Special Colloquium
Cheng Ouyang
Purdue
Differential Geometry, Heat Kernel and Implied Volatility in Local and Stochastic Volatility Models
Abstract: Volatility is a key parameter in the celebrated Black-Scholes model and its later generalizations. A large part of current research on mathematical finance lies on understanding the volatility parameter in various settings. An important concept on this sub ject is the implied volatility, which gives a way to compare prices between different options and to evaluate risk. It also provides a basis for calibrating parameters in various pricing models.
In this talk I will first give a brief introduction of implied volatilities and related background in mathematical finance. Then I show how to use heat kernel expansion to give a near-expiry asymptotic formula for the implied volatility in local volatility models. Once the local volatility case is clear, I will explain what happens in stochastic volatility models and how differential geometry plays a role in this problem. In particular, as we will see, the leading value of the implied volatility could be interpreted as a Riemannian distance under the metric determined by the equation of the price-volatility process.
There will be tea in SEO 300 after the colloquium.
Thursday January 13, 2011 at 3:00 PM in SEO 636