A Partial Differential Equation Approach to Three Problems in Finance: Barrier Option Pricing, Optimal Asset Liquidation and Insider Trading
Barger, Weston David
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We examine three problems in mathematical finance. These problems broadly fall under the sub-disciplines of contract pricing and optimal execution of orders on an exchange under price impact. The first problem deals with the pricing of contingent claims, a classical problem in mathematical finance. After a brief introduction to pricing, we derive asymptotic expansions for the prices of a variety of European and barrier-style claims in a general local-stochastic volatility setting. Our method combines Taylor series expansions of the diffusion coefficients with an expansion in the correlation parameter between the underlying asset and volatility process. We provide rigorous accuracy results for European-style claims which depend explicitly on the correlation parameter. For barrier-style claims, we include several numerical examples to illustrate the accuracy and versatility of our approximations. We then turn our attention to the problem of optimal execution. We assume a continuous-time price impact model similar to that of Almgren-Chriss but with the added assumption that the price impact parameters are stochastic processes modeled as correlated scalar Markov diffusions. For a fixed trading horizon, we perform coefficient expansion on the Hamilton-Jacobi-Bellman equation associated with the trader's value function. The coefficient expansion yields a sequence of partial differential equations that we solve to give closed-form approximations to the value function and optimal liquidation strategy, which is given in feedback form. We examine some special cases of the optimal liquidation problem and give financial interpretations of the approximate liquidation strategies in these cases. We then provide numerical examples to demonstrate the efficacy of our approximations. The third problem we consider deals with insider trading under price impact. We consider an exponentially risk-averse (or risk-neutral) trader who wishes to capitalize on inside information about the true value of an asset that is traded on an exchange. The insider faces a market maker who desires to set a fair price for the asset and attempts to do so by leveraging information contained in the aggregate order flow. We also assume that the insider faces a trading cost that is a linear function of the insider's trading speed. We give equilibrium strategies for the insider and the market maker in both the single auction and continuous-time settings. While the discrete-time equilibrium was given previously, we expand upon it by performing an expansion in small trading cost to analyze the behavior as the cost tends towards zero. A continuous-time equilibrium is given in terms of the solution to a forward-backward ordinary differential equation, which we arrive at by applying nonlinear filtering and dynamic programming in continuous time. We then give some comparative statics about the insider and market maker's strategies and numerically explore the effects of varying model parameters.
- Applied mathematics