Optimal Malliavin weighting functions for the simulations of the Greeks

MC 2000 (July 3-5 2000)

Eric Ben-Hamou

Financial Markets Group

London School of Economics, UK

benhamou_e@yahoo.com

3-5 July 2000 MC 2000 Conference Slide N°2

Outline

• Introduction & motivations

• Review of the literature

• Results on weighting functions

• Numerical results

• Conclusion

3-5 July 2000 MC 2000 Conference Slide N°3

Introduction

• When calculating numerically a quantity– Do we converge? to the right solution?– How fast is the convergence?

• Typically the case of MC/QMC simulations especially for the Greeks important measure of risks, emphasized by traditional option pricing theory.

3-5 July 2000 MC 2000 Conference Slide N°4

– Finite difference approximations: “bump and re-compute”

– Errors on differentiation as well as convergence!

– Theoretical Results: Glynn (89) Glasserman and Yao (92) L’Ecuyer and Perron (94):

– smooth function to estimate:

- independent random numbers: non centered scheme: convergence rate of n-1/4 centered scheme n-1/3

- common random numbers: centered scheme n-1/2

– rates fall for discontinuous payoffs

Traditional method for the Greeks

3-5 July 2000 MC 2000 Conference Slide N°5

How to solve the poor convergence?• Extensive litterature:

– Broadie and Glasserman (93, 96) found, in simple cases, a convergence rate of n-1/2 by taking the derivative of the density function. Likelihood ratio method.

– Curran (94): Take the derivative of the payoff function.– Fournié, Lasry, Lions, Lebuchoux, Touzi (97, 2000)

Malliavin calculus reduces the variance leading to the same rate of convergence n-1/2 but in a more general framework.

– Lions, Régnier (2000) American options and Greeks– Avellaneda Gamba (2000) Perturbation of the vector of

probabilities.– Arturo Kohatsu-Higa (2000) study of variance reduction– Igor Pikovsky (2000): condition on the diffusion.

3-5 July 2000 MC 2000 Conference Slide N°6

Common link:• All these techniques try to avoid differentiating

the payoff function:

• Broadie and Glasserman (93) – Weight = likelihood ratio

– should know the exact form of the density function

WeightPayoffETheGreek *

,ln

,,ln

SpSFE

dSSpSpSFSFE

T

T

,ln TSp

3-5 July 2000 MC 2000 Conference Slide N°7

• Fournié, Lasry, Lions, Lebuchoux, Touzi (97, 2000) : “Malliavin” method

• does not require to know the density but the diffusion.

• Weighting function independent of the payoff.

• Very general framework.

• infinity of weighting functions.

• Avellaneda Gamba (2000) • other way of deriving the weighting function.

• inspired by Kullback Leibler relative entropy maximization.

3-5 July 2000 MC 2000 Conference Slide N°8

Natural questions

• There is an infinity of weighting functions:– can we characterize all the weighting functions?

– how do we describe all the weighting functions?

• How do we get the solution with minimal variance?– is there a closed form?

– how easy is it to compute?

• Pratical point of view: – which option(s)/ Greek should be preferred? (importnace

of maturity, volatility)

3-5 July 2000 MC 2000 Conference Slide N°9

Weighting function description

• Notations (complete probability space, uniform ellipticity, Lipschitz conditions…)

• Contribution is to examine the weighting function as a Skorohod integral and to examine the “weighting function generator”

tttt dWXtdtXtbdX ,,

3-5 July 2000 MC 2000 Conference Slide N°10

Integration by parts

• Conditions…Notations

• Chain rule

• Leading to

TTt XEXDE

TtTTt XDXEXDE '

,, ' TTT XXEXE

3-5 July 2000 MC 2000 Conference Slide N°11

Necessary and sufficient conditions

TtTTT XDXEXXE '' ,

• Condition

• Expressing the Malliavin derivative

TTtTT XXDEXXE ||,

TT

T

ttTT XXY

YXtEXXE |,| 1,

3-5 July 2000 MC 2000 Conference Slide N°12

• Minimum variance of

• Solution: The conditional expectation with respect to

• Result: The optimal weight does depend on the underlying(s) involved in the payoff

Minimal weighting function?

WeightXE T

TX TXWeightE |

3-5 July 2000 MC 2000 Conference Slide N°13

For European options, BS

• Type of Malliavin weighting functions:

TW

SfeE

WT

WSfeEv

WT

W

TxSfeE

Tx

WSfeE

TT

rT

TT

TrT

TT

TrT

TT

rT

1

11

2

2

2

3-5 July 2000 MC 2000 Conference Slide N°14

Typology of options and remarks

• Remarks:– Works better on second order differentiation…

Gamma, but as well vega.– Explode for short maturity.– Better with higher volatility, high initial level– Needs small values of the Brownian motion (so

put call parity should be useful)

Tx

WSfeE TT

rT

3-5 July 2000 MC 2000 Conference Slide N°15

Finite difference versus Malliavin method

• Malliavin weighted scheme: not payoff sensitive

• Not the case for “bump and re-price”– Call option

2/12

KSKSE xT

xT

TWrTxT

xT eESSE

2

2/122

KSEKSE xT

xT

3-5 July 2000 MC 2000 Conference Slide N°16

• For a call

• For a Binary option

O

KSKSE xT

xT

2/12

O

E

E

xT

xT

xT

xT

SKS

KSKS

2/1

2/12

1

11

3-5 July 2000 MC 2000 Conference Slide N°17

Simulations (corridor option)

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Simulations (corridor option)

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Simulations (Binary option)

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Simulations (Binary option)

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Simulations (Call option)

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Simulations (Call option)

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Conclusion

• Gave elements for the question of the weighting function.

• Extensions:– Stronger results on Asian options– Lookback and barrier options– Local Malliavin– Vega-gamma parity

nT n

unuTu duSSMax/1

00 lim