Variance Reduction for Monte Carlo Methods to Evaluate Option Prices under Multi-factor Stochastic Volatility Models

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Variance Reduction for Monte Carlo Methods to
Evaluate Option Prices under Multi-factor
Stochastic Volatility Models
Sean Han Institute of Mathematics and
its Applications, University of Minnesota March
26 2004.
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Part IIntroduction to Stochastic Volatility
Models
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Data SP500 Index
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Modeling Index ProcessesGeometric Brownian
Motion
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Derivatives Pricing Problem

• for example, the price of a derivative is given
  by
• 
  

When the payoff is given as
it defines a European call option.
K strike price r
interest rate (risk free) T maturity date
t current date
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Monte Carlo Simulations
Variance reduction
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Black-Scholes pricing PDE
Numerical PDE Scheme
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Black-Scholes Formula
is a constant

• The Market is complete!

• When

N(.) cumulative standard normal distribution
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SPX Quoted at 03/24/04 and Expired at 04/04/04
Calls Prices Strikes Puts Prices
SPQDO.X 36 1075 SPQPO.X 13.20
SPXDP.X 29 1080 SPQPP.X 14
SPXDR.X 21 1090 SPQPR.X 19
SPTDT.X 16 1100 SPTPT.X 23
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Inverse Problem implied volatility vs
moneyness
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Market Smiles !
Volatility is certainly NOT a constant!
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Implied Vol vs SP 500(fear index!)
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Stylized Facts in modeling (random) Volatility
Process

• Mean reversion (incomplete market!)

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Stylized Facts in modeling (random) Volatility
Process

• Mean reversion
• Leverage effect

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Stylized Facts in modeling (random) Volatility
Process

• Mean reversion
• Leverage effect
• Time scales

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Stylized Facts in modeling (random) Volatility
Process

• Mean reversion
• Leverage effect
• Time scales
• Fatter tailed return distribution

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One-Factor Stochastic Volatility Model
Under the pricing measure (not unique)
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Reproduce Smile from Stochastic Volatility Models