Explicit solutions of discrete-time quadratic optimal hedging strategies for European contingent claims

摘要

We consider the problem of optimally hedging a (path-dependent) European contingent claim (ECC) with its underlying in a discrete-time framework. Specifically, we consider two quadratic optimal hedging strategies : minimum-variance hedging in a risk-neutral measure and optimal local-variance hedging in a market probability measure. The objective function for the former is the variance of the hedging error calculated in a risk-neutral measure and the latter optimizes the variance of the mark-to-market value of the portfolio over a trading interval in a market probability measure. The main aim of the paper is to derive explicit closed form solutions to hedge different types of ECCs using the above mentioned quadratic hedging schemes. To arrive at closed-form solutions, we assume geometric Brownian motion (GBM) as the stochastic model for the underlying asset prices. These explicit solutions when used instead of complex Monte-Carlo based solutions makes the proposed hedging solution well suited for computer implementation. In addition, we outline a mechanism to implement an automated trading position evaluation system based on the proposed hedging solutions.