Equations of Hamilton-Jacobi type and their applications in finance

We investigate, in three parts, the Hamilton-Jacobi (HJ) and Hamilton-Jacobi-Bellman (HJB) equations and their applications in finance. In the first part, we propose a method for calibrating a volatility surface that matches option prices using an entropy-inspired framework. Starting with a stochastic volatility model for asset prices, we cast the estimation problem as a variational one and we derive a Hamilton-Jacobi-Bellman (HJB) equation for the volatility surface. We study the asymptotics of the HJB equation in the fast mean-reversion regime to obtain a volatility surface. We also incorporate uncertainty in quoted derivative prices by softening the constraints in the HJB equation. We present numerical solutions of our estimation scheme and find that, depending on the softness of the constraints, parameters of the volatility surface related to the implied volatility smile can be calibrated so that they are stable over time. These parameters are essentially the ones found in previous fast mean-reversion asymptotics papers by Fouque, Papanicolaou and Sircar.; In the second part, we investigate the effect that portfolio optimizers and hedgers have on the underlying asset price. We find that portfolio optimizers tend to decrease the volatility of the underlying asset as long as the excess returns of the asset are not too high. Analytical approximations are developed for the change in the portfolio optimizers' optimal holdings and their accuracy is confirmed through numerical solutions of an HJB Equation. In the case of hedgers, we find that there is an important distinction between hedgers that are long the option and hedgers that are short the option. In both cases, hedgers change the drift and diffusion of the underlying risky asset. Those who are long the option tend to push the underlying towards the strike. Numerical calculations that solve both the nonlinear option pricing equation and the probability distribution of the underlying are presented.; In the third part, we present a new semi-discrete central scheme for Hamilton-Jacobi equations on triangular meshes. The scheme incorporates local speeds of propagation into the numerical fluxes. Moreover, the scheme is shown to be monotone in certain cases.