Forward Implied Volatility

Farshad Behvand

Christ Church

Oxford University

January 4, 2010

A thesis submitted in partial fulfillment of the requirements for the MSc in

Mathematical Finance

Abstract

We look at the pricing of structures that depend on forward volatility, such as globally

floored cliquets. We start off by analysing market standard models, which prescribe

the dynamics of volatility as either deterministic or instantaneous. We then move

onto to analysing the Discrete Stochastic Implied Volatility (DSIV) model. This

model specifies the dynamics of forward implied volatility directly at discrete times

and as a result has more realism of and control over the dynamics. For each of these

models we explore in detail the implied forward volatility dynamics and how to price

exotic cliquet options. We then compare these prices against Totem marks, which

represent actual market price information.

Acknowledgements

I would like to thank Jochen Theis for all his recommendations, inputs, support

and patience. He has really helped me in developing a greater understanding of this

topic. I would also like to thank my lecturers at Oxford university and colleagues at

Markit, in particular Lukasz Dobrek for their support and input on various issues and

topics in mathematical finance. Finally, my family and fiancee for standing by me.

1

Contents

1 Introduction 4

2 Cliquet Options Market 6

3 Local Volatility 8

3.1 Model Outline and Dynamics . . . . . . . . . . . . . . . . . . . . . . 8

3.2 Model Setup and Pricing . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Stochastic Volatility 15

4.1 Heston Stochastic Volatility Model . . . . . . . . . . . . . . . . . . . 15

4.2 Time Dependent Heston Stochastic Volatility Model . . . . . . . . . 21

4.2.1 Characteristic Function for the Time Dependent Heston Model 21

4.2.2 European & Forward Starting Option Prices . . . . . . . . . . 23

4.2.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2.4 Monte Carlo Numerical Implementation . . . . . . . . . . . . 30

4.3 Exotic Cliquet Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Discrete Stochastic Implied Volatility Model 35

5.1 Model Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2 Model Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.3 Arbitrage Freeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.3.1 Volatility Conditions . . . . . . . . . . . . . . . . . . . . . . . 40

5.3.2 Existence of Risk-Neutral Measure . . . . . . . . . . . . . . . 42

2

5.3.3 Model Completeness . . . . . . . . . . . . . . . . . . . . . . . 44

5.4 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.5 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.6 Exotic Cliquet Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6 Summary 51

A Appendix 54

A.1 SABR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

A.2 Implied Forward and Cliquet Volatility . . . . . . . . . . . . . . . . . 54

A.3 Totem Consensus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

A.4 Exotic Cliquet Totem Results for Heston Model . . . . . . . . . . . . 57

A.5 Monte Carlo Results for Time Dependent Heston Model . . . . . . . 60

A.6 Matlab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3

1 Introduction

In this thesis we address the issue of volatility modelling for derivatives that depend

on “forward volatility”, such as reverse cliquets, Napoleons, accumulators and struc-

tured notes with embedded forward starting features. Traditionally, smile models

have been assessed according to how well they fit the vanilla/spot implied volatility

surface (e.g. the Local Volatility model). However, the pricing of exotic structures

which have forward starting features embedded within them are more dependent on

the assumptions made on the forward implied volatility dynamics than on the price

of today’s vanilla options prices. A good volatility model not only has to be able to

calibrate to the forward starting options but also to the spot implied volatilities. For

example, on the reset date t1 a forward starting option turns into a European option.

Since such options are liquidly traded, the model must be able to internally produce

future implied volatility surfaces that are consistent with historic behaviour.

Many researchers and practitioners have tried to address this issue with stochastic

volatility models. In the literature it is frequently stated that models of this type are

well suited to price forward starting contracts, such as cliquets. However, this is not

easily verified in the markets: figure 1, shows the cliquet implied volatility error (Ap-

pendix A.2 has further details on calculation and table descriptions) when calibrating

the Heston Model (4.1) to the vanilla options market and then pricing quarterly and

annually resetting vanilla resetting cliquets of different maturities (see equation (1)

for payoff definition). Clearly, the model does not match the market prices of the

cliquets. Furthermore, this model with this calibration is clearly not suitable to price

exotic cliquet options as it does not fit the forward implied volatility skew observed

in the market.

We investigate this issue by looking at the dynamics of several different volatility

models in this thesis. We assess the quality of the models on the following criteria

[8]:

• realism of and control over volatility dynamics

• ease of calibration to vanilla and forward starting options

• continuity in time - ability to price all structures for all times

4

• speed and suitability for practical use

In practice, to the author’s understanding no volatility model has all of these prop-

erties. Generally, something needs to give.

This thesis is organised as follows. In sections 3 and 4 we look at three continu-

ous time models: the Local Volatility Model [10], the Heston Model [19] and the

Time dependent Heston Model [11]. In section 5 we construct the Discrete Stochastic

Implied Volatility Model (DSIV Model) [1], a risk-neutral model for the stock price

such that all available information on current and future smiles can be incorporated.

For each of these models we:

• describe the forward volatility dynamics implied by the model

• discuss in detail how to calibrate and price with the model

• compare the model prices of exotic cliquets against Totem market consensus

Finally, we end with a summary of our findings and outline possible future research.

(a) Annual resetting cliquet

(b) Quarterly resetting cliquet

Figure 1: S&P500 Implied cliquet volatility error

( σSV − σMkt)

5

2 Cliquet Options Market

Cliquets are a form of “volatility product” within the equity markets. They are con-

tracts whose payoff depends in some way on the performance of an asset over future

time periods. Cliquets are often referred to as a forward volatility product and for-

ward implied volatility surfaces can be backed out of vanilla cliquet prices (Appendix

A.2). Here are the payoffs of the cliquet options we are going to price in this thesis

[24], [2]:

• Vanilla call cliquet:

n∑i=1

(Sti+1

Sti− α

)+

(1)

• Napoleon:

M∑j=1

{Cj + min

∀ti∈[tj−1,tj ]

(Sti+1

Sti− 1, 0

)}+

(2)

• Reverse cliquets: {C +

n∑i=1

min

(Sti+1

Sti− 1, 0

)}+

(3)

• Globally floored cliquet:{n∑i=1

(r− ∨

(Sti+1

Sti− 1

)∧ r+

)}+

(4)

for C > 0 and α > 0

where S is the stock price, α a constant strike (moneyness), C and Cj are constant

coupons or fixed returns, r+ and r− are a cap and floor respectively, t0, ..., tn are the

reset dates and the final payment is made at tn (it is also possible for the structure

to pay at every reset date). The pricing and risk management of such products is not

straightforward and the market has not yet settled on an agreed reference model.

6

Resetting cliquets on equity indices are liquidly traded as products in their own right;

they are not traded on model marks. It is not common to have cliquets on single stock

equity, however they do exist and pose further problems on modelling the discrete

dividends1 within the exotic models. In this thesis we work with major equity indices

and as a result have market data on both forward starting and spot implied volatil-

ity surfaces. Figure 2 shows the complete range of parameters for which we have

calibration data available for our examples, which are typical of the market. Here

SpotVol refers to the vanilla option data and 12MFwdVol, 6MFwdVol, etc, refers to

the cliquet option data with resetting frequency of twelve months, six months, etc.

Note that moneyness is KSti

and for vanilla options this is based on the spot level.

Figure 2: Instruments for which we have calibration data available

To hedge exotic cliquets we can use a combination of forward starting options, reset-

ting cliquets, vanilla options and the underlying option. Also, to hedge our forward

starting options we use a combination of vanilla options and the underlying. There-

fore, it is vital that our model is able to reproduce the prices of the calibration

instruments to prevent slippage.

1There is no market standard model to factor in discrete dividends into exotic option pricing;

one way is to model the stock process as the pure stock process (Hans Buehler [6])

7

3 Local Volatility

The first model we review is the Local Volatility Model. It was one of the first volatility

models used in practice to price resetting cliquets. Most financial institutions moved

away from using this model with cliquets once they understood the dynamics of

forward volatilities it produces. We start by outlining the model dynamics and the

associated problems with Local Volatility Models. We then show to price with such

models and use the model to price vanilla cliquets and compare these prices against

the market as represented by the Totem consensus.

3.1 Model Outline and Dynamics

The local volatility model describes the stochastic evolution of the underlying stock

price St by means of a volatility term that is a deterministic function of the stock price:

dSt = (rd(t)− q(t))Stdt+ σ(St, t)Stdz (5)

where rd is the instantaneous domestic interest rate and q the continuous dividend

yield. The local volatility function σ(S, t) depends locally on both the time and the

value of the underlying asset. The distribution of the asset price at some time t is

thus no longer log-normal; rather, the goal is to determine the underlying process

from the market-observed smile by determining the local volatility function. Dupire

[10] showed that a unique local volatility surface can be implied by forward induction

from vanilla option prices2. The implied volatility of an option with strike K expiring

at T, σ(K,T ), can be seen as the average over all local volatilities that the underlying

may have as time evolves until the maturity date.

Local volatility models are easy to implement and theoretically self consistent, as

it is possible to perfectly replicate all vanilla option prices and to hedge every prod-

uct with vanilla options. As a result of this, these models are widely used in practice

to price path dependent options, such as barrier and Asian options. This comes at

the cost of having unrealistic spot volatility dynamics; Hagan [26] deduced that the

the dynamic behaviour of spot volatility smile and skews predicted by local volatility

2This is a special application of Gyongy’s theorem [21]

8

models is exactly opposite to the behaviour observed in the market place: when the

price decreases, local vol models predict that the smile shifts to higher prices; when

the price increases, these models predict that the smile shifts to lower prices. In re-

ality, asset prices and market smiles move in the same direction. This contradiction

leads to incorrect hedges and hence incorrect pricing for exotic options, which can be

arbitraged.

Local volatility models are deterministic models and produce sharply flattened fu-

ture volatility smiles and skews as seen in figure 3, which shows the three month

forward implied volatility skew at different reset dates, where 0M refers to the vanilla

three-month option and 60M refers to an option with three months’ expiry starting

in five years’ time. We can see as we go out further in time that the vol skew and

curvature decays. This behaviour is not observed in the market.

A more general criticism of Dupire’s Local Volatility Model was raised by E.Ayache

[12]; he points out that the model gives no explanation for the dynamics of how

volatilities evolve, but is a way of tweaking the diffusion coefficient in the Black-

Scholes model to match a given set of vanilla option prices. Furthermore, he argues

that it is more realistic and informative to explain the evolution of volatility through

jumps or stochastic volatility, rather than having a model which changes volatility

locally at a given point in time to match today’s option prices.

3.2 Model Setup and Pricing

No analytic solution for either vanilla or exotic option exists for the Local Volatility

Model; numerical schemes such as finite difference methods or Monte Carlo must be

adopted. Therefore, in this section we give a brief outline of how to price with this

model in practice.

Fitting to vanilla volatility surfaces

The implementation of Dupire’s Local Volatility Model can be numerically delicate

because of the density-related term,∂CK,T (S,t0)

∂K2 , in the denominator. For values of

the stock price very far away from the money this quantity becomes very small, and

9

(a) Local Volatility

(b) Heston Model

Figure 3: S&P500 3M implied forward volatility skew

10

the local volatility therefore remains finite and well-behaved only if the numerator

approaches zero at the ‘right’ speed. This can lead to serious numerical problems.

Fortunately, it is possible to recast Dupire’s Local Volatility Model in terms of implied

volatilities and their derivatives [30]:

σ(K,T )2 =σ2K,T + 2τ σK,T

∂σK,T

∂T+ 2(rd − q)KτσK,T ∂σK,T

∂K(1 +Kd1τ

∂σK,T

∂K

)2

+K2τ σK,T

(∂2σK,T

∂K2 − d1

(∂σK,T

∂K

)2√τ

) (6)

with

d1 =log(SK

)+ (rd − q + 1

2σ2K,T )τ

σK,T√τ

where τ is the time to maturity (T − t0). In practice option prices are given at a

discrete set of points and in order to use (6) for calibrating the model to market

data, we need to use interpolation and extrapolation in both time and strike. The

local volatility model is arbitrage free given arbitrage free vanilla option prices for

every expiry and maturity, so improper interpolation and extrapolation techniques

may create arbitrage opportunities within the model. To avoid this we interpolate

and extrapolate implied volatilities between strikes using the SABR model (Appendix

A.1) and interpolate and extrapolate implied volatilities between maturities using cu-

bic spline interpolation on the SABR parameters.

Cliquet pricing

We use Monte Carlo to price all cliquet structures with the Local Volatility Model.

To price structures with Monte Carlo we make use of the fundamental theorem of risk

neutral pricing [13]; which states that under the risk-neutral measure Q the derivative

price process satisfies:

V (t) = EQ[e−

∫ Tt rd(s)dsV (F ) | Ft

](7)

or, in other words, that the discounted derivative price process is a martingale under

Q.

11

We calculate the expectation on the RHS of (7) by Monte Carlo. To do this, we

simulate a discretisation of the local volatility process in time into piecewise Gaussian

increments. As a derivative contract terminates with well-determined cash payments

depending only on the stock price at a number of observation times, we can use (7)

directly to calculate the value of the derivative.

The Monte Carlo algorithm we use to calculate the examples in this thesis works

as follows:

• To avoid negative stock prices, we apply Ito’s lemma to log(S) and apply Euler

discretisation (we only require weak convergence):

St+∆t = St exp

{(rd(t)− q(t)−

1

2σ(St, t)

2

)∆t+ σ(St, t)

√∆tεi

}εi

i.i.d∼ N(0, 1)

• For our example calculations we simulate daily on business days and perform

20,000 simulations.

• To improve performance we pre-compute the local volatility function for each

time step as a piecewise linear interpolator function on the state domain from

1% to 400% of the forward (flat extrapolation).

• We generate a simulated time series of stock prices, which is then passed into a

generic payoff to get our final option price.

To test our implementation, we price a range of vanilla options and compare the results

against the analytic prices. This will ensure that we do not have any implementation

or discretisation errors.

Totem Testing

Please refer to A.3 for details on the Totem consensus, definition of % of Totem

spread and table descriptions. Figure 5 shows the results of comparing the model

prices against the market prices for resetting cliquets on the S&P500 (Appendix

A.2). We can clearly see that the model significantly underprices vanilla cliquet op-

tions (figure 5a); we also see that only 16 out of 100 options pass totem consensus (red

12

indicates fail in figure 5b).3 Conversely, the model overprices in-the-money cliquet

options, as the skew on market quoted cliquets is negative whereas the model implied

skew is almost flat (see also figure 3a).

This model clearly fails to match the market and should not be used in practice

to price any structure that depend on forward volatility. We see this again clearly

in figure 4 where we compare the model prices for globally floored cliquet options

against the Totem consensus, where red shade indicates a fail. The model pass rate

is approximately 8% compared to 85% of the Time Dependent Heston model (“TD

model” in figure 4).

Figure 4: S&P500 Quarterly resetting globally floored cliquet Totem test results

3To pass totem consensus the volatility difference must fall between half the total spread observed

in the market.

13

(a) Implied cliquet volatility difference (σLV − σMkt)

(b) Difference as a percentage of Totem spread

Figure 5: S&P500 resetting cliquet Totem test results

14

4 Stochastic Volatility

Stochastic volatility models are a class of models that use an additional stochastic pro-

cess to drive instantaneous variance. They naturally exhibit implied volatility skew

and smiles. Moreover, stochastic volatility models assume more realistic dynamics

than local volatility models for the underlying; they allow for fat tails, high central

peak and volatility clustering, which are all know empirical properties of financial

data.

One of the most commonly used stochastic volatility models is the Heston Model

[19]. Its popularity is mainly due to the existence of a semi-analytic solution for

European option prices, making the calibration fast and practical. We look at two

versions of the Heston Model in this section, one with flat parameters and another

with time-dependent parameters.

4.1 Heston Stochastic Volatility Model

Heston [19] proposed a five parameter stochastic volatility model of the form:

dSt = (rd − q)Stdt+√VtStdW

1t

dVt = κ(θ − Vt)dt+ σ√VtdW

2t (8)

where 〈dW 1t , dW

2t 〉 = ρdt. In this model, κ is the speed of mean reversion,

√θ the

long term volatility, σ is the volatility of variance, ρ correlation and√V0 the short

volatility. We can interpret the parameters σ and ρ as being responsible for the skew,

the volatility of variance controlling the curvature and the correlation the tilt. The

other three parameters control the term structure of the model; where the mean re-

version controls the skewness of the curve from the short vol level to the long vol level.

Note that the effects of the parameters on the skew are strongly interdependent and

it is very difficult to put constraints and structure on these parameters such that the

dynamics of implied and forward implied volatility produced can be controlled.

To analyse the forward volatility dynamics we follow the work of Bergomi [4]. In this

article Bergomi analysed the structural features and dynamics of traditional volatility

15

models. He deduced the following results for the at-the-money forward volatility and

skew via perturbation of the pricing equation:

• T � τ , at order zero in T:

σF =√V0,

dσ

d lnK

∣∣∣∣F

=ρσ

4√V0

(9)

• T � τ , at order 1 in 1T

:

σF =√θ(

1 +ρσ

4κ

)+

√θ

2κT

(V0 − θθ

+ρσ

4κ

V0 − 3θ

θ

),

dσ

d lnK

∣∣∣∣F

=ρσ

2κ√θ

(10)

where τ is the cutoff between the short and long maturities, and is given by 1κ.

The first observation we make is that the short dated skew is inversely proportional

to the at-the-money volatility (9), this is observed in figure 6a. We see in figure 6a

that the observed convexity with respect to the spot implied volatility is larger for

in-the-money strikes than out-of-the money strikes, also we notice the at-the-money

level is lower, too. Furthermore, these effects get more pronounced as the forward

start date increases. We do expect forward implied volatility smiles to be more convex

than today’s4, however, these effects are overemphasized in the Heston model rela-

tive to actual market data. These low at-the-money and high skew scenarios effect

significantly the pricing of short term resetting cliquets, as shown in figures 6b and

6c. These figures clearly show that the forward volatilities produced are significantly

lower than the market except for high strikes as expected.

For short term volatilities, the volatility of variance has two different objectives [4]:

• Static - create skewness in the distribution of lnS so as to match market smiles

• Dynamic - drive the dynamics of implied volatilities in a way which is consistent

with their historical behaviour.

Bergomi observes that the calibrated vol of vol parameter in his examples is a factor

of two larger than the historical value, and calibration of the model can not fulfill

4The price of a call option is an increasing and convex function of implied volatility, therefore

uncertainty in the value of the future price increases the option value

16

(a) Forward 1-month volatility skew

(b) 1 Year, 1-month resetting cliquet volatility smile

(c) 1-month resetting cliquet volatility error

Figure 6: Heston short term dynamics for S&P500

17

both objectives. This may be a reason why the model dynamics appear not to match

the implied forward volatility dynamics as illustrated in figure 7. Figure 7 shows the

one year, three month implied forward skew when calibrating the Heston model to

the vanilla options and the quarterly resetting cliquet options. This figure illustrates

that the volatility of variance is lower when calibrating to the cliquet options; there is

less convexity and the at-the-money volatility is close to today’s implied at-the-money

volatility level.

(a) One year, three month implied volatility skew

(b) Calibrated Heston parameters

Figure 7: S&P500 3-Month implied volatility skew

For long-dated options Bergomi states that all stochastic volatility models with mean

reversion have increments in ln(S) that are stationary and independent at long times.

From (10), we expect the the skewness to decay proportionally to 1T

. We can see

this behaviour in figure 9a; the forward implied volatility skew looks very similar to

today’s smile, there is no excessive convexity and lower at-the-money volatilities. Fur-

thermore, 9b and 9c, show that the implied cliquet smile matches the market better

18

and we are much closer to the Totem consensus5.

Another drawback of the Heston model is that poor fits are observed when calibrating

to the vanilla options; with shallow smiles for short and extremely long maturities.

This is illustrated in figure 8 (Appendix A.2): the table and graph show the implied

volatility calibration error for the Euro Stoxx 50. The results show significant im-

plied volatility differences for short dated options maturing up to six months; with

the model implying much higher at-the-money volatilities and lower skews. This

behaviour can be avoided by putting more weight on the at-the-money volatilities

during calibration, however this will make the fitting of the wings much worse, which

are already several volatility points under the market. This can lead to arbitrage

opportunities as the model can not reprice the hedging instruments, and thus the

hedging portfolio correctly. One way to resolve this problem is to incorporate jumps

or to make the parameters time dependent. We pursue the time dependent approach

in the next section.

Figure 8: Euro Stoxx 50 Heston calibration error

5The results are obtained for the Heston model are at first order in σ and are relevant for equity

smiles. If ρ is small, as is the case for currency smiles, the contribution from terms of order σ2

dominates, altering the conclusions

19

(a) Forward 12-month volatility

(b) 5 Year, 12-month resetting cliquet volatility smile

(c) 12-month resetting cliquet volatility error

Figure 9: Heston long term dynamics for S&P500

20

4.2 Time Dependent Heston Stochastic Volatility Model

We saw in the previous section that the Heston model does not give conclusive fits

to the vanilla option market and stated that this could be resolved by making the

parameters time dependent. Piterbarg [27] points out that “the need to relax the as-

sumption of time-stationarity is real and serious, as models with constant coefficients

are generally incapable of fitting market prices across option expiries. Calibration to

options of multiple expiries is required for consistency if exotics are to be priced. This

is so because exotics do not only depend on the distribution of the underlying at a

particular point in time, but on its dynamics through time”. Therefore, this motivates

us to look at the Time Dependent Heston Model:

dSt = (rd(t)− q(t))Stdt+√vtStdW

1t

dvt = κ(t)(θ(t)− vt)dt+ σ(t)√vtdW

2t (11)

where 〈dW 1t , dW

2t 〉 = ρ(t)dt. We follow closely the work of Elices [11] on how to

calibrate and price within this model assuming piecewise constant parameter func-

tions κ, θ, σ and ρ. In his paper Elices outlines a general methodology to derive the

characteristic function for the distribution of (St, vt) at a time horizon where the pa-

rameters of the underlying process can change and is applied directly to the Heston

model to derive analytic solutions for vanilla and forward starting options. We start

by outlining this method and apply his work to price resetting cliquets and general

forward starting options using Monte Carlo. We then extend the calibration to fit

jointly the vanilla and resetting cliquet options, review the fits of both the Heston

and Time Dependent Heston Models and finally price exotic cliquet options.

4.2.1 Characteristic Function for the Time Dependent Heston Model

Let xt is the logarithm of the stock price and vt, by [11] the time dependent charac-

teristic function is given by

ϕtntn+1(X, V | xtn , vtn) = exp{Ctntn+1(X, V ) +Dtntn+1(X, V )vtn + iXxtn

}(12)

where

21

C(X, V ) = i(rd(t)− q(t))Xτ +κθ

σ2

(ln

(1− ge−dt

1− g

)2

+ (κ− ρσXi− d)τ

)+ C0

D(X, V ) =κ− ρσXi+ d

σ2

(g − ge−dt

1− ge−dt

)g(X, V ) =

κ− ρσXi− d− V σ2

κ− ρσXi+ d− V σ2

g(X, V ) =κ− ρσXi− dκ− ρσXi+ d

d(X, V ) =

√(κ− ρσXi)2 + σ2X(i+X) (13)

with τ = ti+1 − ti. This representation is slightly different from Heston’s original

formulation [19]. Firstly, d appears with the sign change; Albrecher et al [18] prove

that these representations are identical and this representation avoids all of the nu-

merical problems associated with branches of the analytic functions appearing in (13).

The characteristic function ϕtntn+1(X, V | xtn , vtn) is given by the Fourier transform

of the transition density function ftntn+1 ,

ϕtntn+1(X, V | xtn , vtn) = E[ei(X,V )·(xtn+1 ,vtn+1 ) | xtn , vtn

]=

∫R2

ei(X,V )·(xtn+1 ,vtn+1 )ftntn+1(xtn+1 , vtn+1 | xtn , vtn)dxtn+1dvtn+1

The transition density functions can be reconstructed by inverse Fourier transform:

ftntn+1(xtn+1 , vtn+1 | xtn , vtn) =1

2π

∫ ∞−∞

∫ ∞−∞

e−i(xtn+1 ,vtn+1 )·(X,V )ϕtntn+1(X, V | xtn , vtn)dXdV

As the transition density functions combine as

ft0tn+1(xtn+1 , vtn+1 | xt0 , vt0) =

∫R2

ft0tn(xtn , vtn | xt0 , vt0)ftntn+1(xtn+1 , vtn+1 | xtn , vtn)dxtndvtn

we can calculate ϕt0tn+1 from ϕt0tn and ϕtntn+1 using (12):

22

ϕt0tn+1(X, V | xt0 , vt0)

=

∫R2

ei(X,V )·(xtn+1 ,vtn+1 )ft0tn+1(xtn+1, vtn+1 | xt0 , vt0)dxtn+1dvtn+1

=

∫R2

ft0tn(xtn , vtn | xt0 , vt0)∫R2

ei(X,V )·(xtn+1 ,vtn+1 )ftntn+1(xtn+1, vtn+1 | xtn , vtn)dxtn+1dvtn+1dxtndvtn

=

∫R2

ft0tn(xtn , vtn | xt0 , vt0)ϕtntn+1(X, V | xtn , vtn)dxtndvtn

=

∫R2

ft0tn(xtn , vtn | xt0 , vt0) exp{Ctntn+1(X, V ) +Dtntn+1(X, V )vtn + iXxtn

}dxtndvtn

= Ctntn+1(X, V )

∫R2

ei(X,i−1Dtntn+1 (X,V ))·(xtn ,vtn )ft0tn(xtn , vtn | xt0 , vt0)dxtndvtn

= Ctntn+1(X, V )ϕt0tn(X, i−1Dtntn+1(X, V ) | xt0 , vt0

)= exp

{Ctntn+1(X, V ) + Ct0tn(X, i−1Dtntn+1(X, V )) +Dt0tn(X, i−1Dtntn+1(X, V ))vt0 + iXxt0

}In summary the joint characteristic function is given by (12), where

Ct0tn+1(X, V ) = Ctntn+1(X, V ) + Ct0tn(X, i−1Dtntn+1(X, V ))

Dt0tn+1(X, V ) = Dt0tn(X, i−1Dtntn+1(X, V )) (14)

4.2.2 European & Forward Starting Option Prices

In this section we follow the work of Elices [11] to derive semi-analytic prices for

European and forward starting options.

Semi-Analytic European Option Price:

The price of a European option can be broken down into two expectations,

C(K,T ; t0) = P (t0, T )E[(ST −K)+ | Ft0

]= P (t0, T )

{E[exT1{xT>lnK} | Ft0

]−KE

[1{xT>lnK} | Ft0

]}where P (t0, T ) is the discount factor, K is the strike and T is the maturity. To de-

termine these expectations the marginal distribution of xT given xt0 and vt0 must

23

be determined. No analytic expression exists for this distribution, however Heston

[19] derived an analytic expression for the characteristic function of this distribution.

Elices [11] generalised this approach such that the methodology can be applied to the

joint distribution of xT and vT given xt0 and vt0 , from which the marginal character-

istic function xT can be determined by evaluating the joint characteristic function at

V = 0

ϕt0T (X | xt0 , vt0) = E[eiXxT | xt0 , vt0 ]

= E[ei(X,0)(xT ,vT ) | xt0 , vt0 ]

= ϕt0,T (X, 0 | xt0 , vt0)

where the characteristic function is given by (14). The Fourier inversion for the this

characteristic function is well known and given by,

P(xT > lnK) =1

2+

1

π

∫ ∞0

Re

(ϕ(X)e−iX ln(K)

iX

)dX (15)

We can use this result to calculate the second expectation, E[1{xT>lnK} | Ft0

]. To

calculate the first expectation, E[exT1{xT>lnK} | Ft0

], we define a new probability

measure:

PT =exT

E[exT ]PT

then

E[exT1{xT>lnK} | xt0vt0

]= E[exT ]E

[1{xT>lnK} | xt0 , vt0

]= E[exT ]P (xT > lnK)

= FT P (xT > lnK)

where P (xT > lnK) is given by (15) and the characteristic function of PT defined by

ϕt0T (X | xt0 , vt0) =E[eiXxT +xT | xt0 , vt0

]E [exT | xt0 , vt0 ]

24

Now, we have a way of calculating both expectations the final call option price is

given by

C(K,T ; t0) = P (t0, T ){FT P (xT > lnK)−KP (xT > lnK)

}(16)

Semi-Analytic Forward Starting Option Price:

Let Cfwd(α, tn+1; tn) be the price of a forward starting call option, with strike α

and reset date tn expiring on tn+1. This price can be given by:

Cfwd(α, tn+1; tn) = P (t0, tn+1)E[(extn+1 − αextn )+|Ft0

]= P (t0, tn+1)E[extn |Ft0 ]E

[(extn+1−xtn − α)+|Ft0

]= P (t0, tn+1)E[extn |Ft0 ]

[E[ext ]P(xt > α)− αP(xt > α)

]= P (t0, tn+1)Ftn

[Ftn+1

FtnP(xt > α)− αP(xt > α)

](17)

where, xt = xtn+1−xtn when t > tn else equal to zero and Ft is the forward price at t.

To determine the above probabilities, the distribution of (xt, vt) given (xt0 , vt0) must

be defined. We calculate its characteristic function,

ϕ(X, V | 0, vt0) =

∫R2

ei(X,V )·(xt,vt)f(xt, vt | 0, vt0)dxtdvt

=

∫R+

ft0tn(vtn | xt0 , vt0)∫R2

ei(X,V )·(xt,vt)ftntn+1(xt, vt | 0, vtn)dxtdvtdvtn

=

∫R+

ft0tn(vtn | xt0 , vt0)ϕt0,tn(X, V | 0, vtn)dvtn

= eCtn,tn+1 (X,V )

∫R+

ft0tn(vtn | xt0 , vt0)ei(−iDtn,tn+1 (X,V ))vtndvtn

= eCtn,tn+1ϕt0,tn(0,−iDtn,tn+1(X, V )))

= exp{C(X, V ) +D(X, V )vt0

}(18)

where C(X, V ) and D(X, V ) are given by

25

C(X, V ) = Ctn,tn+1(X, V ) + Ct0,tn(0,−iDtn,tn+1(X, V )) (19)

D(X, V ) = Dt0,tn(0,−iDtn,tn+1(X, V ))

Now that we have the modified characteristic function, we can use the same pricing

procedure and results from above to price the forward starting option. Note, we can

also use this result to price vanilla options, by setting the first reset date to t0. This

is a very useful and simple test to check that our pricing algorithm is correct. Fur-

thermore, we can price vanilla resetting cliquets as a sum of forward starting options:

CRC(α, tn+1; t0) =N∑i=1

Cfwd(α, ti−1, ti)P (ti, t0) (20)

4.2.3 Calibration

In the above sections we outlined semi-analytic solutions for both vanilla and forward

starting options; this makes the calibration process simple, efficient and practical.

We can either calibrate to vanilla options, the forward starting options or a mixture

of both. The model does not have sufficient degrees of freedom in order for us to

perform a global calibration to all available market data. Therefore, we calibrate

locally to the market data which is relevant to our hedging strategy. E.g. to price

a semi-annual resetting reverse cliquet option, we could calibrate to the semi-annual

resetting cliquets or a combination of cliquets and vanilla options, if vanilla options

are used as part of the hedging strategy.

Calibration routine

The parameters in the time-dependent Heston model are not changing continuously

but piecewise constant between different parameter sets (calibration node). To cal-

ibrate this model, we use a bootstrapping algorithm, where we fit each calibration

node independently to all the selected vanilla and forward starting options in each

period, starting from the first and iterating to the last node, solving a minimisation

problem on the objective function:

26

min

{N∑i=1

wi(Cmodelfwd (i)−Cmarket

fwd (i))2

+M∑i=1

wi(Cmodelspot (i)−Cmarket

spot (i))2

}

The weights wi > 0 can be chosen freely, e.g. to give the highest priority to the at-the-

money options. We keep this function constant at one in our example calculations.

When the parameters of the first period has been calibrated, they are fixed and the

parameters of the next calibration node are calibrated; this process continues to the

last calibration node. Note that we fix the initial volatility at the first calibration node.

In our example computations we calibrate our model to three or four calibration

nodes; the first is used to fit very short maturity options (1M - 6M), then next one

or two to calibrate short to long dated options (12M to 60M) and the final one to

calibrate very long dated options (84M-120M). We could have a calibration node for

each period in our calibration instrument set, but then we would lose intuition of what

our parameters imply on the dynamics of the model. To calibrate a flat parameter

Heston model we simply set our calibration node to be the last maturity of all our

selected calibration instruments.

Calibration Results

We define a good calibration fit as half a vol point difference or less; see Appendix A.2

for calculation details and table descriptions. Figures 10, 11 and 12 show the results

of calibrating the flat and piecewise constant parameter Heston models to vanilla op-

tions, semi-annual resetting cliquets and a mixture of semi-annual resetting cliquets

and at-the-money vanilla options on the FTSE 100 index. We can see, as expected,

that the time dependent Heston model fits all the instruments significantly better,

both for short and long term maturities.

Figure 10 shows the absolute calibration error, model implied volatility minus market

implied volatility, to the vanilla options. The flat parameter model does not produce

an adequate fit that could be used in practice; the differences are very significant

and can lead to substantial slippage (hedging errors that arise from mispricing of the

hedging instruments). However, we can see that the piecewise model with three sets

of parameters produces a very good fit to the market which would be acceptable in

27

practice.

(a) Flat parameters

(b) Time dependent parameters

Figure 10: FTSE 100: Vanilla option calibration error

Figure 11 shows the absolute calibration error, model implied cliquet volatility minus

market implied cliquet volatility, when fitting to the six monthly resetting cliquets.

We again observe that the flat parameter model does not fit the market where the

time-dependent model shows promising results. The strong fits to the vanilla cliquet

market allow us to incorporate forward skew into our pricing and risk management

of exotic cliquet structures.

Another desirable feature we wanted was the ability to calibrate to both vanilla and

forward starting options. This is not achievable as shown in figure 12. Just adding

the ATM spot volatilities has significant impact on the dynamics of the model and

ultimately how well the model fits the market.

28

(a) Flat parameters

(b) Time dependent parameters

Figure 11: FTSE 100: 6M resiting cliquet calibration error

(a) Flat parameters

(b) Time dependent parameters

Figure 12: FTSE 100: 6M RC & ATM vanilla option calibration error

29

4.2.4 Monte Carlo Numerical Implementation

To price exotic cliquet structures with the Heston Models we use Monte Carlo simula-

tion. Simulating the Heston Model using a simple discretisation scheme, such as the

Euler scheme, can result in negative variance paths if the Feller condition 2κθ > σ2 is

not satisfied; this leads to bias and ultimately mispricing. Various variance discreti-

sation schemes have been proposed, Roger et al [28] analyse the bias associated for a

range of these schemes. We follow the work of Leif Andersen [3] and use a quadratic-

exponential (QE) scheme. The basic idea is to determine the distribution of the

variance process V (t + ∆t) using moment matching and approximation techniques.

It is well known that the variance process of the Heston model is proportional to a

non-central chi-squared distribution with non-centrality parameter V (t) ·n(t, t+ ∆t),

where n is independent of V (t) [9]. Andersen develops on this idea and states that

the distribution of V (t + ∆t) can be approximated by a Gaussian variable for large

values of V (t) and as a central chi-squared distribution for small values of V (t) (see

[3] for full details).

We give a very brief outline of the discretisation of the stock and variance processes

modifying results from [3] to incorporate time dependence. The discretisation of the

variance process (QE scheme) is given by:

V (t+ ∆t) =

{a(t) (b(t) + εi)

2 when Ψ(t) ≤ Ψc

Ψ−1 (ui; p(t), β(t)) else(21)

where

a(t) =m(t)

1 + b(t)2, b(t)2 = 2Ψ(t)−1−1+

√2Ψ(t)−1

√2Ψ(t)−1 − 1 , Ψ(t) =

m(t)2

s(t)2

with

m(t) = E[V (T ) | V (t)] = θ(t) + (V (t)− θ(t))e−κ(t)(T−t)

s(t)2 = Var[V (T ) | V (t)] =V (t)σ(t)2e−κ(t)(T−t)

κ

(1− e−κ(t)(T−t))+

θ(t)σ(t)2

2κ(t)

(1− e−κ(t)(T−t))2

30

and finally

Ψ−1(ui; p(t), β(t)) =

0 0 ≤ ui ≤ p

β(t)−1 ln(

1−p(t)1−ui

)p ≤ ui ≤ 1

with

ui ∼ Uni[0, 1] , p(t) =Ψ(t)− 1

Ψ(t) + 1, β(t) =

1− p(t)m(t)

, Ψc = 1.5

Ψc is the critical level controlling when to switch between the two distributions. An-

dersen states that the choice of Ψc has very small relative effects on the quality of the

overall simulation scheme and is set to 1.5. We also see this in our numerical tests

and get extremely good results using this scheme (see Appendix A.5 for results). For

the stock discretisation we use the following discretisation scheme:

S(t+ ∆t) =

S(t) exp{K0 +K1V (t) +K2V (t+ ∆t) +

√K3V (t) +K4V (t+ ∆t) · εi

}(22)

where

K0 = −ρ(t)κ(t)θ(t)

σ(t)∆t , K1 = γ1∆t

(κ(t)ρ(t)

σ(t)− 1

2

)− ρ(t)

σ(t),

K2 = γ2∆t

(κ(t)ρ(t)

σ(t)− 1

2

)+ρ(t)

σ(t), K3 = γ1∆t(1− ρ(t)2) ,

K4 = γ2∆t(1− ρ(t)2)

Note that K0, ..., K4 are time dependent and γ1 = γ2 = 0.5. This defines our discreti-

sation schemes for both the stock and variance process. For our examples we simulate

daily on business days, using 20, 000 paths and pass our simulated market into our

generic payoff to get a price.

31

4.3 Exotic Cliquet Pricing

In this section we compare the model prices against the Totem consensus for a series

of globally floored and reverse cliquets on the S&P500. To produce these results we

calibrate the time dependent Heston model to the resetting cliquet options and the

flat parameter Heston model to the vanilla options and use the Monte Carlo scheme

outlined in section 4.2.4 to calculate prices.

Figure 14 shows how far the model prices differ from the Totem globally floored

cliquet consensus (see Appendix A.3 for further details), where the red shade indi-

cates a price which is more than half the total market spread (fail). We see that

the time dependent Heston model passes nearly all the marks, only 13 marks out

of a possible 312 fail (96% pass rate). The standard Heston model does well, too,

especially for options with higher reset tenors, with a pass rate of 87.5%.

Figure 13 shows the results of the reverse cliquets. These results are surprising as

the flat parameter Heston model has a pass rate of 75% compared to a pass rate of

12.5% for Heston model.6 We expected the time dependent Heston model to mach

the market much closer, as this model fits the forward volatility skew observed in the

market (Figure 6). A closer inspection of the individual contributions revealed that

several contributors’ prices are consistently lower than the consensus and these marks

are consistent with the time dependent Heston model prices. Furthermore, we notice

that a majority of contributors prices are higher and in line with the flat parameter

Heston model 7.

If we take a closer look at the payoff definition of the reverse cliquet (3) and in

particular the term

n∑i=1

min

(Sti+1

Sti− 1, 0

)

we notice that this term will more negative if the volatility is higher. Therefore, any

6The Totem spreads are very tight for annual resetting cliquets, this is why we are observing such

large differences. We believe the spreads in the market are wider than this as these structures are

illiquid and traders will take provisions against potential risks.7Confidentiality of the data has prevented us from including these results in this section.

32

model which exhibits higher volatility will produce lower prices for reverse cliquets.

In section 4.1 we outlined that the standard Heston model calibrated to the vanilla

options produces forward volatility smiles which have high skews with low at-the-

money volatilities. This leads to the underpricing of resetting cliquet options with

short reset tenors. As a result, this model will produce higher reverse cliquet prices.

Moreover, a model which is correctly calibrated to the forward volatility smiles will

produce lower reverse cliquet prices. Therefore, we believe the Totem consensus is

split with the majority of the contributors pricing this structure using a stochastic

volatility model calibrated to vanilla options and several sophisticated contributors

incorporating forward volatility risk within their models. This is not observed in the

globally floored cliquets as these options are less sensitive to extreme volatility move-

ments.

Similar results where also found for the FTSE100 (figures 23 and 21) and Euro Stoxx

50 (figures 24 and 22) and can found in Appendix A.4.

Figure 13: S&P500 reverse cliquet Totem test results

33

(a) Monthly resetting

(b) Quarterly resetting

(c) Semi-annual resetting

(d) Annual resetting

Figure 14: S&P500 globally floored cliquet Totem test results

34

5 Discrete Stochastic Implied Volatility Model

Standard stochastic volatility models such as the Heston Model prescribe dynamics

for the instantaneous volatility, via stochastic differential equations. In such models,

the level of forward skew is predominately determined by the correlation between spot

and volatility at t0. The DSIV model makes the implied volatility for each time step

stochastic, therefore taking into account the effects of stochastic future volatility and

the joint distribution of the underlying prices for all time periods. This allows the

DSIV model to have more control over the dynamics of future skew and smiles, i.e. it

actually specifies the dynamics for both the future-at-the-money volatility (FATMV)

and future skew given the FATMV. However, the drawback of the model is that one

gives up control of the behaviour of the model between periods. In particular, it is

necessary to prove that there is no-arbitrage between periods.

As liquid price information for options is typically only available for particular time

periods, the basic idea is to start with a discrete (finite) structure of times and to

prescribe the propagation of the stock price between these times. This is similar to

the idea of market models8 in interest rates, except that we have a single underlying

asset, so there are more constraints to be satisfied. This will constrain our choices for

the dynamics as we have to ensure that the model is arbitrage free.

To construct our model between our chosen times, we only need to define the transi-

tion kernel for those times, i.e. Pti,tj for i < j. To give a consistent transition kernel,

these functions have to satisfy the composition rule (which follows from the Tower

Law) ∫Ps,t(x, y)Pt,u(y, z)dy = Ps,u(x, z)

The simplest choice is to assume that the stock price and volatility level are Markov

variables for the model. Under this assumption, at each time ti the implied volatility

σi(S,K) for options of strike K expiring at ti+1 is a function of Sti and of the ATM

vol level σti . Specifying this function is equivalent to specifying the transition kernel

between the times ti - knowing the prices of options expiring at ti+1 for all strikes K

8In fixed-income the Libor rates are the traded underliers over which options are written on,

therefore models such as LMM control the dynamics of each Libor rate

35

given the stock price Sti and the at-the-money volatility σti allows us to determine

the probability distribution of(Sti+1

, σti+1

)given (Sti , σti), and hence the transition

kernel Pti,ti+1.

By the same argument all transition kernels from the initial time, Pt0,ti , are deter-

mined by the spot vanilla option prices. The composition rule prescribes that∫Pt0,ti(x, y)Pti,ti+1

(y, z)dy = Pt0,ti+1(x, z) (23)

This is not enough to fully determine Pti,ti+1, but it places a constraint on it9.

5.1 Model Outline

We will assume that we have a discrete set of times 0 = t0 < t1 < t2 < ... < tn−1 <

tn = T for which we have information on spot-starting vanilla options and resetting

cliquets; these will provide information on the spot and forward implied volatility

skews. The basic set-up is to have two processes in the model, one for the stock

and one for implied volatility for each time step. The model will be Markov with

respect to these two processes: the level of the stock price and the level of the implied

volatility fully determine the state of the model at any time.

Volatility Dynamics

To properly manage forward volatility risk we need a model which allows for more

realism of and control over the implied forward volatility dynamics. The aim of the

DSIV model is to address this issue, by allowing traders and risk managers to specify

directly the implied forward volatility dynamics at each discrete step in the model. To

have this flexibility the at-the-money volatilities and volatility skew must be modelled.

We start by considering the at-the-money implied volatility dynamics first. This

9If the transition kernel only depends on the spot price and we let the time step go to zero, i.e.

assume that we have option prices for all expiries and all strikes, then the probability kernel becomes

completely determined by spot volatilities, and we end up with a Dupire local volatility model (if

the spot process is continuous).

36

is typically negatively correlated with relative stock price movements, mean revert-

ing, positive and should not get too large; an exponential Gaussian process will have

these properties. The following dynamics are proposed for the at-the-money volatility

dynamics [1]:

σti+1=

1

αti

(σti+1

−(1− e−κ(ti+1−ti)

) (σti+1

− αtiαti+1

))σti+1

= αti+1σti exp

(ν√ti+1 − tiXi −

1

2ν2 (ti+1 − ti)

)(Sti+1

Sti

)γ(24)

where,

• Xi ∼ N (0, 1),

• γ is the correlation between the stock price movement and at-the-money volatil-

ity, used to control the short term skew,

• αti is the mean reversion level of volatility process between ti and ti+1

• κ is a mean reversion speed, used to to control/fit the long dated skew and

curvature, and

• ν is the volatility of volatility

Each of the above parameters controls the dynamics of the at-the-money implied

volatility for the period [ti+1, ti+2]. Note that the impact of γ and ν is conditional on

the value of κ; if κ is large then this reduces the impact of γ and ν.

We still need to specify the dynamics of the implied volatility skew for each time

step. We outlined earlier that local volatility models produce flattening skews and

smiles and stochastic volatility models produce very convex smiles with low at-the-

money volatilities. One way to avoid this in the discrete setting is to impose time-

homogeneity on the forward skews, i.e. to require the forward skew of a given tenor

at any point to be similar to the spot skew. This can be achieved by making the

skew scale invariant, i.e. a function of moneyness. Let ∆i = ti+1− ti and Fti+1be the

forward price at ti+1, we hence require:

σti+1(x) = qti+1

(x,∆i)σti+1(0)

37

where

x =log(

KFti+1

)σti+1

(0)√

∆i

(25)

The volatility function skew qti+1(x,∆i) is deterministically given as an input

qti+1(x,∆i) =

σfwd(x,∆i)

σfwd(0,∆i)+

(σspot(x,∆i)

σspot(0,∆i)− σfwd(x,∆i)

σfwd(0,∆i)

)exp(−λti+1

) (26)

where

• σfwd is the forward implied volatility taken from a model that is calibrated to

the forward starting options, i.e. Heston or SABR.

• σspot is the spot implied volatility

• λti is a predetermined dampening factor.

This means that the transition kernels for different time steps are essentially the same

as a function of moneyness. In general by fixing the forward volatility skew, it makes

it difficult to simultaneous calibrate to implied volatilities, as there are not sufficient

free parameters available.

This allows for full control of the the implied forward volatility dynamics at each

time step. We can control the shape of the forward volatility skew and how much

spot skew we would like to factor for. This level of flexibility puts arbitrage constraints

on the model, which we address in section 5.3. In our construction and examples, we

do not know how to calibrate/determine λti+1and therefore only work with forward

volatility, this reduces qti+1(x,∆i) to:

qti+1(x,∆i) =

σfwd(x,∆i)

σfwd(0,∆i)(27)

Spot Dynamics

Once the implied volatility function σti+1(K) is fixed, it determines the transition

kernel for the stock price from ti to ti+1 and allows us to determine the cumulative

38

distribution function from the market

C(K, ti+1;Sti , σti(K), ti) =

∫ ∞K

(S −K)p(S, ti+1;Sti , σti(K), ti)dS (28)

dC(K, ti+1;Sti , σti(K), ti)

dK= −

∫ ∞K

p(S, ti+1;Sti , σti(K), ti)dS

This gives:

Fti+1(K;Sti , σti(K)) = P(Sti+1

< K) (29)

= 1 +dC(K, ti+1;Sti , σti(K), ti)

dK

where C(K, ti+1;σti(K), Sti , ti) is the value of a call option expiring ti+1 and K =

Fti exp{x · σti(0)√

∆i} . Let us elaborate on this: at t0 we know σt0(K) (implied

volatility) and St0 (spot), we can use these to solve (numerically) for the distribution

S1. Once we have the distribution for S1 we can determine the distribution of σt1(K)

and repeat the process again to get the distribution of S2. By repeating this process

we can evolve numerically the distribution of S for all periods. This completes our

definition of the model for the given times t0, ..., tn.

5.2 Model Restrictions

We construct the model at discrete times and model the implied volatility directly.

This has the advantage of allowing more controllable and realistic dynamics for the

volatility process, but comes at the cost of not being continuous in time. Consequently,

this model is set up to price and risk manage a narrow range of exotic options, specif-

ically exotic products that depend on forward volatility such as Napoleon options. It

should not be used to price structures where the reset periods do not coincide with

the discrete model periods. Therefore, to price within this model we must carry out

a local calibration to the instruments which will be used in the hedging strategy and

make sure that all reset and calibration periods are consistent.

One potential approach to avoiding this restriction is to extend the dynamics to

39

continuous time as in Schonbucher [29]. Schonbucher outlines a continuous stochastic

implied volatility model by deriving the dynamics of the volatility and spot process

from the no-arbitrage constraints the model has to satisfy. The problem with this

approach is that the dynamics are limited and one loses the ability to control the

volatility dynamics, which is the point of the DSIV model. In section 5.3 we explore

how one can extend the dynamics of the DSIV model to continuous time and what

constraints this imposes on the stock process.

5.3 Arbitrage Freeness

So far we have outlined the model and discussed its limitations. In this section we

look at arbitrage freeness and model completeness.

To prove no-arbitrage we make use of the First and Second Fundamental Theorems

of Asset Pricing [13]:

First Fundamental Theorem of Asset Pricing: A market model has a risk-neutral

probability measure if, and only if it does not admit arbitrage.

Second Fundamental Theorem of Asset Pricing: An arbitrage free market model

is complete if, and only if the risk-neutral probability measure is unique.

We show no arbitrage by extending the dynamics of the stock process to continuous

time, and constructing a martingale measure. Subsequently we discuss model com-

pleteness.

A pre-condition for the existence of an arbitrage free extension of the model to con-

tinuous time is the absence of arbitrage in the discrete time model. As we model

implied volatility this is not obvious and needs verification. We start off by outlining

the conditions that must be met by the volatility process to avoid arbitrage.

5.3.1 Volatility Conditions

For the implied volatility process not to admit arbitrage in discrete time, we need

[14]:

40

• the at-the-money implied volatility to be non-negative: σti+1> 0

• call and put spreads to be non-negative:

dC

dK≤ 0 ,

∂P

∂K≥ 0

• calls and puts to have positive convexity:

d2C

dK2≥ 0 ,

d2P

dK2≥ 0

For the first no-arbitrage condition

σti+1> 0

⇔ 1

αti

(σti+1

−(1− e−κ∆i

) (σti+1

− αtiαti+1

))> 0

⇔ 1

αti

(σti+1

e−κ∆i + αtiαti+1

(1− e−κ∆i

))> 0

⇔ 1

αtiσti+1

e−κ∆i > −αti+1

(1− e−κ∆i

)αti>0⇐⇒ σti+1

> −αtiαti+1

(eκ∆i − 1

)αti+1>0⇐⇒ σti exp

(ν√

∆iXi −1

2ν2∆i

)(Sti+1

Sti

)γ> −αti

(eκ∆i − 1

)︸ ︷︷ ︸<0

This is clearly satisfied if σti and Sti , Sti+1are positive, which we can assume by in-

duction on i. The second and third conditions limit the freedom in choosing the func-

tion qti+1(x,∆i) in (27). To construct qti+1

(x,∆i), more specifically, σfwd(x,∆i) and

σfwd(0,∆i), we calibrate the Heston model to forward starting options with reset date

ti and expiry date ti+1. Once the model is calibrated we can compute σfwd(x,∆i) from

the model prices of forward starting options with strikeK = Fti exp{x·σfwd(0,∆i)√

∆i}.

[23] and [14] show that stochastic volatility models such as the Heston model produce

arbitrage free volatility smiles, so σfwd(x,∆i) fulfils conditions one to three. Fur-

thermore, σti+1(x), is given by scaling σfwd(x,∆i) by a positive constant

σfwd(0,∆i)

σti+1 (0);

therefore to show that σti+1(x) meets conditions two and three, we must show that the

scaled implied volatility surface is a valid implied volatility surface. Time constraints

of this thesis have prevented us from doing so, but we have provided a numerical

41

illustration (figure 15) that σti+1(x) is a valid volatility surface when qti+1

(x,∆i) is

represented by the Heston model. Figure 15 shows that even under extreme scaling,

10% and 300%, we satisfy conditions two and three.

(a) dCdK ≤ 0 (b) d2C

dK2 ≥ 0

(c) dPdK ≥ 0 (d) d2P

dK2 ≥ 0

Figure 15: No arbitrage conditions for scaled one year implied volatility surface

5.3.2 Existence of Risk-Neutral Measure

We start by extending the dynamics of the stock price process to a continuous process.

The existence of a continuous extension can be shown in two different ways:

(i) As the implied volatility is fixed at time ti for expiry ti+1, we can construct

a local volatility function σi (St, t;σti), by extending the implied volatilities for

ti+1 to an implied volatilities surface for the period [ti, ti+1]. We can represent

St as a continuous time stochastic process:

42

dStSt

= σ (St, t;σt1 , ..., σtn) (30)

where σ (St, t;σt1 , ..., σtn) = σi (St, t;σti) for ti ≤ t < ti+1.

The extension to an implied volatility surface, and hence the local volatility

function is clearly not unique, but for our choice of the Heston model to gener-

ate forward smile we can use the Heston model to construct an arbitrage free

volatility surface and thus obtain a natural extension.

We have omitted the drift for notational simplicity, this can easily be included

by considering a numeraire-adjustment.

(ii) The other possibility is to the use Clark-Ocone formula [25]

Sti+1= E

[Sti+1

| Fti]

+

∫ ti+1

ti

E[DtSti+1

| Ft]dW (t)

where Dt is the Malliavin derivative. For simplicity we assume Sti+1has zero

drift, then E[Sti+1

| Fti]

= Sti , we can now define the continuous process by

St = E [Sti | Fti ] +

∫ t

ti

E[DtSti+1

| Ft]dW (t) (31)

for ti ≤ t ≤ ti+1. Unfortunately the Malliavin derivative in (31) cannot be

carried out analytically or even easily on a computer, so we do not pursue this

any further within the constraints of this thesis.

Using (i) to construct the continuous stock price process and definingXi = 1√∆i

∫ ti+1

tidWs,

we can re-write (24) and (30) as:

Sti+1= Sti exp

(∫ ti+1

ti

σi (St, t;σti)dW1s −

1

2

∫ ti+1

ti

σi (St, t;σti)2dt

)σti+1

=1

αti

(σti+1

−(1− e−κ∆i

) (σti+1

− αtiαti+1

))σti+1

= αti+1σti exp

(ν

∫ ti+1

ti

dW 2s −

ν2

2∆i

)(Sti+1

Sti

)γ

43

where d〈W 1,W 2〉 = 0.

To construct the risk-neutral measure all derivative prices must be martingales [13].

This can be achieved by constructing a portfolio Π containing the derivatives, which

has price process V , a quantity Θ1 of the stock and a quantity Θ2 of another asset

whose value V1 depends with non-vanishing derivative on the implied volatility (for-

ward starting options or resetting cliquet options):

Π = V −Θ1S −Θ2V1

dΠ = dV −Θ1dS −Θ2dV1

where

dVIto=dV

dtdt +

dV

dSdS +

1

2

d2V

dS2〈dS, dS〉 +

dV

dσdσ +

1

2

d2V

dσ2〈dσ, dσ〉 +

d2V

dSdσ〈dS, dσ〉

and dσ = dσti+1for ti ≤ t < ti+1. We can explicitly construct our risk-neutral

process by equating the change in the portfolio to the risk free rate, dΠ = rdΠdt,

and determining the market price of volatility risk [15] as long as dVdσ6= 0. However,

this is complicated to do analytically in our setup and can easily be done numerically

during the calibration process. Effectively, we assume that we are imputing the risk-

neutral measure directly by fitting the parameters of the process to the option prices

[15]. This concludes the existence of the risk-neutral measure and from the First

Fundamental Theorem of Asset Pricing that the model does not admit arbitrage.

5.3.3 Model Completeness

For the model to be complete, all the associated risks must be hedged within the

model framework. The DSIV model has two factors of randomness; one on the stock

process, the other on the implied volatility process. We can hedge the randomness

on the stock price process using the stock and we need a form of volatility product to

hedge out the randomness on the implied volatility process. This can be achieved by

using either or a combination of vanilla options, forward starting options and cliquet

options. So the choice of options used for hedging determines the risk-neutral measure

used in pricing.

44

5.4 Pricing

So far we have outlined the model, but not discussed how to price with this model.

No analytic solutions exist for vanilla or exotic options and the DSIV model must be

implemented numerically. We opt for Monte Carlo due to time constraints. However,

a calculation method based on representations of conditional expectation function

such a tree or a grid should be viable as well and would be more appropriate for

calibration purposes, as it should be considerably faster.

Monte Carlo Algorithm

Once the model is fully calibrated, the Monte Carlo algorithm works as follows. We

start by setting the time grid (t0, t1, ..., tn) to exactly match the cliquet reset fre-

quency, either monthly, quarterly, semi-annual or annual. At time t0; σt0(K) (implied

volatility) and St0 are known, from (29) we know the cumulative distribution of the

stock price at time t1:

Ft1(K;St0 ;σt0(K)) =

1 +dC(K, t1;St0 , σt0(K), t0)

dK=

1 +C(K + ∆K; t1;St0 , σt0(K + ∆K), t0)−C(K −∆K, t1;St0 , σt0(K −∆K), t0)

2∆K

To determine this distribution we generate a uniform random U1 and by inversion

solve for St1 such that Ft1(St1) = U1. Once we have St1 , we then generate a Gaussian

random number to deduce σt1(0)

σt1 =1

αt0

(σt1 −

(1− e−κ∆1

)(σt1 − αt0αt1)

)σt1 = αt1σt0 exp

(ν√

∆1ε1 −ν2

2∆1

)(St1St0

)γ

and multiplying the at-the-money implied volatility by the forward skew qt1(x,∆1),

we get σ1(K). We repeat the above process to determine S2 and carry on iteratively

to generate the full path of the underlying.

45

Numerical Enhancements

To use Monte Carlo in the calibration process; the path generation must be highly

efficient for the calibration to be achievable in acceptable time. We immediately see

that the most time consuming part of the algorithm is the computation of the distri-

bution of Sti+1for each sample of the at-the-money volatility. This is not practical

as we have to solve numerically for the distribution of S for each path and period.

Therefore, we pre-compute the distribution ofSti+1

Fti+1

10 for all periods given a particular

at-the-money level and then use interpolation to do the numerical inversion.

We assume the at-the-money volatility belongs to an interval [σmin, σmax] and dis-

cretise this interval into N subsets. Once we know the at-the-money volatility level

we can compute the distribution ofSti+1

Fti+1:

Fti+1(K, σti(K)) = P

(Sti+1

Fti+1

< K

)= 1 +

dC(K, ti+1;σti(K), ti)

dK

= 1 +∂C(K, ti+1;σti(K), ti)

∂K+∂C(K, ti+1;σti(K), ti)

∂σ

∂σti(K)

∂K

= N(−d1) +√

∆iN′(d2)

∂σti(K)

∂K

where

d1 =ln(K) + 1

2σti(K)2∆i

σti(K)√

∆i

d2 = d1 − σti(K)√

∆i

N(ξ) =

∫ K

−∞

1√2πe−

12ξ2dξ

where C(K, ti+1;σti(K), ti) is the Black call option price function; based on the for-

ward which is equal to one in this setup.

10If we pre-computed the distribution of Sti+1 , this will be dependent on today’s spot and forward

level; but our forward is stochastic so we model the stock level relative to the forward and multiply

back with the simulated forward.

46

We pre-compute the distribution by diving the probability interval [0, 1] into N subin-

tervals and then using a root finding method to solve for eachSti+1

Fti+1. We can then

use interpolation to calculate intermediary points and multiplying with the simulated

Fti+1to get Sti+1

. This will obviously lead to some numerical error, but improves

significantly the speed of calibration and makes the calibration of many reset dates

practically possible.

5.5 Calibration

We get our market data from the vanilla call cliquets and spot vanilla options, see

figure (2) for a full set of calibration instruments. Ideally, we will have market prices

for forward starting options for each date on the time grid, we do not have this data

and must imply the forward volatility dynamics from the cliquet option prices11. We

calibrate each period independently to the cliquet options, starting from the first to

last solving a minimisation problem on the objective function:

min

{N∑i=1

wi(σmodelfwd (i)− σmarketfwd (i)

)2

}

The weights wi are chosen to give the highest priority to the at-the-money options.

When the parameters of the first period are calibrated, they are fixed and the pa-

rameters of the next period are calibrated, this process continues until the last period.

This calibration algorithm is quite primitive as we fit all parameters without any

constraints to the data and parameters. Further improvements can be made by es-

timating some of the parameters from the market data, this will reduce the dimen-

sionality of the optimisation problem. Also, due to the complexity of this model and

data requirements, we cannot calibrate the model on the fly, it will have to be done

once a day with traders having a view on how the parameters change intra-day.

In our examples we use 20,000 simulation paths in the calibration and 100,000 paths

when pricing the exotic cliquet structures. Also, we only price structures up to three

years in maturity and the forward skew qti+1(x,∆i) only changes three times during

this period. The profile of qti+1(x,∆i) changes as follows:

11We can compute forward starting options for annual resetting cliquets

47

• For implied forward volatility smiles less than one year in maturity we use the

spot implied volatility skew. For example, for a quarterly resetting cliquet, we

will fix the three month implied volatility skew for all implied forward volatility

skews less than one year of maturity, i.e. the six month, three month implied

forward volatility skew will look like today’s implied volatility skew.

• For implied forward volatility skews greater than and equal to one year and less

than two years in maturity we use the one year reset implied forward volatility

skew (i.e one year, three month implied forward volatility skew).

• Finally for implied forward volatility smiles greater than equal to two years we

use the two year reset implied forward volatility skew.

Calibration Results

Figures 16, 17 and 18 show the calibration error, model cliquet volatility minus im-

plied cliquet volatility, for annual and semi-annual resetting cliquets. The calibration

fits seem good and would be acceptable in practice, although the fits are not as good

as the time dependent Heston model. This could be improved by increasing the num-

ber of simulations and changing qti+1(x,∆i) more often. Also, time limitations of the

thesis have prevented us from looking at the joint calibration of forward and vanilla

options and we leave this for future research.

(a) Annual resetting (b) Semi-annual resetting

Figure 16: S&P500 resetting cliquet calibration error

48

(a) Annual resetting (b) Semi-annual resetting

Figure 17: FTSE100 resetting cliquet calibration error

(a) Annual resetting (b) Semi-annual resetting

Figure 18: Euro Stoxx 50 resetting cliquet calibration error

5.6 Exotic Cliquet Pricing

Figures 19 and 20 show that the exotic cliquet prices for both the globally floored and

reverse cliquets are in line with the Totem consensus. For globally floored cliquets we

have a pass rate of 91% across the three major equity indices and no price being over a

whole spread out. The reverse cliquets have a pass rate of 50%, which is satisfactory,

however for higher reset frequency the results seem better with only two quotes being

a whole spread out.

In summary this model is well suited to price such structures as indicated by the

Totem tests. More research and testing is required to improve the calibration and

performance of the model if it is to be used in practice.

49

(a) Quarterly resetting

(b) Semi-annual resetting

(c) Annual resetting

Figure 19: Globally floored cliquet Totem test results

50

Figure 20: Reverse cliquet Totem test results

6 Summary

In this thesis we have looked at modelling forward implied volatility by analysing three

different volatility models. We started off, by stating that the ideal model would give

us control over volatility dynamics, calibrate to both forward and vanilla options, be

continuous and practical. For each of the models, we discussed these criteria and

found that none could satisfy all:

• The local volatility model: a continuous time deterministic volatility model

which has the ideal quality of perfectly replicating the vanilla options. But this

model incorrectly specifies the dynamics of forward volatility and as a result

misprices the resetting and exotic cliquets.

• The Time Dependent Heston Model: a continuous time model with an addi-

tional stochastic process to drive the instantaneous variance. This model gave

good fits to both the vanilla and resetting cliquets market data, but was unable

to calibrate to both sets of data. Moreover, model prices for the exotic cliquet

markets matched Totem concensus reasonably well. However, the limitation of

this model is that we do not have control over the dynamics of forward volatility

and are model implied.

51

• The DSIV model: models the implied volatility for each time step as stochas-

tic; we specify the dynamics of the at-the-money implied volatility and have

the flexibility of specifying the volatility skew. This gives very good control

over the dynamics of forward implied volatility. The model has the limitation

that it can only price a very limited range of exotic structures as it is not con-

tinuous. In our implementation, we used Monte Carlo and this obviously has

practical issues, but we believe a grid implementation would resolve these issues.

The DSIV model, like the time dependent Heston model, gave very good fits to

the resetting cliquet market data and performed well in Totem tests.

Our summary of the models above indicate that we have two potential models to use

in pricing of exotic structures. To further test these models, we must look at the

behaviour of the Greeks. These ultimately define our hedging strategy and how well

suited the model is for risk managing an exotic structure.

There have also been recent developments in modelling volatility. H.Buehler [7],

J.Gatheral [16] and L.Bergomi [5] have worked on a new breed of models, known as

“Variance Swap Models”. Variance swaps are contracts on realised variance and inher-

ently strike independent, which makes them natural instruments to hedge volatility

products. Furthermore, forward variance swaps can be used as natural hedging in-

struments for cliquets and other forward starting exotic products. One particular

model that we would like discuss briefly at is Bergomi variance swap model [5].

Bergomi argues that variance swaps should be considered as a hedging instrument

and have their own dynamics. He proposes a model that allows independently set the

requirements for:

• The dynamics of variance swap volatilities.

• The level of short-term forward skew.

• The correlation between the underlying and the short and long variance swap

volatilities.

The model setup is quite similar to DSIV model instead of modelling the implied

volatility it models the forward variance (ξit) at each time step as stochastic:

52

ξit = ξit(0) exp

(ω[e−k1(Ti−t)Xt + θe−k2(Ti−t)Yt

]−

ω2

2

[e−2k1(Ti−t)E[X2

t ] + θ2e−2k2(Ti−t)E[Y 2t ] + 2θe−(k1+k2)(Ti−t)E[XtYt]

])(32)

where ξit is a random process until t = ti+1, when t reaches ti+1 the Variance Swap

variance is known and is equal to ξit+1.

This model would have the same benefits as the DSIV model but would have the

advantage that variance swaps are more liquidly traded and are natural hedging in-

struments. Also, as realised variance rather than implied volatility is modelled the

model is simpler to handle and no arbitrage conditions are easier to satisfy.

53

A Appendix

A.1 SABR Model

The SABR model [26] is a stochastic volatility model on the forward Ft:

dFt = σtFβt dWt

dσt = ασtdZt (33)

where Wt and Zt are Brownian motions such that 〈dWt, dZt〉 = ρdt, α ≥ 0 is the

volatility of volatility and β ∈ [0, 1] the skewness. Note that β can also be used to

control the distribution of the forward process, i.e. if it is set to 1 we have a log-

normal model.

The SABR model is a stochastic volatility model used in the interest rate and other

derivative markets for pricing vanilla options. Its main application to date has been

in the fitting and description of volatility skews, because it allows Black’s formula

still to be used to value European options, but with a volatility value which depends

on strike and the model’s other parameters. The advantage of the SABR model over

other stochastic volatility models is that it is the simplest stochastic volatility model

which is homogenous in the forward rate and the volatility. The SABR model can

be used to accurately fit the implied volatility curves observed in the marketplace for

any single exercise date. More importantly, the delta and vega risks are in line with

market observations [26], making the SABR model an effective means to manage the

smile risk in markets where each asset only has a single exercise date.

A.2 Implied Forward and Cliquet Volatility

If we work in the Black Scholes (BS) framework, we can easily derive an analytic

solution for a forward starting call option:

Cfwd(α, ti+1; ti),= erd(ti+1−ti)C

(Sti+1

Sti, α, σti,ti+1

, ti+1; ti

)(34)

54

where C is the Black call option price function and Sti+1and Sti are given by the

forward price at time ti+1 and ti. It is common practice to quote vanilla, option and

cliquet prices in terms of volatility, as it is more intuitive to understand. The implied

forward volatility σfwd, is the fitted forward volatility σti+1,ti+2which matches forward

starting option prices Cfwd:

σfwd = minσ

{Cfwd −Cfwd(σti,ti+1

)}

(35)

Similarly, we can also derive analytic resetting cliquet (RC) prices in the Black Sc-

holes framework:

CRC =tn∑i=0

Cfwd(α, ti+1; ti)

and the implied cliquet volatility σRC is given by:

σRC = minσ

{CRC −CRC(σt0,tn)

}(36)

The following terminology is used in Figures: 1, 5(a), 6(c), 8, 9(c), 10, 11, 12, 16, 17

and 18:

• Moneyness α is the ratio KSti

.

• Term refers to is the expiry of the option in months.

• Implied vanilla/cliquet volatility error means the difference in volatility points

between the model price and the market: σMdl − σMkt.

• Two different shades are used in the table to highlight any errors that are greater

than half a volatility point. A light yellow shade for anything greater than 0.5%

and a light blue shade for anything lower than (-0.5%). We believe any errors

less than half a volatility point is reasonable.

55

A.3 Totem Consensus

The Totem consensus is a monthly service which provides the major market makers

in OTC derivatives with definitive consensus market prices. More than fifty of the

leading banks, trading houses and other investment professionals use the Totem ser-

vice as the key independent check of their trading book valuations.

The Totem service generates a mid market price and spread for a particular structure

by collating quotes from leading investment banks that contribute to this service.

Any outliers are removed and it is ensured that the market data which has gone into

the pricing of these structures is consistent among all contributors; this avoids any

ambiguity in the prices.

The confidentiality of the data prevents us from disclosing actual contributor marks

or the consensus and are limited to disclosing a measure called % of Totem spread:

% of Totem spread =|ModelPrice− TotemConsensus|

TotemSpread(37)

where

TotemSpread = MaxAcceptedPrice−MinAcceptedPrice

If this measure is below 50% then the model price is within the Totem acceptance

tolerance and can be considered a pass. If this measure is significantly above 50%,

then this is not satisfactory to pass the Totem consensus. The spreads of Totem

are extremely tight for some exotic products and this may not be observed in actual

quoted dealer prices as traders will take provisions against model and future risks.

The following terminology is used in Figures: 4, 5, 13, 14, 19, 20, 21, 22, 23 and

24:

• LV Model price - the local volatility model price.

• Flat model price - the flat parameter Heston model price. This model is cali-

brated to the vanilla options market data.

• TD model price - the time dependent Heston model price. This model is cali-

brated to the resetting cliquets market data.

56

• A red shade indicates a point that falls outside Totem consensus.

A.4 Exotic Cliquet Totem Results for Heston Model

Note, to reduce the calibration error of the flat parameter Heston Model, we do not

include the four and five year vanilla option prices in the calibration.

Figure 21: FTSE100 reverse cliquet Totem test results

Figure 22: Euro Stoxx 50 reverse cliquet Totem test results

57

(a) Monthly resetting

(b) Quarterly resetting

(c) Semi-annual resetting

(d) Annual resetting

Figure 23: FTSE100 globally floored cliquet Totem test results

58

(a) Monthly resetting

(b) Quarterly resetting

(c) Semi-annual resetting

(d) Annual resetting

Figure 24: Euro Stoxx 50 globally floored cliquet Totem test results

59

A.5 Monte Carlo Results for Time Dependent Heston Model

Figure 25 shows the comparison of the semi-analytic price against the Monte Carlo

price for a series of three month forward starting options on the S&P500. Using daily

time steps and 20,000 simulations the Monte Carlo prices are in line with the semi-

analytic prices (all within two Monte Carlo standard errors).

(a) Semi-analytic forward price

(b) Monte Carlo forward price

(c) Monte Carlo standard error

Figure 25: S&P500 three month forward numerical price comparison

60

A.6 Matlab Code

We have attached a sample of the main classes at the end of this document for

reference purposes. For all the code used in this thesis please refer to supplementary

CD.

61

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