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Smart Monte Carlo:
Various Tricks Using Malliavin Calculus
Quantitative Finance, NY, Nov 2002
Eric Benhamou
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
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
Slide 3
I. Motivation for Fast Monte Carlo Engines
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
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
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
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?
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
Slide 6
II. Smart Computation of the Greeks
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
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
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
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
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
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)
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
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
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
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
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
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)
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
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
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
Slide 14
How to Derive the Malliavin Weights?
Integration by parts:
Chain rule
Greeks is to compute
,, ' TTT XXEXE
TtTTt XDXEXDE '
TtT XDEXE
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
Slide 15
Necessary and Sufficient Conditions
Condition
Expressing the Malliavin derivative
TtTTT XDXEXXE '' ,
TTtTT XXDEXXE ||,
TT
T
ttTT XXY
YXtEXXE |,| 1,
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
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 |
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
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
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
Slide 18
II. Typology of Options and Practical Use
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
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
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
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
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
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
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
Slide 22
Simulations (Corridor Option)
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
Slide 23
Simulations (Binary Option)
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
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
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
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
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
Slide 26
II. Other Developments: Smart Calibration, Conditional
Expectations and Design of Efficient Monte Carlo Engines
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
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
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
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|
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
Slide 29
Conditional Volatility in Heston Model SSE TT |2
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
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
Quantitative Finance, Risk Conference, NY, November 3-4, 2002
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