volatility of any (anti-)displaced process.

In contrast to the short-expiry limiting implied volatility, the large-expiry limiting implied

volatility of the displaced lognormal is also analyzed. We first give the approximation of the

implied volatility in the large-strike case (Theorems 14, 15, 17 and 18 where K = S0exT ;

Theorem 20 where K = S0exTα). Secondly, we give the approximation of the implied

volatility in the fixed-strike case (Theorems 21 and 22).

Finally, we discuss some issues in the discretisation scheme of the Monte Carlo simulation.

Under some regularity conditions, we prove the partial strong convergency of the discretized

stochastic volatility process to the continuous process (Theorem 12). We also prove a strong

convergency result of the mean-reverting CEV process (Theorem 13).

62

APPENDIX A

A.1 Appendix: Proof of Theorem 1–Global Behavior of

Displaced Lognormal

Define for all K > θ+

d2(K) :=log[(S0 − θ)/(K − θ)]

σ√T

− σ√T

2.

Let KS(T ) := inf KS(T ). Our notation may suppress the S or T . Note that K = (K,∞).

If θ ≥ 0 then K = θ. If θ < 0 then K > 0 because CBS(S0−θ, 0−θ, σ) > S0−θ+θ = S0.

Proposition A.1.1. For all T > 0 and K ∈ K(T ):

If θ > 0 then σimp(K) < σ. If θ < 0 then σimp(K) > σ.

Proof. For all K > θ and S0 > θ,

d

dθCBS(S0 − θ,K − θ, σ) = −∂C

BS

∂S− ∂CBS

∂K= −N(d2 + σ

√T ) +N(d2) < 0 (A.1)

so, according as θ ≷ 0, we have CBS(S0 − θ,K − θ, σ) ≶ CBS(S0, K, σ). By (2.9),

σimp(K) ≶ σ. (A.2)

for all K ∈ K.

Proposition A.1.2. For all T > 0, we have σimp(K)→ σ as K →∞.

Hence supK∈K(T ) σimp(K) = σ if θ > 0; and infK∈K(T ) σimp(K) = σ if θ < 0.

Proof. We prove for θ > 0. The proof for θ < 0 is similar.

By Proposition A.1.1, σimp(K) < σ for all K ∈ K. So it suffices to show that for all

δ > 0, there exists k such that for all K > k,

σ − δ < σimp(K)

63

or equivalently

CBS(S0, K, σ − δ) < CBS(S0, K, σimp(K)) = CBS(S0 − θ,K − θ, σ).

Hence it suffices that for K large enough,

0 < M(K) := CBS(S0 − θ,K − θ, σ)− CBS(S0, K, σ − δ), (A.3)

which we verify as follows. For all K > θ+,

∂M

∂K= −N

(log[(S0 − θ)/(K − θ)]

σ√T

− σ√T

2

)+N

(log(S0/K)

(σ − δ)√T− (σ − δ)

√T

2

). (A.4)

So

sgn∂M

∂K= sgn

[log(S0/K)

(σ − δ)√T− log[(S0 − θ)/(K − θ)]

σ√T

+δ√T

2

](A.5)

= sgn

[log(S0/K)σ − log[(S0 − θ)/(K − θ)]σ−δ

(σ − δ)σ√T

+δ√T

2

](A.6)

= sgn

[log[(S0 × (K − θ))/(K × (S0 − θ))]σ − log[(K − θ)/(S0 − θ)]δ

(σ − δ)σ√T

+δ√T

2

].

(A.7)

The part inside the sgn in (A.7) approaches −∞ as K →∞.

So for K sufficiently large, ∂M/∂K(K) < 0.

Moreover limK→∞M(K) = 0− 0 = 0.

Therefore M(K) > 0 for K sufficiently large, which proves (A.3).

Lemma A.1.3. For all K ∈ K,

sgn∂σimp

∂K(K) = sgnF (σimp(K), K)

where F : R× (θ+,∞)→ R is defined by

F (y,K) := −y2T/2− y√Td2(K) + log(S0/K).

64

Proof. Take the K derivative of (2.9),

∂CBS

∂K(S0, K, σimp(K)) +

∂CBS

∂σ(S0, K, σimp(K))

∂σimp

∂K(K) =

∂CBS

∂K(S0 − θ,K − θ, σ).

(A.8)

Therefore

sgn∂σimp

∂K(K) = sgn

[∂CBS

∂K(S0 − θ,K − θ, σ)− ∂CBS

∂K(S0, K, σimp(K))

]= sgn

[(log(S0/K)

σimp

√T−σimp

√T

2

)− d2(K)

]= sgn

[− σ2

imp(K)T/2− σimp(K)√Td2(K) + log(S0/K)

]= sgnF (σimp(K), K),

as claimed.

Lemma A.1.4. There exists K ∈ K such that

sgn∂σimp

∂K(K) = sgn θ.

Proof. If θ = 0 then this is obvious.

If θ < 0 then as K ↓ K we have CBS(S0− θ,K − θ, σ)→ S0 hence σimp →∞. So there

exists K > K such that ∂σimp/∂K(K) < 0, as claimed.

If θ > 0 then the at-the-money strike K = S0 satisfies the conclusion, because

sgn∂σimp

∂K(S0) = sgn

[− σimp(S0)

√T/2 + σ

√T/2

]> 0

by Proposition A.1.1.

Proposition A.1.5. For all T > 0 and K ∈ K(T ),

sgn∂σimp

∂K(K) = sgn θ. (A.9)

Proof. If θ = 0 then this is obvious. Otherwise, by Lemma A.1.4 the conclusion holds

for at least one K ∈ K. By continuity of∂σimp∂K on K, the conclusion will hold for all

K ∈ K, if we can show that ∂σimp/∂K 6= 0 on K, or equivalently (by Lemma A.1.3) that

65

F (σimp(K), K) 6= 0 for all K ∈ K. The remaining lemmas complete the proof, by verifying

that F (σimp(K), K) 6= 0.

For K > θ+ define

∆(K) := d22(K) + 2 log(S0/K) (A.10)

h(K) := d2(K) + σ√TK − θK

. (A.11)

Let

D := K > θ+ : ∆(K) ≥ 0,

and for all K ∈ D, define

y±(K) :=−d2(K)±

√∆(K)√

T. (A.12)

Lemma A.1.6. For (y,K) ∈ R × K, we have F (y,K) = 0 if and only if K ∈ D and

y = y±(K).

Proof. If K /∈ D then F (·, K) is a quadratic with no real roots. If K ∈ D then F (·, K) has

roots y = y±(K).

Lemma A.1.7. For all K ∈ D define

H±(K) := S0N(±√

∆(K))− (S0 − θ)N(d2(K) + σ√T )− θN(d2(K))

Then for all K ∈ D ∩K,

H±(K) = CBS(y±(K))− CBS(σimp(K))

where CBS(·) is shorthand for CBS(S0, K, ·).

Hence y+(K) ≷ σimp(K) if H+(K) ≷ 0. Likewise y−(K) ≷ σimp(K) if H−(K) ≷ 0.

66

Proof. Using log(S0/K)/(y±(K)√T )− y±(K)

√T/2 = d2(K) and (2.9), we have

CBS(y±(K))− CBS(σimp) =

[S0N

(log(S0/K)

y±(K)√T

+y±(K)

√T

2

)−KN(d2(K))

]−[(S0 − θ)N(d2(K) + σ

√T )− (K − θ)N(d2(K))

]= S0N(d2(K) + y±(K)

√T )− (S0 − θ)N(d2(K) + σ

√T )− θN(d2(K))

= H±(K)

The remaining conclusion is by monotonicity of CBS .

Lemma A.1.8. If θ ≷ 0 then for all K ∈ D we have√

∆(K) ≷ |h(K)|.

Proof. Consider only the case θ > 0. The proof for θ < 0 is similar.

We need to show that for all K ∈ D,

d22(K) + 2 log(S0/K) > d2

2(K) +

(K − θK

)2

σ2T + 2d2(K)σ√TK − θK

or equivalently that

((K − θK

)2

−(K − θK

))σ2T + 2

K − θK

log[(S0− θ)/(K − θ)]− 2 log(S0/K) < 0. (A.13)

The first term is negative, because 0 < (K − θ)/K < 1; so it suffices to show that for all

K > θ,

L(K) :=K − θK

log[(S0 − θ)/(K − θ)]− log(S0/K) ≤ 0.

This is verified by L(S0) = 0 and ∂L/∂K = (θ/K2) log[(S0−θ)/(K−θ)] ≶ 0 for K ≷ S0.

Lemma A.1.9. For all K > θ+:

• If θ > 0 then K ∈ D and∂H−∂K > 0,

∂H+∂K > 0.

• If θ < 0, h(K) > 0 then ∂∆∂K (K) < 0. If moreover K ∈ D then

∂H−∂K (K) > 0,

∂H+∂K (K) <

0.

• If θ < 0, h(K) < 0 then ∂∆∂K (K) > 0. If moreover K ∈ D then

∂H−∂K (K) < 0,

∂H+∂K (K) >

0.

67

Proof. The ∆ conclusions hold because

∂∆

∂K(K) = − 2h(K)

(K − θ)σ√T.

If θ > 0, the K ∈ D conclusion clearly holds for K ∈ (S0,∞), and also holds for K ∈ (θ, S0]

because

log[(S0 − θ)/(K − θ)]/(σ√T )− σ

√T/2 < log(S0/K)/(σ

√T )− σ

√T/2 < 0

implies

∆(K) > (log(S0/K)/(σ√T )−σ

√T/2)2+2 log(S0/K) = (log(S0/K)/(σ

√T )+σ

√T/2)2 ≥ 0.

Lastly, in all cases, the H± conclusions hold because of

∂H±∂K

(K) =1√2π

Ke−d22(K)/2

(K − θ)σ√T√

∆(K)

(√∆(K)∓ h(K)

)and Lemma A.1.8.

Lemma A.1.10. If θ > 0 then for all K ∈ K = (θ,∞), we have K ∈ D and

y1(K) < σimp(K) < y2(K).

Proof. By Lemma A.1.9 we have K ∈ D and y±(K) are well-defined.

To prove σimp < y+, note that as K ↓ θ we have d2(K) → ∞ hence H+(K) → 0.

Moreover, ∂H+/∂K > 0 on K, by Lemma A.1.9. So on K we have H+ > 0, hence σimp < y+

by Lemma A.1.7.

To prove y− < σimp, note that as K → ∞ we have d2(K) → −∞ hence H−(K) → 0.

Moreover, ∂H−/∂K > 0 on K, by Lemma A.1.9. So on K we have H− < 0, hence y− < σimp

by Lemma A.1.7.

Lemma A.1.11. If θ < 0 then for each K ∈ K ∩ D we have

σimp(K) /∈ [y−(K), y+(K)].

68

Proof. Because θ < 0, it is clear that h is decreasing on (0,∞).

Because K ∈ D, we have h(K) 6= 0 by Lemma A.1.8.

If h(K) < 0, then for all k > K we have h(k) < 0 and ∂∆/∂K(k) > 0 by Lemma

A.1.9. So for all k > K we have k ∈ D and∂H−∂K (k) < 0 by Lemma A.1.9. Moreover

limk→∞H−(k) = 0. Therefore H−(K) > 0, hence σimp(K) < y−(K) by Lemma A.1.7.

If h(K) > 0, then for all k ∈ (0, K) we have h(k) > 0 and ∂∆/∂K(k) < 0 by Lemma

A.1.9. So for all k ∈ (0, K) we have k ∈ D and∂H+∂K (k) < 0 by Lemma A.1.9. Moreover, as

k ↓ 0, we have ∆(k)→∞, hence

H+(k)→ S0 − (S0 − θ)N(

log[(S0 − θ)/(−θ)]σ√T

+σ√T

2

)− θN

(log[(S0 − θ)/(−θ)]

σ√T

− σ√T

2

)= S0 − CBS(S0 − θ,−θ, σ) < S0 − (S0 − θ + θ) = 0.

Therefore H+(K) < 0, hence y+(K) < σimp(K) by Lemma A.1.7.

Lemma A.1.12. For all K ∈ K we have F (σimp(K), K) 6= 0.

Proof. Combine Lemmas A.1.10, A.1.11, and A.1.6.

A.2 Appendix: Proof of Theorem 2–At-the-money Behavior of

Displaced Lognormal

Proof of Theorem 2.1. Taking K = S0 in (2.9), we have

CBS(S0, S0, σatm) = CBS(S0 − θ, S0 − θ, σ), (A.14)

hence

S0[2N(σatm

√T/2)− 1] = (S0 − θ)[2N(σ

√T/2)− 1], (A.15)

and

N(σatm

√T/2) =

S0 − θ2S0

[2N(σ√T/2)− 1] +

1

2

= (1− θ

S0)N(σ

√T/2) +

θ

S0N(0) ≤ N((1− θ

S0)d1)

(A.16)

for θ ≥ 0, because N(x) is concave on x ≥ 0. Monotonicity of N implies σatm ≤ (1−θ/S0)σ.

Similarly, for θ ≤ 0, we have σatm ≥ (1− θ/S0)σ.

69

Proof of Theorem 2.2. Using (A.8), we have∣∣∣∣∂ log σimp

∂ logK

∣∣∣∣K=S0

=

∣∣∣∣ S0

σimp

∂σimp

∂K

∣∣∣∣K=S0

=N(σatm

√T/2)−N(σ

√T/2)

φ(σatm√T/2)σatm

√T

, (A.17)

because θ < 0 implies σatm > σ. By concavity of N on [0,∞),

φ(σatm

√T/2) ≤ N(σatm

√T/2)−N(σ

√T/2)

σatm√T/2− σ

√T/2

≤ φ(σ√T/2). (A.18)

Combining (A.17), (A.18), and Theorem 2.1 produces the lower bound∣∣∣∣∂ log σimp

∂ logK

∣∣∣∣K=S0

≥ σatm − σ2σatm

≥ 1

2

(1− 1

1− θ/S0

)=

|θ|2(S0 + |θ|)

. (A.19)

Combining (A.17) and (A.18) produces the upper bound∣∣∣∣∂ log σimp

∂ logK

∣∣∣∣K=S0

≤ σatm − σ2σatm

× φ(σ√T/2)

φ(σatm√T/2)

≤ 1

2eσ

2atmT/8. (A.20)

as claimed.

A.3 Appendix: Proof of Theorems 3 and 6–Short-expiry

Behavior of DL

The notation A(T ) ∼ B(T ) means that A(T )/B(T )→ 1 as T ↓ 0.

Proof of Theorem 3. For all K > θ+, we have CBS(S0 − θ,K − θ, σ, T ) ↓ (S0 − K)+ as

T ↓ 0, so for all T sufficiently small, CBS(S0 − θ,K − θ, σ, T ) < S0 hence K ∈ KS(T ).

Applying Roper-Rutkowski (2007) Proposition 5.1 to the displaced lognormal model, we

have, as T ↓ 0,

σimp(K,T ) ∼ | log(S0/K)|√−2T log(CBS(S0 − θ,K − θ, σ, T )− (S0 −K)+)

. (A.21)

On the other hand, applying it to the Black-Scholes model, we have, as T ↓ 0,

σ ∼ | log((S0 − θ)/(K − θ))|√−2T log(CBS(S0 − θ,K − θ, σ, T )− (S0 −K)+)

. (A.22)

70

Therefore

σimp(K,T ) ∼ |σ|| log(S0/K)|| log((S0 − θ)/(K − θ))|

=σ log(S0/K)

log((S0 − θ)/(K − θ))(A.23)

as T ↓ 0.

Proof of Theorem 6. For all K ∈ (0, θ), we have CBS(θ −K, θ − S0,−σ, T ) ↓ (S0 −K)+ as

T ↓ 0, so for all T sufficiently small, CBS(θ − K, θ − S0,−σ, T ) < S0 hence K ∈ KS(T ).

Applying Roper-Rutkowski (2007) Proposition 5.1 to the displaced anti-lognormal model,

we have, as T ↓ 0,

σimp(K,T ) ∼ | log(S0/K)|√−2T log(CBS(θ −K, θ − S0,−σ, T )− (S0 −K)+)

. (A.24)

On the other hand, applying it to the Black-Scholes model, we have, as T ↓ 0,

−σ ∼ | log((θ −K)/(θ − S0))|√−2T log(CBS(θ −K, θ − S0,−σ, T )− (S0 −K)+)

. (A.25)

Therefore (A.23) holds.

A.4 Appendix: Proof of Theorem 4–Global Behavior of

Displaced anti -Lognormal

Define for all K < θ

d2(K) :=log[(θ − S0)/(θ −K)]

|σ|√T

− |σ|√T

2.

Lemma A.4.1. For all K ∈ KADL,

sgn∂σimp

∂K(K) = sgnF (σimp(K), K)

where F : R× (0, θ)→ R is defined by

F (y,K) := −y2T/2 + y√Td2(K) + log(S0/K).

71

Proof. Taking the K derivative of (2.19),

∂CBS

∂K(S0, K, σimp(K))+

∂CBS

∂σ(S0, K, σimp(K))

∂σimp

∂K(K) = −1−∂C

BS

∂K(θ−S0, θ−K, |σ|).

(A.26)

Therefore

sgn∂σimp

∂K(K) = sgn

[− 1− ∂CBS

∂K(θ − S0, θ −K, |σ|)−

∂CBS

∂K(S0, K, σimp(K))

]= sgn

[− 1 +N(d2(K)) +N

(log(S0/K)

σimp(K)√T−σimp(K)

√T

2

)]= sgn

[(log(S0/K)

σimp(K)√T−σimp(K)

√T

2

)+ d2(K)

]= sgn

[− σ2

imp(K)T/2 + σimp(K)√Td2(K) + log(S0/K)

]= sgnF (σimp(K), K),

as claimed.

Lemma A.4.2. θ > S0 > 0, there exists K ∈ KADL such that

sgn∂σimp

∂K(K) = − sgn θ.

Proof. The at-the-money strike K = S0 satisfies the conclusion, because

sgn∂σimp

∂K(S0) = sgn

[− σimp(S0)

√T/2− |σ|

√T/2

]< 0.

Proposition A.4.3. For all T > 0, K ∈ KADL(T ) and θ > S0,

sgn∂σimp

∂K(K) = − sgn θ. (A.27)

Proof. By Lemma A.4.2 the conclusion holds for at least one K ∈ KADL. By continuity of∂σimp∂K on KADL, the conclusion will hold for all K ∈ KADL, if we can show that ∂σimp/∂K 6=

0 on KADL, or equivalently (by Lemma A.4.1) that F (σimp(K), K) 6= 0 for all K ∈ K. The

remaining lemmas complete the proof, by verifying that F (σimp(K), K) 6= 0.

72

For K < θ define

∆(K) := d22(K) + 2 log(S0/K) (A.28)

h(K) := d2(K)− |σ|√Tθ −KK

. (A.29)

Let

D := 0 < K < θ : ∆(K) ≥ 0,

and for all K ∈ D, define

y±(K) :=d2(K)±

√∆(K)√

T. (A.30)

Lemma A.4.4. For (y,K) ∈ R × KADL, we have F (y,K) = 0 if and only if K ∈ D and

y = y±(K).

Proof. If K /∈ D then F (·, K) is a quadratic with no real roots. If K ∈ D then F (·, K) has

roots y = y±(K).

Lemma A.4.5. For all K ∈ D define

H±(K) := −S0N(∓√

∆(K))− (θ − S0)N(d2(K) + |σ|√T ) + θN(d2(K)) (A.31)

Then for all K ∈ D ∩KADL,

H±(K) = PBS(y±(K))− PBS(σimp(K))

where PBS(·) is shorthand for PBS(S0, K, ·).

Hence y+(K) ≷ σimp(K) if H+(K) ≷ 0. Likewise y−(K) ≷ σimp(K) if H−(K) ≷ 0.

Proof. Using log(S0/K)/(y±(K)√T )− y±(K)

√T/2 = −d2(K) and (2.18), we have

PBS(y±(K))− PBS(σimp) =

[KN

(− log(S0/K)

y±(K)√T

+y±(K)

√T

2

)− S0N

(− log(S0/K)

y±(K)√T− y±(K)

√T

2

)]−[(θ − S0)N(d2(K) + |σ|

√T )− (θ −K)N(d2(K))

]= KN(d2(K))− S0N(d2(K)− y±(K)

√T )− (θ − S0)N(d2(K) + |σ|

√T )

+ (θ −K)N(d2(K))

= H±(K)

73

The remaining conclusion is by monotonicity of PBS .

Lemma A.4.6. θ > S0, for all K ∈ D we have√

∆(K) < |h(K)|.

Proof. We want to show:

h2(K)−∆(K) =

((θ −KK

)2

+

(θ −KK

))σ2T−2

θ −KK

log[(θ−S0)/(θ−K)]−2 log(S0/K) > 0.

(A.32)

The first term is positive, because (θ −K)/K > 0; so it suffices to show that for all K > θ,

L(K) := −θ −KK

log[(θ − S0)/(θ −K)]− log(S0/K) ≥ 0.

This is verified by L(S0) = 0 and ∂L/∂K = (θ/K2) log[(θ−S0)/(θ−K)] ≶ 0 for K ≶ S0.

Lemma A.4.7. For all 0 < K < θ:

• If h(K) > 0 then ∂∆∂K (K) > 0. If moreover K ∈ D then

∂H−∂K (K) < 0,

∂H+∂K (K) > 0.

• If h(K) < 0 then ∂∆∂K (K) < 0. If moreover K ∈ D then

∂H−∂K (K) > 0,

∂H+∂K (K) < 0.

Proof. The ∆ conclusion hold because:

∂∆(K)

∂K=

2h(K)

(θ −K)|σ|√T. (A.33)

The H± conclusions hold because of

∂H±∂K

(K) =1√2π

Ke−d22(K)/2

(θ −K)|σ|√T√

∆(K)

(√∆(K)± h(K)

).

Lemma A.4.8. If θ > 0 then for each K ∈ KADL ∩ D we have

σimp(K) /∈ [y−(K), y+(K)].

Proof. Because θ > 0 and∂h(K)∂K = 1

θ−K1

|σ|√T

+|σ|√Tθ

K2 > 0, h(K) is monotonically in-

creasing on (0, θ). Furthermore h(0) = −∞, h(θ) = ∞, so there is a point k0 ∈ (0, θ) such

that h(k0) = 0. By (A.33), ∆(K) is decreasing on K : 0 < K < k0 and increasing on

K : k0 < K < θ.

74

Because K ∈ D, we have h(K) 6= 0 by Lemma A.4.6.

For a particular K ∈ D, if h(K) < 0, then for all k < K, we have h(k) < 0 and

∂∆/∂K(k) < 0 by Lemma A.4.7. So for all k < K and k ∈ D, we have∂H+∂K (k) < 0 by

Lemma A.4.7. Moreover, k ↓ 0, ∆(k)→∞, hence by (A.31)

limk→0

H+(k) = 0− (θ − S0)N(d2(0) + |σ|√T ) + θN(d2(0)) = 0− CBS(θ − S0, θ, |σ|, T ) < 0.

Therefore H+(K) < 0 hence y+(K) < σimp(K) when K ∈ KADL by Lemma A.4.5.

If h(K) > 0, then for all k ∈ (K, θ), we have h(k) > 0 and ∂∆/∂K(k) > 0 by Lemma

A.4.7. So for all k > K and k ∈ D, we have∂H−∂K (k) < 0 by Lemma A.4.7.

k = θ ⇒ d2(k) =∞⇒ ∆(k) =∞,

⇒ H−(θ) = −S0N(−√

∆(k))−(θ−S0)N(d2(k)+|σ|√T )+θN(d2(k)) = −S0−(θ−S0)+θ = 0.

Therefore H−(K) > 0 hence y−(K) > σimp(K) when K ∈ KADL by Lemma A.4.5.

Lemma A.4.9. For all K ∈ KADL we have F (σimp(K), K) 6= 0.

Proof. Combine Lemmas A.4.4, and A.4.8.

A.5 Appendix: Proof of Theorem 5–At-the-money Behavior of

Displaced anti -Lognormal

Proof of Theorem 5.1. Consider at-the-money in (2.19), we have,

S0[2N(σatm

√T/2)− 1] = (θ − S0)[2N(|σ|

√T/2)− 1], (A.34)

and

N(σatm

√T/2) =

θ − S0

2S0[2N(|σ|

√T/2)− 1] +

1

2

= (θ

S0− 1)N(|σ|

√T/2) + (2− θ

S0)N(0).

(A.35)

75

If θS0− 1 ∈ [0, 1], then

(θ

S0− 1)N(|σ|

√T/2) + (2− θ

S0)N(0) ≤ N

((θ

S0− 1)|σ|√T

2

). (A.36)

Because N(x) is concave on x ≥ 0. Monotonicity of N implies σatm ≤ (θ/S0 − 1)|σ|.Similarly, if θ

S0− 1 ∈ [1,∞), then

(θ

S0− 1)N(|σ|

√T/2) + (2− θ

S0)N(0) ≥ N

((θ

S0− 1)|σ|√T

2

). (A.37)

Using the monotonicity of N , we have σatm ≥ (θ/S0 − 1)|σ|.

Proof of Theorem 5.2. Re-order the entries in (A.26) and plug in (A.34), we have,∣∣∣∣∂ log σimp

∂ logK

∣∣∣∣K=S0

= =

∣∣∣∣−1 +N(−|σ|√T/2) +N(−σatm

√T/2)


√T

∣∣∣∣K=S0

(A.38)

=

∣∣∣∣N(|σ|√T/2) +N(σatm

√T/2)− 1


√T

∣∣∣∣K=S0

(A.39)

=

∣∣∣∣θ

θ−S0

(N(σatm

√T/2)− 1/2

)φ(σatm

√T/2)σatm

√T

∣∣∣∣K=S0

. (A.40)

By the concavity of N on [0,∞):

φ(σatm

√T/2) ≤ N(σatm

√T/2)−N(0)

σatm√T/2

≤ φ(0). (A.41)

Combine (A.40) and (A.41), we have

θ

2(θ − S0)≤∣∣∣∣∂ log σimp

∂ logK

∣∣∣∣K=S0

≤ θ

2(θ − S0)exp(σ2

atmT/8) (A.42)

76

APPENDIX B

B.1 Appendix: Proof of Theorem 7–At-the-money Behavior of

Displaced Independent Stochastic Volatility

Proof of Theorem 7. Define the average volatility of the DISV process:

Σ =

√∫ T0 σ2

t dt

T. (B.1)

Since conditioning on the average volatility Σ, the process ST − θ is lognormal process,

E(ST −K)+ = EE[(ST − θ)− (K − θ)]+|Σ = E(CBS(S0 − θ,K − θ,Σ)).

For K ∈ KDISV , the implied volatility σimp satisfies

CBS(S0, K, σimp) = E(CBS(S0 − θ,K − θ,Σ)). (B.2)

Differentiating with respect to K in (B.2) and exchanging the expectation and the differen-

tiation (validate later), we have

∂CBS(S0, K, σimp)

∂σimp

∂σimp

∂K= E

(∂CBS(S0 − θ,K − θ,Σ)

∂K−∂CBS(S0, K, σimp)

∂K

). (B.3)

Since∂CBS(S0,K,σimp)

∂σimp> 0, we have

sgn∂σimp

∂K

∣∣∣∣K=S0

= sgnE(∂CBS(S0 − θ,K − θ,Σ)

∂K− ∂CBS(S0, K, σatm)

∂K

)K=S0

(B.4)

= sgnE(N(−σatm

√T

2)−N(−Σ

√T

2)

)(B.5)

= sgn

(EN(

Σ√T

2)−N(

σatm√T

2)

). (B.6)

77

What remains is to show θ ≷ 0 ⇒ N(σatm√T

2 ) ≶ EN(Σ√T

2 ). Assuming the exchange-

ability of the expectation and the differentiation, we have

∂E(CBS(S0 − θ,K − θ,Σ))

∂θ= E

∂CBS(S0 − θ,K − θ,Σ)

∂θ= E(−N(d1) +N(d2)) ≤ 0,

where d1,2 =log((S0−θ)/(K−θ))±Σ2T/2

Σ√T

. So E(CBS(S0 − θ,K − θ,Σ)) is monotonically de-

creasing as a function of θ. Thus,

θ ≷ 0⇒ E(CBS(S0 − θ,K − θ,Σ)) ≶ E(CBS(S0, K,Σ)),

⇒ CBS(S0, K, σimp) ≶ E(CBS(S0, K,Σ)).

Particularly when at-the-money, we have

N(σatm

√T

2) ≶ EN(

Σ√T

2). (B.7)

To complete the proof, we need to validate the exchangeability of the expectation and

the differentiation. Since∣∣∣∣∂CBS(S0 − θ,K − θ,Σ)

∂θ

∣∣∣∣ = | −N(d1) +N(d2)| ≤ 2,

∣∣∣∣∂CBS(S0 − θ,K − θ,Σ)

∂K

∣∣∣∣ = | −N(d2)| ≤ 1,

we can use dominant convergent theorem to exchange the expectation and the differentiation.

B.2 Appendix: Proof of Theorems 8 and 9– Short-expiry

Behavior of DH

Proof of Theorem 8. Denote CDH(S0, K, T ) the call option price under the DH process,

with strike K, maturity T and initial value S0; and similarly CH(S0, K, T ) the call option

price under the Heston process. We have CDH(S0, K, T ) = CH(S0− θ,K− θ, T ). Applying

78

Roper-Rutkowski (2007) corollary 5.1 to the displaced Heston model, we have, as T ↓ 0,

σDHimp (K,T ) ∼

| log(S0/K)|√

−2T log(CDH (S0,K,T )−(S0−K)+)if K 6= S0√

2πCDH(S0, K, T )/(S0√T ) if K = S0

(B.8)

On the other hand, applying it to the Heston model, we have, as T ↓ 0,

σHimp(K − θ, T ) ∼

| log((S0−θ)/(K−θ))|√

−2T log(CH (S0−θ,K−θ,T )−(S0−K)+)if K 6= S0√

2πCH(S0 − θ,K − θ, T )/((S0 − θ)√T ) if K = S0

(B.9)

By comparing (B.8) and (B.9), and using (3.12), we have

limT→0

σDHimp (K,T ) =

limT→0

σHimp(K − θ, T )× log(S0/K)

log((S0 − θ)/(K − θ))if K 6= S0

limT→0

σHimp(K − θ, T )× (1− θ/S0) if K = S0.(B.10)

Proof of Theorem 9. Denote CADH(S0, K, T ) the call option price under the displaced anti -

process, with strike K, maturity T and initial value S0; and similarly CH(S0, K, T ) the call

option price under the Heston process. Let S = θ − S, K = θ −K. Using put-call parity,

we have

E(ST −K)+− (S0−K)+ = E(K− ST )+− (K− S0)+ = E(ST − K)+− (S0− K)+. (B.11)

If at-the-money, then,

E(ST −K)+ = E(ST − K)+. (B.12)

Applying Roper-Rutkowski (2007) corollary 5.1 to the displaced anti-Heston model, we

have, as T ↓ 0,

σADHimp (K,T ) ∼

| log(S0/K)|√

−2T log(CADH (S0,K,T )−(S0−K)+)if K 6= S0√

2πCADH(S0, K, T )/(S0√T ) if K = S0

(B.13)

79

Applying it to the Heston model, we have, as T ↓ 0,

σHimp(θ −K,T ) ∼

| log((S0−θ)/(K−θ))|√

−2T log(CH (θ−S0,θ−K,T )−((θ−S0)−(θ−K))+)if K 6= S0√

2πCH(θ − S0, θ −K,T )/((θ − S0)√T ) if K = S0

(B.14)

By comparing (B.13) and (B.14), and using (B.11) and (B.12), we have

limT→0

σADHimp (K,T ) =

limT→0

−σHimp(θ −K,T )× log(S0/K)

log((S0 − θ)/(K − θ))if K 6= S0

limT→0

−σHimp(θ −K,T )× (1− θ/S0) if K = S0.(B.15)

B.3 Appendix: Proof of Propositions 3.1.2 and 3.2.1–Level,

Slope and Convexity of DH Short-expiry Implied Volatility

Proof of Proposition 3.1.2. We utilize the short-expiry approximation formula of the implied

volatility given by Durrleman (2004) and the relationship between the implied volatilities of

the DH model and the Heston model (Theorems 8 and 9).

Denote St = St − θ, K = K − θ. Durrleman (2004) shows that, the near-money short-

expiry implied volatility with strike K, maturity T and initial value S0 is

σHimp(K, T ) =

√σ2

0 + a0 log(S0/K) + b0T

2+c02

log2(S0/K) +O(log(S0/K)T + T 2).

(B.16)

It can be approximated as

σHimp(K, T ) ∼√σ2

0 + a0 log(S0/K) + b0T

2+c02

log2(S0/K), (B.17)

where A(T ) ∼ B(T ) means A(T )/B(T )→ 1 as T ↓ 0 and

a0 = −ερ2, c0 =

ε2

6σ20

(1− 7ρ2

4), b0 = κ(µ− σ2

0) +ερ

2σ2

0 −ε2

6(1− ρ2/4). (B.18)

Since we have specified that ρ = 0 for the Heston process, we have a0 = 0. In the following,

we give a limit approximation of the level, slope and convexity as T ↓ 0. Though we do not

80

explicitly write the limT→0 out.

i. Intercept

Using (3.13)-(3.14), the near-expiry at-the-money implied volatility of the displaced

Heston is:

(σDHimp (S))2 = (σHimp(S0−θ)(1−θ/S0))2 = (σ20+κ(µ−σ2

0)T

2−ε

2T

12)×(1−θ/S0)2+O(T 2).

(B.19)

ii. Slope

∂σDHimp (K)

∂K|K=S0

= limK→S0

σDHimp (K)− σDHimp (S0)

K − S0

= limK→S0

σHimp(K − θ) log(S0/K)log((S0−θ)/(K−θ))

− σHimp(S0 − θ)(1− θ/S0)

K − S0

= limK→S0

∂σHimp(K − θ)∂K

log(S0/K)

log((S0 − θ)/(K − θ))+

σHimp(K − θ)− 1K log((S0 − θ)/(K − θ)) + 1

K−θ log(S0/K)

(log((S0 − θ)/(K − θ)))2(L’Hopital’s rule)

= limK→S0

∂σHimp(K − θ)∂K

(1− θ/S0) + σHimp(S0 − θ)θ

2S20

where in the last step we use

limK→S0

log(S0/K)

log((S0 − θ)/(K − θ))= 1− θ/S0

and

limK→S0

− 1K log((S0 − θ)/(K − θ)) + 1

K−θ log(S0/K)

(log((S0 − θ)/(K − θ)))2=

θ

2S20

.

Applying (B.17) for the Heston model, we have,

∂σHimp(K)

∂K

∣∣∣∣K=S0

=1

2(σHimp(K))−1

(− a0

1

K− c0 log(S0/K)

1

K

)∣∣∣∣K=S0

= 0 +O(T ).

(B.20)

81

So∂σDHimp (K)

∂K|K=S0

= σHimp(S0 − θ)θ

2S20

+O(T ). (B.21)

Consequently the slope is

∂ log σDHimp (K)

∂ logK

∣∣∣∣K=S0

=θ

2(S0 − θ)+O(T ). (B.22)

iii. Convexity

For the second derivative, using (3.14), we have

∂2σDHimp (K)

∂K2=

∂2σHimp(K − θ)∂K2

log(S0K )

log(S0−θK−θ )

+ 2∂σHimp(K − θ)

∂K

∂

∂K

(log(

S0K )

log(S0−θK−θ )

)+ σHimp(K − θ) ∂2

∂K2

(log(

S0K )

log(S0−θK−θ )

).

Using∂σHimp(K−θ)

∂K

∣∣∣∣K=S0

= O(T ) in (B.20), we only need to calculate the first and

third term. The first term is straightforward to calculate, and the third term requires

more attention.

(a) The first term

Combining

limK→S0

∂2σHimp(K)

∂K2=

1

2

(σHimp(S0−θ)

)−1 c0(S0 − θ)2

and limK→S0

log(S0K )

log(S0−θK−θ )

=S0 − θS0

,

we have

limK→S0

∂2σHimp(K − θ)∂K2

log(S0K )

log(S0−θK−θ )

=1

2

(σHimp(S0 − θ)

)−1 c0(S0 − θ)S0

. (B.23)

(b) The third term

82

With some calculation, we have

H(S0, θ) := limK→S0

∂2

∂K2

(log(

S0K )

log(S0−θK−θ )

)=

4θ2 − 5θS0

6S30(S0 − θ)

.

⇒ limK→S0

σHimp(K − θ) ∂2

∂K2

(log(

S0K )

log(S0−θK−θ )

)= σHimp(S0 − θ)H(S0, θ). (B.24)

Combining (B.23) and (B.24) gives the second derivative of the implied volatility

∂2σDHimp (K)

∂K2

∣∣∣∣K=S0

=1

2

(σHimp(S0−θ)

)−1 c0(S0 − θ)S0

+σHimp(S0−θ)H(S0, θ)+O(T ).

(B.25)

Proof of Proposition 3.2.1. We will use the relationship between the implied volatilities of

the displaced anti -Heston model and the Heston model. Other parts of the proof are similar

to those in Proposition 3.1.2. Denote K = θ −K and S = θ − S0.

i. Intercept:

The short-expiry at-the-money implied volatility is easy to get:

(σADHimp (S0))2 = (σHimp(θ−S0)(1−θ/S0))2 = (σ20+κ(µ−σ2

0)T

2−ε

2T

12)×(1−θ/S0)2+O(T 2).

(B.26)

ii. Slope

The first derivative∂σDHimp (K)

∂K is calculated as in the case of displaced Heston, which is

∂σADHimp (K)

∂K

∣∣∣∣K=S0

= −σHimp(θ − S0)θ

2S20

+O(T ). (B.27)

Consequently, we have

∂ log σADHimp (K)

∂ logK

∣∣∣∣K=S0

=θ

2(S0 − θ)+O(T ). (B.28)

iii. Convexity

83

Use (3.14), we have

∂2σADHimp (K)

∂K2= −

[∂2σHimp(θ −K)

∂K2

log(S0K )

log(S0−θK−θ )

+ 2∂σHimp(θ −K)

∂K

∂

∂K

(log(

S0K )

log(S0−θK−θ )

)+ σHimp(θ −K)

∂2

∂K2

(log(

S0K )

log(S0−θK−θ )

)].

Again since∂σHimp(θ−K)

∂K

∣∣∣∣K=S0

= O(T ), we only need to calculate the first and the

third term.

(a) The first term

Combining

limK→S0

∂2σHimp(K)

∂K2=

1

2

(σHimp(θ−S0)

)−1 c0(θ − S0)2

and limK→S0

log(S0K )

log(S0−θK−θ )

=S0 − θS0

,

we have

limK→S0

∂2σHimp(θ −K)

∂K2

log(S0K )

log(S0−θK−θ )

=1

2

(σHimp(θ − S0)

)−1 c0(S0 − θ)S0

. (B.29)

(b) The third term

This is exactly the same as in the displaced Heston case:

limK→S0

σHimp(θ −K)∂2

∂K2

(log(

S0K )

log(S0−θK−θ )

)= σHimp(θ − S0)H(S0, θ). (B.30)

Combing (B.29) and (B.30) gives the convexity as

∂2σADHimp (K)

∂K2

∣∣∣∣K=S0

= −1

2

(σHimp(θ−S0)

)−1 c0(S0 − θ)S0

−σHimp(θ−S0)H(S0, θ)+O(T ).

(B.31)

84

APPENDIX C

C.1 Appendix: Proof of Theorem 12–Partial Strong

Convergency of Stochastic Volatility Process

Proof. The proof here has the same flavor as the proof of Theorem 1 in Zhang et al (2004),

except for two major differences:

1. In Zhang et al(2004) Theorem 1, their stopping time is τ = ρ ∧ θ where ρ = inft ≥0 : S∆t

t /∈ Ω1, θ = inft ≥ 0 : St /∈ Ω1. We add one more piece to the stopping

time τ , which is τ = ρ ∧ θ ∧ γ, where γ = inft ≥ 0 : σt /∈ Ω2.

2. We substitute the local Lipschitz condition (*) with their local Lipschitz condition

(15).

To make it complete, we sketch the outline of the proof below.

First, we can write a continuous version of the Euler approximation defined in (6.6) as

follows.

S∆tt = S0 +

∫ t

0A(u, S∆t

u , σ∆tu )du+

∫ t

0B(u, S∆t

u , σ∆tu )dWu, (C.1)

where we introduce the piecewise constant process

S∆tt = S∆t

tn , for t ∈ [tn, tn+1),

σ∆tt = σ∆t

tn , for t ∈ [tn, tn+1).(C.2)

Note that S∆tt and S∆t

t coincide at the discrete points tn = n∆t.

Let T1 ∈ [0, T ] be an arbitrary time. For any t ∈ [0, τ ∧ T1], we have

E(

sup0≤t≤τ∧T1

|S∆tt − St|2

)≤ 2E sup

0≤t≤τ∧T1

∣∣∣∣ ∫ t

0(A(u, S∆t

u , σ∆tu )− A(u, Su, σu))du

∣∣∣∣2+ 2E sup

0≤t≤τ∧T1

∣∣∣∣ ∫ t

0(B(u, S∆t

u , σ∆tu )−B(u, Su, σu))dWu

∣∣∣∣2.(C.3)

85

Applying the Holder inequality to the first term of (C.3) gives

2E sup0≤t≤τ∧T1

∣∣∣∣ ∫ t

0(A(u, S∆t

u , σ∆tu )− A(u, Su, σu))du

∣∣∣∣2≤ 2TE sup

0≤t≤τ∧T1

∫ τ∧T1

0

∣∣∣∣(A(u, S∆tu , σ∆t

u )− A(u, Su, σu))

∣∣∣∣2du.

(C.4)

Applying the Doob inequality to the second term of (C.3) gives

2E sup0≤t≤τ∧T1

∣∣∣∣ ∫ t

0(B(u, S∆t

u , σ∆tu )−B(u, Su, σu))du

∣∣∣∣2≤ 8C1E sup

0≤t≤τ∧T1

∫ τ∧T1

0

∣∣∣∣(B(u, S∆tu , σ∆t

u )−B(u, Su, σu))

∣∣∣∣2du.

(C.5)

If A(.) and B(.) satisfy the local Lipschitz condition (*), we have

|(A(u, S∆tu , σ∆t

u )− A(u, Su, σu))|2 ∨ |(B(u, S∆tu , σ∆t

u )−B(u, Su, σu))|2

≤ K1(Ω1 × Ω2)|S∆tu − Su|2 +K2(Ω1 × Ω2)|σ∆t

u − σu|2.(C.6)

Substituting (C.4)-(C.6) into (C.3) and use condition (6.11), we have

E(

sup0≤t≤τ∧T1

|S∆tt − St|2

)≤ 2(T + 4C1)K1(Ω1 × Ω2)E

∫ τ∧T1

0|S∆tu − Su|2du+K4(Ω1 × Ω2)∆t

= 2(T + 4C1)K1(Ω1 × Ω2)E∫ τ∧T1

0|S∆tu − S∆t

u + S∆tu − Su|2du

+K4(Ω1 × Ω2)∆t

≤ 4(T + 4C1)K1(Ω1 × Ω2)E∫ τ∧T1

0(|S∆t

u − S∆tu |2 + |S∆t

u − Su|2)du

+K4(Ω1 × Ω2)∆t

≤ 4(T + 4C1)K1(Ω1 × Ω2)E∫ τ∧T1

0|S∆tu − S∆t

u |2du+K4(Ω1 × Ω2)∆t

+4(T + 4C1)K1(Ω1 × Ω2)

∫ T1

0E(

sup0≤u′≤τ∧u

|S∆tu′ − Su′ |

2)

du,

(C.7)

where K4(Ω1 × Ω2) = 2(T + 4C1)K2(Ω1 × Ω2)D(Ω2)T .

86

If the sum of the first term and the second term on the right-hand side is bounded, then we

could apply the Gronwall inequality which leads to a bound on E(

sup0≤t≤τ∧T1|S∆tt −St|2

).

So all we need to prove now is the first term on the right-hand side is bounded. By the

definition of S∆tt , we know that S∆t

t = S∆t[t/∆t]∆t

, where [t/∆t] is the integer part of t/∆t.

Use (C.1), we have

E|S∆tt − S∆t

t |2 = E|S∆t[t/∆t]∆t − S

∆tt |2

= E(∣∣∣∣ ∫ [t/∆t]∆t

0A(u, S∆t

u , σ∆tu )du−

∫ t

0A(u, S∆t

u , σ∆tu )du

+

∫ [t/∆t]∆t

0B(u, S∆t

u , σ∆tu )dWu −

∫ t

0B(u, S∆t

u , σ∆tu )dWu

∣∣∣∣2)≤ 2E

(∣∣∣∣ ∫ t

[t/∆t]∆tA(u, S∆t

u , σ∆tu )du

∣∣∣∣2 +

∣∣∣∣ ∫ t

[t/∆t]∆tB(u, S∆t

u , σ∆tu )dWu

∣∣∣∣2).(C.8)

Applying (6.10) to the first term on the right-hand side and using the Holder inequality leads

to ∣∣∣∣ ∫ t

[t/∆t]∆tA(u, S∆t

u , σ∆tu )du

∣∣∣∣2 ≤ K3(Ω1 × Ω2)∆t2. (C.9)

Applying (6.10) to the second term on the right-hand side and using the Ito isometry

(E(∫ t

0 f(Xu)dWu)2 =∫ t

0 Ef2(Xu)du) leads to

E∣∣∣∣ ∫ t

[t/∆t]∆tB(u, S∆t

u , σ∆tu )dWu

∣∣∣∣2 ≤ K3(Ω1 × Ω2)∆t. (C.10)

If T∆t < 1, (C.9) and (C.10) leads to

E|S∆tt − S∆t

t |2 ≤ 2K3(Ω1 × Ω2)∆t2 + 2K3(Ω1 × Ω2)∆t. (C.11)

Thus

E∫ t∧T1

0|S∆tu − S∆t

u |2du ≤ 2T1K3(Ω1 × Ω2)∆t2 + 2T1K3(Ω1 × Ω2)∆t = C0(Ω1 × Ω2)∆t,

(C.12)

87

where C0(Ω1 × Ω2) = 2K3(Ω1 × Ω2)(1 + T1)∆t. Plugging (C.12) into (C.7), we have

E(

sup0≤t≤τ∧T1

|S∆tt −St|2

)≤ C1(Ω1×Ω2)∆t+C2(Ω1×Ω2)E

∫ T1

OE(

sup0≤u′≤τ∧u

|S∆tu′ −Su′|

2)

du

(C.13)

where C1(Ω1 ×Ω2) = C0(Ω1 ×Ω2)4(T + 4C1)K1(Ω1 ×Ω2) +K4(Ω1 ×Ω2), C2(Ω1 ×Ω2) =

4(T + 4C1)K1(Ω1 × Ω2). Now applying the Gronwall inequality to (C.13), we have

E(

sup0≤t≤τ∧T

|S∆tt − St|2

)≤ C1(Ω1 × Ω2)eC2(Ω1×Ω2)T∆t = C(Ω1 × Ω2)∆t. (C.14)

Hence the claim is proved.

C.2 Appendix: Coefficients of SABR Satisfy the Local Lipschtiz

Condition (*)

The σt of the SABR process can be simulated exactly, so σ∆ttn

= σtn . Using this, we have

∣∣∣∣σtSβt − σ∆tt (S∆t

t )β∣∣∣∣2 =

∣∣∣∣σtSβt − σ∆ttn (S∆t

tn )β∣∣∣∣2

=

∣∣∣∣σtSβt − σtn(S∆ttn )β

∣∣∣∣2=

∣∣∣∣σtSβt − σt(S∆ttn )β + σt(S

∆ttn )β − σtn(S∆t

tn )β∣∣∣∣2

≤ 2

∣∣∣∣σtSβt − σt(S∆ttn )β

∣∣∣∣2 + 2

∣∣∣∣σt(S∆ttn )β − σtn(S∆t

tn )β∣∣∣∣2

= 2σ2t

(Sβt − (S∆t

tn )β)2

+ 2(σt − σtn)2(S∆ttn )2β .

(C.15)

When Ω1 is bounded compact set and does not contain 0, for any x, y ∈ Ω1 there ∃z ∈ (x, y),

such that

(xβ − yβ)2 = (βzβ−1)2(x− y)2 ≤ C(Ω1)(x− y)2. (C.16)

Thus when Ω1, Ω2 are bounded, B(t, St, σt) = σtSβt satisfies the local Lipschitz condition

(*).

We can show that the coefficients of the logarithm of the SABR process satisfy the local

Lipschitz condition (*) in a similar way.

88

C.3 Appendix: Proof of Proposition 6.2.2 and Theorem

13–Strong Convergence of Mean-reverting CEV Process

Proof of Proposition 6.2.2. Equation (6.16) gives the recursive relationship of the second

moment of V ∆tn :

E(V ∆tn+1)2 = (1−λ∆t)2E(V ∆t

n )2+2λ∆tµ(1−λ∆t)EV ∆tn +λ2∆t2µ2+σ2∆tE|V ∆t

n |2β . (C.17)

For β < 1, the Holder inequality gives

(E|V ∆tn |2β)

12β ≤ (E|V ∆t

n |2)12 ⇒ E|V ∆t

n |2β ≤ (E|V ∆tn |2)β . (C.18)

Plugging the EV ∆tn from (6.17) into (C.17) and use (C.18), we have

E(V ∆tn+1)2 ≤ (1−λ∆t)2E(V ∆t

n )2+2λ∆tµ(1−λ∆t)

[(1−λ∆t)n(EV ∆t

0 −µ)+µ

]+λ2∆t2µ2+σ2∆t(E(V ∆t

n )2)β .

(C.19)

When 0 < β < 1, f(x) = xβ is a concave function; so ∀x, the tangent line of f(x) is above

itself. Thus we can choose the tangent line of f(x) which pass the point (1, 1) as

g(x) = β(x− 1) + 1,

such that xβ < β(x− 1) + 1. Let Z0 = E(V ∆t0 )2 and Zn evolves by the following schemes:

Zn+1 = (1−λ∆t)2Zn+2λ∆tµ(1−λ∆t)

[(1−λ∆t)n(EV ∆t

0 −µ)+µ

]+λ2∆t2µ2+σ2∆t

(β(Zn−1)+1

),

(C.20)

The Zn derived in this way will be bigger than E(V ∆tn )2 for all n > 1.

Simplify the notation and write Zn as:

Zn+1 = aZn + b+ crn, (C.21)

where

a = (1− λ∆t)2 + βσ2∆t, b = 2λ∆tµ2(1− λ∆t) + λ2∆t2µ2 + (1− β)σ2∆t,

c = 2λ∆tµ(1− λ∆t)E(V ∆t0 − µ).

(C.22)

89

It is well established that if |a| < 1, |r| < 1, then

limn→∞

Zn =b

1− a. (C.23)

So when |(1 − λ∆t)2 + βσ2∆t| < 1 and |1 − λ∆t| < 1, the limit of Zn exists, therefore the

Zn will be bounded and E(V ∆tn )2 will be bounded.

Proof of Theorem 13. The proof is in similar spirit to the ones in Higham and Mao (2004).

We use ak, Φk, Ψk as defined by them:

1. a0 = 1, ak = e−k(k+1)/2.

2. For each k ≥ 1, Ψk(µ) is a continuous function with support in (ak, ak−1) such that

0 ≤ Ψk(µ) ≤ 2kµ , for ak < µ < ak−1. Also we have

∫ ak−1ak

Ψk(µ)dµ = 1.

3. Φk(x) :=∫ |x|

0 dy∫ y

0 Ψk(µ)dµ. Three properties of Φk(x) will be used:

(a)

|Φ′k(x)| ≤ 1. (C.24)

(b)

|Φ′′k(x)|

≤2k|x| , for ak < |x| < ak−1

= 0, otherwise(C.25)

(c)

|x| − ak−1 ≤ Φk(x) ≤ |x|, for all x ∈ R. (C.26)

First, we denote a continuous version of the Euler approximation (6.16) as

V ∆tt := V0 +

∫ t

0λ(µ− V ∆t

u )du+

∫ t

0σ|V ∆t

u |βdWu (C.27)

where we introduce the piecewise constant process

V ∆tt := V ∆t

tn , for t ∈ [tn, tn+1). (C.28)

Using (C.27), we have

Vt − V ∆tt = −λ

∫ t

0(Vu − V ∆t

u )du+ σ

∫ t

0(|Vu|β − |V ∆t

u |β)dWu. (C.29)

90

where the first and second inequality use |x + y|β ≤ C(|x|β + |y|β) as proved in Lemma

C.3.1; the second to last and third to last inequality uses (C.25) and the last inequality use

Lemma C.3.2.

Plugging the above results into (C.32), we have

EΦk(Vt − V ∆tt ) ≤ λ

∫ t

0E|Vu − V ∆t

u |du+ λ

∫ t

0E|V ∆t

u − V ∆tu |du+ C

σ2T

k+ C

σ2T

kak(∆tD)β

≤ λ

∫ t

0E|Vu − V ∆t

u |du+ λ(∆tD)1/2 + Cσ2T

k+ C

σ2T

kak(∆tD)β ,

where the second inequality uses Lemma C.3.2 again. Combining this with the left-hand

side of (C.26) gives

E|Vt− V ∆tt | ≤ ak−1 +C

σ2T

k+C

σ2T

kak(∆tD)β + λ(∆tD)1/2 + λ

∫ t

0E(Vt− V ∆t

t )du. (C.33)

Applying the Gronwall’s inequality to (C.33) gives the upper bound for E|Vt − V ∆tt |,

sup0≤t≤T

E|Vt − V ∆tt | ≤ eλT

(ak + C

σ2T

k+ C

σ2T

kak(∆tD)β + λ(∆tD)1/2

). (C.34)

The terms in the parenthesis went to 0 as ∆t→ 0. This proves (6.22).

Lemma C.3.1. ∀β > 0, there is a constant C(β), such that ∀X, Y ,

|X + Y |β ≤ C(β)(|X|β + |Y |β). (C.35)

Proof. Note that

|X + Y | ≤ |X|+ |Y | ⇒ ∀β > 0, |X + Y |β ≤ (|X|+ |Y |)β .

So if we can show 1C(β)

≤ (|X|

|X|+|Y |)β + (

|Y ||X|+|Y |)

β , we are done. Let p =|X|

|X|+|Y | , then

1− p =|Y |

|X|+|Y | . The function f(p) = pβ + (1− p)β is continuous on the interval [0, 1], so it

can achieve the minimum. Moreover, the minimum is bigger than 0. So there is C(β) such

that the inequality holds.

92

APPENDIX D

D.1 Appendix: Proof of Theorem 14 –Large-strike and

Large-expiry Asymptotic of Displaced Lognormal

Lemma D.1.1. For A defined in (7.3), T ∈ Kx, define

σ2∞(x, T ) :=

2(2A+ x− 2√A2 + Ax) if x ∈ R/[−1

2σ2, 1

2σ2]

2(2A+ x+ 2√A2 + Ax) if x ∈ [−1

2σ2, 1

2σ2]

(D.1)

we have

1

2σ2∞(x, T )

≤ x if x >1

2σ2

≥ x if x ∈ [0,1

2σ2]

≥ (−x) if x ∈ [−1

2σ2, 0]

≤ (−x) if x < −1

2σ2

(D.2)

For each x, the equalities in the first and last equation in (D.2) hold only at most finite T ,

so for T large enough, the strict inequalities hold and we have

1

2σ2∞(x, T )

< x if x >

1

2σ2

< (−x) if x < −1

2σ2

(D.3)

Proof. We suppress the T in σ2∞(x, T ) to be σ2

∞(x). Note that σ2∞(x) defined in (D.1) is in

fact the solution of the function

A =(−x+ σ∞(x)2/2)2

2σ∞(x)2. (D.4)

Denote

V ∗BS(x, σ) =(x+ σ2/2)2

2σ2, (D.5)

94

then A = V ∗BS(x, σ)− x and (D.4) can be written as

V ∗BS(x, σ)− x = V ∗BS(x, σ∞(x))− x. (D.6)

When x ∈ R/[−12σ

2, 12σ

2], we choose σ∞(x) to be the smaller root which satisfies (D.4),

so

σ2∞(x) = 2(2A+ x− 2

√A2 + Ax). (D.7)

If x > 12σ

2 > 0, using A =(−x+σ2/2)2

2σ2 ≥ 0, we have

1

2σ2∞(x)− x = 2A− 2

√A2 + Ax ≤ 0. (D.8)

The equality holds if and only if x = σ2/2, which happens at most at one T . If x < −12σ

2 < 0,

using A+ x =(x+σ2∞/2)2

2σ2∞≥ 0 and 0 ≤ A+ x < A, we have

1

2σ2∞(x)− (−x) = 2(A+ x)− 2

√A2 + Ax ≤ 0. (D.9)

The equality hold if and only if A+ x =(−x+σ2/2)2

σ2 + x = 0, which again, happens at most

at two T ’s.

When x ∈ [−12σ

2, 12σ

2], we choose σ∞(x) to be the larger root which satisfies (D.4), so

σ2∞(x) = 2(2A+ x+ 2

√A2 + Ax). (D.10)

If 0 ≤ x ≤ 12σ

2, using A ≥ 0 gives

1

2σ2∞(x)− x = 2A+ 2

√A2 + Ax ≥ 0. (D.11)

If −12σ

2 ≤ x < 0, using A+ x ≥ 0 gives

1

2σ2∞(x)− (−x) = 2(A+ x) + 2

√A2 + Ax ≥ 0. (D.12)

Proof. [Proof of Theorem 14.]

95

We first show the case when x > σ2/2. For S from displaced lognormal dynamics, denote

S = S − θ, then S will be a lognormal process and

1

S0E(ST − S0e

xT )+ = N(z+)− exTN(z−), (D.13)

where z± =(−x±σ2/2)

√T

σ .

Recall an approximation of the cumulative density function of standard normal N(z)

from Olver(1974):

N(−z) = 1−N(z) =exp(−z2/2)

z√

2π(1 +O(1/z2)), as z → +∞. (D.14)

When x > σ2/2, there is a T0 such that when T > T0, x > σ2/2. Then as T → ∞, we

have z± → −∞. Using (D.14) and ABS in (7.7), we have

N(z+)− exTN(z−) =1√2πT

e−1

2 ( x2

σ2−x+σ2

4 )T(

σ

x− σ2/2− σ

x+ σ2/2

)(1 +O(

1

T))

=1√2πT

e− (x−σ2/2)2

2σ2 TABS(x, σ, 0)(1 +O(

1

T))

=1√2πT

e−(V ∗BS(x,σ)−x)TABS(x, σ, 0)(1 +O(1

T)).

(D.15)

Consequently,

1

S0E(ST − S0e

xT )+ =S0

S0

1

S0E(ST − S0e

xT )+

=S0

S0

1√2πT

e−(V ∗BS(x,σ)−x)TABS(x, σ, 0)(1 +O(1

T)).

(D.16)

We choose σ∞(x) and a1 so that

V ∗BS(x, σ)− x = V ∗BS(x, σ∞(x))− x, (D.17)

S0

S0ABS(x, σ, 0) = ABS(x, σ∞(x), a1(x)). (D.18)

96

Plugging (D.17) and (D.18) into (D.16), we have

1

S0E(ST−S0e

xT )+ =1√2πT

e−(V ∗BS(x,σ∞(x))−x)TABS(x, σ∞(x), a1(x))(1+O(1

T)). (D.19)

Now if we could show that −x + (σ∞(x)2 + a1(x)/T )/2 > 0 for large enough T , then we

could apply (D.14) to show that (D.19) is an asymptotic formula for the Black-Scholes call

price with volatility√σ2∞(x) + a1(x)/T .

Denote σ2T = σ2

∞(x)+a1(x)T . By Lemma D.1.1 we know that x > σ2

∞(x)/2. limT→∞ a1(x)

converges to a constant. So for T large enough, we have x > σ2T /2. Using (D.14), the Black-

Scholes call price formula can be approximated as

CBS(S0, S0exT , σT , T ) =

1√2πT

exp

(− 1

2(x2

σ2T

− x+σ2T

4)T

)(σT

x− σ2T /2− σTx+ σ2

T /2

)(1 +O(1/T ))

=1√2πT

e−(V ∗BS(x,σ∞(x))−x)TABS(x, σ∞(x), a1(x))(1 +O(1

T)).

(D.20)

Intuitively, comparing (D.19) and (D.20), we can see that the implied volatility of the

displaced lognormal process is

σ2imp(x, T ) = σ2

∞(x) + a1(x)/T + o(1/T ). (D.21)

The rigorous proof is as follows. From (D.19), ∀ε > 0,

1

S0E(ST − S0e

xT )+ =ABS(x, σ∞, a1)√

2πTexp(−(V ∗BS(x, σ∞)− x)T )(1 +O(1/T ))(D.22)

≤ ABS(x, σ∞, a1)√2πT

exp(−(V ∗BS(x, σ∞)− x)T )eε. (D.23)

For any δ > 0, Lemma D.1.1 shows that ε(δ) = ( 4x2

σ4∞(x)− 1) δ16 > 0. Thus, we can choose

ε > 0 in the previous step so that ε− ε(δ) < 0, and

exp

(1

8a1(x)(

4x2

σ2∞− 1) + ε

)≤ exp

(1

8(a1(x) + δ)(

4x2

σ2∞− 1) + ε− ε(δ)

).

97

Using (D.23) and (D.20), ∀δ > 0, there exists T1, such that when T > T1,

1

S0E(ST − S0e

xT )+ ≤ ABS(x, σ∞, a1 + δ)√2πT

exp

(− (V ∗BS(x, σ∞)− x)T + ε− ε(δ)

)≤ CBS(S0, S0e

xT , σ2∞(x) + (a1(x) + δ)/T, T )

Using the monotonicity of the Black-Scholes formula, the implied volatility of the displaced

lognormal process σimp(x) is

σ2imp(x, T ) ≤ σ2

∞(x) + (a1(x) + δ)/T.

Similarly, we can prove the lower bound, so the equality in (7.5) holds.

For the x < −σ2/2 case, we consider the put price on the displaced lognormal dynamics,

and apply similar methods.

D.2 Appendix: Proof of Theorem 15–First Order Approximation

of Large-strike Large-expiry Implied Volatility

Lemma D.2.1. For x ∈ (−σ2/2, σ2/2), we have

limT→∞

E(ST − S0exT )+ = S0 − θ. (D.24)

Proof. Apply the asymptotic approximation of N(.) to get the results. More specifically,

1

S0E(ST − S0e

xT )+

=S0

S0

1

S0E(ST − S0e

xT )+

=S0

S0

(1 +

1√2πT

e−1

2 ( x2

σ2−x+σ2

4 )T(

σ

x− σ2/2− σ

x+ σ2/2

)(1 +O(

1

T))

)= 1− θ

S0+S0

S0

1√2πT

e−1

2 ( x2

σ2−x+σ2

4 )T(

σ

x− σ2/2− σ

x+ σ2/2

)(1 +O(

1

T))

(D.25)

Proof. [Proof of Theorem 15.]

98

When θ < 0, limT→∞ E(S0−S0exT ) = S0− θ > S0, so σimp(T ) is not well defined when

T is large.

When θ > 0, denote B := N−1(S0−θS0

). Define σ(T ) as

σ(T ) :=B +

√B2 + 2xT√T

.

Abbreviate σ(T ) as σ, it satisfies the equation:

−x+ σ2/2

σ

√T = B, (D.26)

and limT→∞(−x−σ2/2)

√T

σ = limT→∞−√B2 + 2xT = −∞ . So

limT→∞

CBS(S0, S0exT , σ, T ) = lim

T→∞S0N

((−x+ σ2/2)

√T

σ

)−S0e

xTN

((−x− σ2/2)

√T

σ

)= S0−θ.

(D.27)

Recall that limT→∞CBS(S0, S0exT , σimp(T )) = S0 − θ, so

limT→∞

σimp(x, T ) = limT→∞

σ(T ) = 2x. (D.28)

D.3 Appendix: Proof of Theorem 16–Asymptotic Formula of

Black-Scholes Call Option

Proof.

1

S0CBS(S0, S0e

xT ,

√2x+ 2B

√2x

T+a(T )

T, T )

=

[N

( (B√

2xT +

a(T )2T )√T√

2x+ 2B√

2xT +

a(T )T

)−N(B)

]+N(B)− exTN

((−2x−B√

2xT −

a(T )2T )√T√

2x+ 2B√

2xT +

a(T )T

)

(D.29)

99

Applying Taylor expansion to the first term, we have

N

( (B√

2xT +

a(T )2T )√T√

2x+ 2B√

2xT +

a(T )T

)−N(B)

=1√2πT

exp

(− 1

2

( (B√

2xT +

a(T )2T )√T√

2x+ 2B√

2xT +

a(T )T

)2)( (B√

2xT +

a(T )2T )√T√

2x+ 2B√

2xT +

a(T )T

−B)√

T (1 +O(1

T))

=1√2πT

exp(−1

2B2)

a(T )/2−B2√

2x(1 +O(1/

√T )).

(D.30)

For the last term, using (D.14) gives

exTN

((−2x−B√

2xT −

a(T )2T )√T√

2x+ 2B√

2xT +

a(T )T

)

=1√2πT

exp

(− 1

2

( (B√

2xT +

a(T )2T )√T√

2x+ 2B√

2xT +

a(T )T

)2)√2x+ 2B√

2xT +

a(T )T

2x+B√

2x/T +a(T )2T

(1 +O(1

T)

=1√2πT

exp(−1

2B2)

1√2x

(1 +O(1/√T )).

(D.31)

Plugging (D.30) and (D.31) into (D.29), we have

1

S0CBS(S0, S0e

xT ,

√2x+ 2B

√2x

T+a(T )

T, T )

= N(B) +1√2πT

e−12B

2 1√2x

(a(T )/2−B2 − 1)(1 +O(1/√T )).

(D.32)

100

D.4 Appendix: Proof of Theorems 17 and 18– Second and Third

Order Approximation of Large-strike Large-expiry Implied

Volatility

Proof of Theorem 17. Using (D.25) and the definition of a(T ) and B, we have ∀ε > 0,

1

S0E(ST − S0e

xT )+

= 1− θ

S0+S0

S0

1√2πT

e−1

2 ( x2

σ2−x+σ2

4 )T(

σ

x− σ2/2− σ

x+ σ2/2

)(1 +O(

1

T))

≤ 1− θ

S0+S0

S0

1√2πT

e−1

2 ( x2

σ2−x+σ2

4 )T(

σ

x− σ2/2− σ

x+ σ2/2

)eε

= N(B) +1√2πT

exp(−1

2B2)

1√2x

(a(T )/2−B2 − 1)eε

(D.33)

From the definition of a(T ), we know that a(T ) > B2 + 1. ∀ε > 0, ∀δ > 0, there ∃ε(δ) > 0,

such that ∃T0, ∀T > T0

(a(T )/2−B2 − 1)eε < ((a(T ) + δ)/2−B2 − 1)e−ε(δ). (D.34)

Using (D.33), (D.34) and proposition 16, we have

1

S0E(ST − S0e

xT )+

≤ N(B) +1√2πT

exp(−1

2B2)

1√2x

(a(T )/2−B2 − 1)eε

≤ N(B) +1√2πT

exp(−1

2B2)

1√2x

((a(T ) + δ)/2−B2 − 1)e−ε(δ)

≤ 1

S0CBS(S0, S0e

xT ,

√2x+ 2B

√2x

T+a(T ) + δ

T, T ).

(D.35)

So

σ2imp(x, T ) ≤ 2x+ 2B

√2x

T+a(T ) + δ

T.

Similarly, we can show that

σ2imp(x, T ) ≥ 2x+ 2B

√2x

T+a(T )− δ

T.

101

This completes the proof.

Proof of Theorem 18. Similar to the proof of Theorem 17.

D.5 Appendix: Proof of Theorem 19–Large-strike Large-expiry

At-the-money Implied Volatility

Proof. Suppress the 0 in σimp(0, T ) to be σimp(T ). When at-the-money, for the displaced

lognormal, we have

CBS(S0, S0exT , σimp(T ), T ) = S0(2N(σimp

√T/2)− 1),

On the other hand, S − θ is lognormal process, so

E(ST −K)+ = E((ST − θ)− (K − θ))+ = (S0 − θ)(2N(σ√T/2)− 1).

So

S0(2N(σimp

√T/2)− 1) = (S0 − θ)(2N(σ

√T/2)− 1),

⇒ N(σimp(T )√T/2) =

S0 − θS0

N(σ√T/2) +

θ

S0;

⇒ σimp(T ) =2√TN−1

(S0 − θS0

N(σ√T/2) +

θ

2S0

).

Taking the limit of T , we have (7.14).

D.6 Appendix: Proof of Theorem 20–Approximation of Implied

Volatility when K = S0 exp(xT α)

Proof. Here we sketch the outline of the proof. The rigorous proof is similar to the proof of

Theorem 17. We prove the x > 0 case. Denote x = 1Tα log(

S0exTα−θS0−θ

), M = xTα−1, M =

xTα−1 and A =

(−M+σ2/2

σ

)2

. σ2∞ satisfies

(−M+σ2∞/2

σ∞

)2

=

(−M+σ2/2

σ

)2

. Choose the

smaller root σ2∞ = 2(M + A) −

√2MA+ A2 so that

σ2∞2 −M < 0. Abbreviate σimp(x, T )

102

to be σimp. Using the approximation of N(.), we have

1

S0E(ST − S0e

xTα)+

=S0

S0

1

S0E(ST − S0e

xTα)+

=S0

S0

(N

((−xTα−1 + σ2/2)

√T

σ

)− exT

αN

((−xTα−1 − σ2/2)

√T

σ

))=S0

S0

1√2πT

exp

(− 1

2

(−xTα−1 + σ2/2

σ

)2

T

)(σ

xTα−1 − σ2/2− σ

xTα−1 + σ2/2

)(1 + o(1/T ))

=S0

S0

1√2πT

exp

(− 1

2

(−M + σ2

∞/2σ∞

)2

T

)ABS(M, σ, 0)(1 + o(1/T ))

=1√2πT

exp

(− 1

2

(−M + σ2

∞/2σ∞

)2

T

)exp

(a

8(4M2

σ2∞)− 1

)ABS(M,σ∞, 0)(1 + o(1/T ))

=1√2πT

exp

(− 1

2

(−M + σ2imp/2

σimp

)2

T

)(σimp

xTα−1 − σ2imp/2

−σimp

xTα−1 + σ2imp/2

)(1 + o(1/T ))

→ 1

S0CBS(S0, S0e

xTα , σimp, T ).

(D.36)

The x < 0 case can be similarly proved.

D.7 Appendix: Proof of Theorem 21–Fixed-strike Large-expiry

Implied Volatility

D.7.1 Asymptotic behavior

Proof. Suppress the x in the σimp(x, T ) to be σimp(T ). On the one hand, we have

limT→∞

E(ST−K)+ = limT→∞

S0×N(

log(S0/K)

σimp(T )√T

+σimp(T )

√T

2

)−K×N

(log(S0/K)

σimp(T )√T−σimp(T )

√T

2

).

(D.37)

103

On the other hand,

limT→∞

E(ST −K)+ = limT→∞

E((ST − θ)− (K − θ))+

= limT→∞

(S0 − θ)N(log((S0 − θ)/(K − θ)) + σ2T/2

σ√T

)

− (K − θ)N(log((S0 − θ)/(K − θ))− σ2T/2

σ√T

)

= S0 − θ.

(D.38)

Comparing (D.37) and (D.38), we know that the limit of σimp(T )√T exists and satisfy

the following condition

limT→∞

S0×N(

log(S0/K)

σimp(T )√T

+σimp(T )

√T

2

)−K×N

(log(S0/K)

σimp(T )√T−σimp(T )

√T

2

)= S0−θ.

(D.39)

Recall the definition of σblsimpv(S0, K, P ) in (7.21) is

S0×N(

log(S0/K)

σblsimpv(S0, K, P )+σblsimpv(S0, K, P )

2

)−K×N

(log(S0/K)

σblsimpv(S0, K, P )−σblsimpv(S0, K, P )

2

)= P,

(D.40)

comparing (D.39) and (D.40), we have (7.20).

D.7.2 Monotonicity

Proof. Recall the σimp(T ) is defined as follows

CBS(S0, K, σimp(T ), T ) = CBS(S0 − θ,K − θ, σ, T ). (D.41)

Differentiating both sides with respect to T gives us

∂CBS(S0, K, σimp(T ), T )

∂T+∂CBS(S0, K, σimp(T ), T )

∂σimp(T )

∂σimp(T )

∂T=∂CBS(S0 − θ,K − θ, σ)

∂T.

(D.42)

104

Using∂CBS(S0,K,σimp(T ),T )

∂σimp(T )> 0, we have

sgn∂σimp(T )

∂T= sgn

(∂CBS(S0 − θ,K − θ, σ)

∂T−∂CBS(S0, K, σimp(T ))

∂T

)= sgn

((S0 − θ)N

′(

log((S0 − θ)/(K − θ))σ√T

+σ√T

2

)σ

− S0N′(

log(S0/K)

σimp(T )√T

+σimp(T )

√T

2

)σimp(T )

)= sgn

[(S0 − θ) exp

(− 1

2

(log((S0 − θ)/(K − θ))

σ√T

+σ√T

2

)2)σ

− S0 exp

(− 1

2

(log(S0/K)

σimp(T )√T

+σimp(T )

√T

2

)2)σimp(T )

].

(D.43)

From (D.43), we have

limT→∞

sgn∂σimp(T )

∂T= limT→∞

sgn

[(S0 − θ) exp

(− 1

2

(log((S0 − θ)/(K − θ))

σ√T

+σ√T

2

)2)σ√T

− S0 exp

(− 1

2

(log(S0/K)

σimp(T )√T

+σimp(T )

√T

2

)2)σimp(T )

√T

]

= limT→∞

sgn

[(K − θS0 − θ

)1/2

exp

(− 1

2

σ2T

4

)σ√T

−(K

S0

)1/2

exp

(− 1

2

σ2impT

4

)σimp(T )

√T

].

(D.44)

Note that exp

(− 1

2σ2T

4

)σ√T as T →∞ is dominated by the exponential term, so the first

term goes to 0. From Theorem 21.1, limT→∞ σimp(T )√T exists, so the second term goes to

some positive constant. Therefore,

limT→∞

sgn∂σimp(T )

∂T< 0.

105

D.8 Appendix: Proof of Theorem 22–Approximation of

Fixed-Strike Large-expiry Implied Volatility.

Proof. Denote a =σ2blsimpv(S0,K,S0−θ)

2 . From Theorem 21, we know that the first term of

σimp(x, T ) is σblsimpv(S0, K, S0 − θ)/√T = 2a/T . Here we want a more refined approxima-

tion of the residual term. So denote σ2imp(x, T ) = 2a

T + 2r. Denote S0 = S0 − θ and use the

Taylor approximation of N(.) to the second order, we have

1

S0E(S0 −K)+

=S0

S0+

(N(−x+ a+ rT√

2a+ 2rT)−N(

−x+ a√2a

)

)+ ex

(N(−x− a− rT√

2a+ 2rT)−N(

−x− a√2a

)

)=S0

S0+

1√2π

exp(−1

2(−x+ a√

2a)2)

[(√2a+ 2rT −

√2a

)+

(√2a+ 2rT −

√2a

)2( x2

4a(a+ rT )

(√a+ rT

a− 1

2

)− 1

8

)]+ o((rT )2)

(D.45)

Let√

2a+ 2rT −√

2a = z, then

1

S0E(S0 −K)+

=S0

S0+

1√2π

exp(−1

2(−x+ a√

2a)2)

[z + z2

(x2

2a(z +√

2a)2

(z√2a

+1

2

)− 1

8

)]+ o((rT )2)

≈ S0

S0+

1√2π

exp(−1

2(−x+ a√

2a)2)

[(−1

8+

x2

8a2)z2 + z

](D.46)

On the other hand, we have

1

S0E(S0 −K)+ =

1

S0CBS(S0 − θ,K − θ, σ, T ) (D.47)

Compare (D.46) with (D.47) and solve for z.

106

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