Smart Monte Carlo: Various Tricks Using Malliavin Calculus

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Smart Monte CarloVarious Tricks Using Malliavin
Calculus Quantitative Finance, NY, Nov 2002 Eric
Benhamou eric.benhamou_at_gs.com Goldman Sachs
International
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Agenda

1. Motivation for Fast Monte Carlo Engines
2. Smart Computation of the Greeks
3. Typology of Options and Practical Use
4. Other Developments Smart Calibration,
   Conditional Expectations and Design of Efficient
   Monte Carlo Engines

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I. Motivation for Fast Monte Carlo Engines
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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

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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?

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II. Smart Computation of the Greeks
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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

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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

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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)

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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

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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

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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)

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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

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How to Derive the Malliavin Weights?

• Integration by parts
• Chain rule
• Greeks is to compute

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Necessary and Sufficient Conditions

• Condition
• Expressing the Malliavin derivative

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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

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For European Options, BS

• Type of Malliavin weighting functions

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II. Typology of Options and Practical Use
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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

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Finite Difference Versus Malliavin Method

• Malliavin weighted scheme not payoff sensitive
• Not the case for bump and re-price
• Call option

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Comparison Call and Digital

• For a call
• For a Binary option

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Simulations (Corridor Option)
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Simulations (Binary Option)
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Simulations (Call Option)
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Industrial Use

• Fast Greeks formulae can be derived easily in the
  case of
• Market models (with payoff like Asian cap
  knock-out, Asian digital capetc)
• Stochastic volatility models homogeneous (like
  Heston model)
• Fast Greeks particularly useful for
  path-dependent payoffs

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II. Other Developments Smart Calibration,
Conditional Expectations and Design of Efficient
Monte Carlo Engines
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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

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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

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Conditional Volatility in Heston Model
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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

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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