Local Vol Delta-Hedging

8/13/2019 Local Vol Delta-Hedging

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IV. Volatility Expansion


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Bruno Dupire 3

Introduction

This talk aims at providing a betterunderstanding of:

How local volatilities contribute to the value ofan option

How P&L is impacted when volatility ismisspecified

Link between implied and local volatility Smile dynamics

Vega/gamma hedging relationship


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Bruno Dupire 4

In the following, we specify the dynamicsof the spot in absolute convention (as

opposed to proportional in Black-Scholes)

and assume no rates:

: local (instantaneous) volatility

(possibly stochastic) Implied volatility will be denoted by

dS dW t t t

Framework & definitions


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Bruno Dupire 5

P&L

St t

St

Break-even

points

t

t

Option Value

St

Ct

Ct t

S

Delta

hedge

P&L of a delta hedged option


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Bruno Dupire 6

P&L Histogram P&L Histogram

Expected P&L = 0 Expected P&L > 0

Correct volatility Volatility higher than expected

St tStSt t St

Ito: When , spot dependency disappearst 0

1std 1std

1std 1std

P&L of a delta hedged option (2)


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P&L is a balance between gain from Gand loss from :

P L dtt t dt & ,

2

0 02 G

=> discrepancy if different from expected:

gain over dt 1

22

0

2

0 Gdt

0

00

:

:

Profit

Loss Magnified by G

G00

2

02

From Black-Scholes PDE:

Black-Scholes PDE


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Total P&L over a path

= Sum of P&L over all small time intervals

P L

dtT

&

( )

122

0

2

00

G

Spot

Time

Gamma

No assumption is made

on volatility so far

P&L over a path


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The probability density may correspond to the density of

a NON risk-neutral process (with some drift) with volatility .

Terminal wealth on each path is:

Taking the expectation, we get:

E X E S dS dtT

wealthT

( ) [ ( )| ] G0 0 2 0200

12

( is the initial price of the option)

wealth =T X dtT

G02

0

2

00

12 ( )

X0

General case


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In a complete model (like Black-Scholes), the drift doesnot affect option prices but alternative hedging strategies

lead to different expectations

Example: mean reverting process

towards L with high volatility around L

We then want to choose K (close to L) T

and 0(small) to take advantage of it.

L

Profile of a call (L,T) for

different vol assumptionsIn summary: gamma is a volatility

collector and it can be shaped by:

a choice of strike and maturity,

a choice of 0 , our hedging volatility.

Non Risk-Neutral world



X X E S dS dtT

( ) ( ) [ ( )| ] G

0 02 02

0012

X X E S dS dt( ) ( ) ( ) [ | ] G 0 2 02 012

C C E S dS dt ( ) ( ) ( [ | ] ) G 0 2 02 012

From now on, will designate the risk neutral densityassociated with .

In this case, E[wealthT] is also and we have:

Path dependent option & deterministic vol:

European option & stochastic vol:

dS dW t t

X

Average P&L



Buy a European option at 20% implied vol

Realised historical vol is 25%

Have you made money ?

Not necessarily!

High vol with low gamma, low vol with high gamma

Quiz



An important case is a European option withdeterministic vol:

C C dS dt ( ) ( ) ( ) G 02

0

2

012

The corrective term is a weighted average of the volatility

differences

This double integral can be approximated numerically

Expansion in volatility


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Bruno Dupire 14

t

Delta = 100%

KDelta = 0%

S

C S K K t K t dt T

( ) ( ) , , 02

0

12

Extreme case: 0 00 G K

This is known as Tanakas formula

P&L: Stop Loss Start Gain


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Bruno Dupire 15

13

15

17

19

21

23

Implied volatility

strike maturity

11

15

19

23

27

31

Local volatility

spot time

Aggregation

Differentiation

Local / Implied volatility relationship


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Bruno Dupire 16

2

2

2

2( , )S T

C

TC

K

Stripping local vols from implied vols is the

inverse operation:

Involves differentiations

(Dupire 93)

Smile stripping: from implied to local


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Bruno Dupire 17

Let us assume that local volatility is a

deterministic function of time only:

dS t dW t t In this model, we know how to combine local

vols to compute implied vol:

Tt dt

T

T

2

0

Question: can we get a formula with ? S t,

From local to implied: a simple case


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Bruno Dupire 18

Implied Vol is a weighted average of Local Vols

(as a swap rate is a weighted average of FRA)

When = implied vol0

solve by iterations depends onG0 0

1

202

0

2

0( ) G dSdt

0

2

2

0

0

G

G

dSdt

dSdt

From local to implied volatility


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Bruno Dupire 19

Weighting Scheme: proportional to G0

At the

moneycase:

S0=100

K=100

Out of the

moneycase:

S0=100

K=110

tS

G0

Weighting scheme


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Bruno Dupire 20

Weighting scheme is roughly proportional to thebrownian bridge density

Brownian bridge density:

BB x t P S x S KK T t T , ,

Weighting scheme (2)


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Bruno Dupire 21

( ) ( ) 2 2 S S dS

small

T

large T

S0

S

(S) S0

S

K(S)

S0

S

(S) S0

S

K(S)

ATM (K=S0) OTM (K>S0)

S dtdSdt

G

G0

0

Time homogeneous case


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Bruno Dupire 22

and are

averages of the same local

vols with different weightingschemes

K1

K2

=> New approach gives us a direct expression for

the smile from the knowledge of local volatilities

But can we say something about its dynamics?

S0

K2

K1

t

Link with smile


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Bruno Dupire 23

23.5

24

24.5

25

25.5

26

S0 K

t

S1Weighting scheme imposessome dynamics of the smile for

a move of the spot:

For a given strike K,

(we average lower volatilities)S K

Smile today (Spot St)

StSt+dt

Smile tomorrow (Spot St+dt)

in sticky strike model


if ATM=constant


in the smile model

&

Smile dynamics


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Bruno Dupire 25

C C dSdt ( ) ( ) ( ) G 0 2 02 01

2

If & are constant 0

2

0

2

G dSdtCC 02020 21

)()(

Vega =

2

2

C

C C C 2

2

2 2

Vega analysis


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Bruno Dupire 28

Local change of implied volatility is obtained by combininglocal changes in local volatility according a certain weighting

dC

d

dC

d

d

d

2 2

2

2

sensitivity in

local vol

weighting obtain

using stripping

formula

Thus:cancel sensitivity to any move of implied vol

cancel sensitivity to any move of local vol

cancel all future gamma in expectation

Superbuckets: local change in implied vol


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Bruno Dupire 29

This analysis shows that option prices arebased on how they capture local volatility

It reveals the link between local vol andimplied vol

It sheds some light on the equivalence

between full Vega hedge (superbuckets)

and average future gamma hedge

Conclusion


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Dual Equation

The stripping formula

can be expressed in terms of

When

solved by

KK

T

CK

C2

2 2

:

KXK

1

0T

K

S uu

du

X

0,


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Bruno Dupire 31

Large Deviation Interpretation

The important quantity is

If then satisfies:

and

K

S uudu

0,

dWxadx

x

x ua

duxy

0

dWdtdy

KS

K

S

K

S

uu

du

SK

uu

du

u

du

0,

lnln

0,

KyWKx tt


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Bruno Dupire 32

Delta Hedging

We assume no interest rates, no dividends, and absolute(as opposed to proportional) definition of volatility

Extend f(x) to f(x,v) as the Bachelier (normal BS) price of

f for start price x and variance v:

with f(x,0) = f(x)

Then,

We explore various delta hedging strategies

dyeyfvXfEvxf v

xy

vx2

)(

,

2

)(2

1)]([),(

),(2

1),( vxfvxf xxv


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Bruno Dupire 33

Calendar Time Delta Hedging

Delta hedging with constant vol: P&L depends on thepath of the volatility and on the path of the spot price.

Calendar time delta hedge: replication cost of

In particular, for sigma = 0, replication cost of

t

uxx dudQVfTXf0

2

,0

2

0 )(

2

1).,(

)).(,(2

tTXf t

t

uxxdQVfXf0

,002

1)(

)( tXf


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Bruno Dupire 34

Business Time Delta Hedging

Delta hedging according to the quadratic variation:

P&L that depends only on quadratic variation and

spot price

Hence, for

And the replicating cost of is

finances exactly the replication of f until

txtxxtvtxtt dXfdQVfdQVfdXfQVLXdf ,0,0,021),(

,,0 LQV T

t

t

uuxtt dXQVLXfLXfQVLXf 0 ,00,0 ),(),(),(

),( ,0 tt QVLXf ),( 0 LXf

),( 0 LXf LQV ,0:


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Bruno Dupire 35

Daily P&L Variation


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Bruno Dupire 36

Tracking Error Comparison


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Bruno Dupire 38

Stochastic volatility model Hull&White (87)

P

tt

P

tt

t

t

dZdtd

dWrdtS

dS

Incomplete model, depends on risk premium

Does not fit market smile

Strike price

Impliedvola

tility

Strike price

Impliedvolatility

Z W,

0Z W, 0

Hull & White


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Bruno Dupire 39

Role of parameters

Correlation gives the short term skew Mean reversion level determines the long term

value of volatility

Mean reversion strength Determine the term structure of volatility

Dampens the skew for longer maturities

Volvol gives convexity to implied vol

Functional dependency on S has a similar effect

to correlation


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Bruno Dupire 40

Heston Model

dtdZdWdZvdtvvdv

dWvdtS

dS

,

Solved by Fourier transform:

,,,,,,

ln

01, vxPvxPevxC

tTK

FWDx

x

TK


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Bruno Dupire 46

2 ways to generate skew in a stochastic vol model

-Mostly equivalent: similar (St,t) patterns, similar

futureevolutions

-1) more flexible (and arbitrary!) than 2)

-For short horizons: stoch vol modellocal vol model

+ independent noise on vol.

Spot dependency

0,)2

0,,,)1

ZW

ZWtSfxtt

0S

STST0S


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Bruno Dupire 47

Convexity Bias

0,

?| 2022

ZW

KSEdZd

dWdS

ttt

t

2

0

2

onlyNO! tE

tlikely to be high if

00or SSSS tt

KSE tt |2

0S

2

0

K


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Bruno Dupire 48

Risk Neutral drift for instantaneous forwardvariance

Markov Model:

fits initial smile with local vols

Impact on Models

dWtSfSdS

t, tS,

]|[

,,

2

2

SSE

tStSf

tt


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Bruno Dupire 49

Skew case (r


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Bruno Dupire 50

Smile dynamics: Stoch Vol Model (2)

Pure smile case (r=0)

ATM short term implied follows the local vols

Future skews quite flat, different from local volmodel

Again, do not forget conditioning of vol by S

Local vols

SSmile

0Smile S

SSmile

S 0S S

K


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Forward Skew


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Bruno Dupire 52

Forward Skews

In the absence of jump :

model fits market

This constrains

a) the sensitivity of the ATM short term volatility wrt S;

b) the average level of the volatility conditioned to ST=K.

a) tells that the sensitivity and the hedge ratio of vanillas depend on the

calibration to the vanilla, not on local volatility/ stochastic volatility.To change them, jumps are needed.

But b) does not say anything on the conditional forward skews.

),(][, 22 TKKSETK locTT


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Sensitivity of ATM volatility / S

S 2

dSS

tS

tSttSSSSSE loc

tloctloctttttt

),(

),(),(][

22222

2

ATM

At t, short term ATM implied volatility ~ t.

As tis random, the sensitivity is defined only in average:

In average, follows .

Optimal hedge of vanilla under calibrated stochastic volatility corresponds to

perfect hedge ratio under LVM.

2

loc

Local Vol Delta-Hedging

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