Variance Reduction for Monte Carlo Methods to Evaluate Option Prices under Multi-factor Stochastic Volatility Models : 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.
Part I: Introduction to Stochastic Volatility Models : Part I: Introduction to Stochastic Volatility Models
Data: S&P500 Index : Data: S&P500 Index
Modeling Index Processes: Geometric Brownian Motion : Modeling Index Processes: Geometric Brownian Motion
Derivatives Pricing Problem : Derivatives Pricing Problem for example, the price of a derivative is given by
K: strike price r : interest rate (risk free)
T : maturity date t : current date When the payoff is given as it defines a European call option.
Monte Carlo Simulations : Monte Carlo Simulations Variance reduction
Black-Scholes pricing PDE : Black-Scholes pricing PDE Numerical PDE Scheme
Black-Scholes Formula : Black-Scholes Formula N(.): cumulative standard normal distribution When is a constant The Market is complete!
^SPX Quoted at 03/24/04 and Expired at 04/04/04 : ^SPX Quoted at 03/24/04 and Expired at 04/04/04
Inverse Problem: implied volatility vs moneyness : Inverse Problem: implied volatility vs moneyness
Market Smiles ! : Market Smiles ! Volatility is certainly NOT a constant!
Implied Vol vs S&P 500 (fear index!) : Implied Vol vs S&P 500 (fear index!)
Stylized Facts in modeling (random) Volatility Process : Stylized Facts in modeling (random) Volatility Process Mean reversion (incomplete market!)
Stylized Facts in modeling (random) Volatility Process : Stylized Facts in modeling (random) Volatility Process Mean reversion
Leverage effect
Stylized Facts in modeling (random) Volatility Process : Stylized Facts in modeling (random) Volatility Process Mean reversion
Leverage effect
Time scales
Stylized Facts in modeling (random) Volatility Process : Stylized Facts in modeling (random) Volatility Process Mean reversion
Leverage effect
Time scales
Fatter tailed return distribution
One-Factor Stochastic Volatility Model : One-Factor Stochastic Volatility Model Under the pricing measure (not unique)
Reproduce Smile from Stochastic Volatility Models : Reproduce Smile from Stochastic Volatility Models