Smart Monte Carlo: Various Tricks Using Malliavin Calculus Quantitative Finance, NY, Nov 2002 Eric Benhamou [email protected] Goldman Sachs International

Smart Monte Carlo:

Various Tricks Using Malliavin Calculus

Quantitative Finance, NY, Nov 2002

Eric Benhamou

[email protected]

Goldman Sachs International

Quantitative Finance, Risk Conference, NY, November 3-4, 2002

Slide 2

Agenda

I. Motivation for Fast Monte Carlo Engines

II. Smart Computation of the Greeks

III. Typology of Options and Practical Use

IV. Other Developments: Smart Calibration, Conditional

Expectations and Design of Efficient Monte Carlo Engines


Slide 3

I. Motivation for Fast Monte Carlo Engines


Slide 4

Multi-Asset Products

Growing demand of multi-asset products have urged to develop generic pricing engines (often using Monte Carlo):

—Parser to enter tailor made complex payoffs

—Ability to design easily multi-asset models

—Modelling components easy and fast to calibrate

—Powerful risk engine

—Stability of prices and risks

—Fast pricing and generation of risk reports


Slide 5

Computing Challenge of Monte Carlo Trading Book

The two most time-consuming steps are:

—Calibration

—Risk

How can we create generic smart Monte Carlo engines to speed up calibration and Greek computation?


Slide 6

II. Smart Computation of the Greeks


Slide 7

The Challenge of Fast Greeks

Price sensitivities required for:

—Pricing (measure of the error and price charge)

—Estimation of the risk of the book (hedging)

—PNL explanation and back testing

—Credit valuation adjustment and VAR


Slide 8

Traditional Method for the Greeks

Finite difference approximation: “bump and re-price”

Two types of errors:

—Differentiation

—Convergence

Obviously very inefficient for payoffs containing discontinuities like binary, corridor, range accrual, step-up, cliquet, ratchet, boost, scoop, altiplano, barrier and other types of digital options for example


Slide 9

How to Avoid Poor Convergence?Avoid Differentiating

Take the derivative of the payoff function

Pathwise method (Broadie Glasserman (93))

Take the derivative of the probability function

Likelihood ratio method (Broadie Glasserman (96))

Do an integration by parts

Compute a weighting function using Malliavin calculus (Fournié et al. (97), Benhamou (00))

Compute the Vector of perturbation numerically

Work of Avellaneda, Gamba (00)


Slide 10

Comparison of the Methods

All these techniques try to avoid differentiating the payoff function:

Likelihood ratio

—Weight = likelihood ratio

—Advantage: easy to use

—Drawback: requires to know the exact form of the density function

WeightPayoffETheGreek *

,ln TSp

,ln

,,ln

SpSFE

dSSpSpSFSFE

T

T


Slide 11

Comparison of the MethodsContinued

Malliavin method:

– Does not require knowing the density only the diffusion

– Weighting function independent of the payoff

– Very general framework

– Infinity of weighting functions

Numerical estimation of the weighting function

– Other way of deriving the weighting function

– Inspired by Kullback Leibler relative entropy maximization

Spirit close to importance sampling


Slide 12

The Best Weighting Function?

There is an infinity of weighting functions:

—Can we characterize all the weighting functions?

—Can 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?

Practical point of view:

—Which option(s)/ Greek should be preferred? (importance of maturity, volatility)


Slide 13

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”

Notations: general diffusion

first variation process

Malliavin derivative

Skorohod integral

tttt dWXtdtXtdX ,,

tttt dWXtx

dtXtx

dY ,,

TtXD


Slide 14

How to Derive the Malliavin Weights?

Integration by parts:

Chain rule

Greeks is to compute

,, ' TTT XXEXE

TtTTt XDXEXDE '

TtT XDEXE


Slide 15

Necessary and Sufficient Conditions

Condition

Expressing the Malliavin derivative

TtTTT XDXEXXE '' ,

TTtTT XXDEXXE ||,

TT

T

ttTT XXY

YXtEXXE |,| 1,


Slide 16

Minimal Weighting Function?

Minimum variance of

Solution: The conditional expectation with respect to :

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

WeightXE T

TX TXWeightE |


Slide 17

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


Slide 18

II. Typology of Options and Practical Use


Slide 19

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)

—Use of localization formula to target the discontinuity point

Tx

WSfeE TT

rT


Slide 20

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


Slide 21

Comparison Call and Digital

For a call

For a Binary option

OKSKSE xT

xT

2/12

OEE xT

xT

xT

xT SKSKSKS

2/12/12 111


Slide 22

Simulations (Corridor Option)


Slide 23

Simulations (Binary Option)


Slide 24

Simulations (Call Option)

0.015

0.0175

0.02

0.0225

1 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Simulations Number

Gam

ma

Val

ue

Malliavin Simulation

Finite Difference

Exact value 0.02007


Slide 25

Industrial Use

Fast Greeks formulae can be derived easily in the case of:

—Market models (with payoff like Asian cap knock-out, Asian digital cap…etc)

—Stochastic volatility models homogeneous (like Heston model)

Fast Greeks particularly useful for path-dependent payoffs


Slide 26

II. Other Developments: Smart Calibration, Conditional

Expectations and Design of Efficient Monte Carlo Engines


Slide 27

Smart Calibration

When using calibration algorithms, one needs to compute gradient with respect to various model parameters

One can use localization formula to isolate the discontinuity of the payoff function to get faster estimate of the gradient


Slide 28

Conditional Expectation

Conditional expectation can be seen as a Dirac function in one point. To smoothen payoff, one can do integration by parts like for the Greeks

Typical example is in Heston model, to compute the conditional volatility

T

TTTT XGE

XGXFEXGXFE

0

00|


Slide 29

Conditional Volatility in Heston Model SSE TT |2


Slide 30

Design of a Generic Risk Engine for Monte Carlo Trades

According to the payoff profile, at parsing time, should branch or not on Malliavin calculus weighting formula and use a localization formula

When distributing the various trades across the different computers of the pool, should aggregate them according to trades requiring same Malliavin weighting


Slide 31

Conclusion

Malliavin weights enable to derive weights knowing only the diffusion coefficients

Combined with the localization of the discontinuity, method quite powerful

Extensions:

—Use of vega-gamma parity in homogeneous models

—Extension to jump diffusion models

Documents

Smart Monte Carlo: Various Tricks Using Malliavin Calculus Quantitative Finance, NY, Nov 2002 Eric Benhamou [email protected] Goldman Sachs International