##### Department of Mathematics,

University of California San Diego

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### Math 288 - Probability

## Knut Solna

#### UC Irvine

## Multiscale Stochastic Volatility Asymptotics

##### Abstract:

We consider the problem of pricing derivative securities in an environment of uncertain and changing market volatility. The popular Black-Scholes model relates derivative rices to current stock prices through a constant volatility parameter. The natural extension of this approach is to model the volatility as a stochastic process. In a regime with a multiscale or bursty stochastic volatility we derive an generalized pricing theory that incorporates the main effects of a stochastic volatility. We consider high frequency S&P 500 historical pricing data and analyze these with a view toward identifying important time scales and systematic features. The data shows a periodic behavior that depends on both maturity dates and also the trading hour. We examine the implications of this for modeling and option pricing.

Host: Ruth Williams

### February 27, 2003

### 9:00 AM

### AP&M 6438

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