Longer Research Statement

My research interests are in stochastic processes and mathematical finance. In particular, my dissertation provides a weak existence result which has application to the financial engineer’s calibration problem and essentially generalizes an earlier result of Gyöngy [9]. In this research statement I will briefly review the calibration problem, show how my work fits into the literature, and discuss future research goals. As is well known, the Black-Scholes model does not do a very good job of capturing the riskneutral dynamics for many price processes. Local volatility models attempt to correct this shortcoming by replacing the constant volatility parameter with a function of time and level. Although Dupire [8] has shown that this function may be implied from the full term structure of European option prices across all strikes, in practice the financial engineer has only a finite number of prices, and she must exercise care when interpolating these prices to avoid introducing arbitrage. As an alternative to this approach, Brigo and Mercurio [3, 4] show how to choose the local volatility function so that European option prices have a particularly nice form. Using the Fokker-Planck or Kolmogorov forward equation, Brigo and Mercurio choose the local volatility function in such a way that the risk-neutral one-dimensional marginal distributions of the price process are mixtures of lognormal distributions and European option prices are convex combinations of Black-Scholes prices. Related work by Derman & Kani and Britten-Jones & Neuberger [5, 7] has shown how to construct a local volatility model with the same European option prices as a given stochastic volatility model. This line of research essentially culminated in the rediscovery of the following theorem of Gyöngy.