Explicit Solutions of Discrete-Time Hedging Strategies for Multi-Asset Options

We consider the use of discrete-time quadratic optimal hedging strategies for hedging Multi-asset options. Specifically, the objective of the current work is to provide specialized closed form solutions to hedge Exchange options using two quadratic hedging schemes. The two quadratic optimal hedging strategies that we consider are mean-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 single trading interval in a market probability measure. To arrive at closed-form solutions, we assume geometric Brownian motion (GBM) as the stochastic model for the underlying asset prices. The hedging solutions are expressed in terms of the pricing function of the hedged ECC and the prices of the underlying assets. The motivation to provide these explicit solutions is that the proposed hedging solutions are well suited for computer implementation and reduce compute time and complexity unlike the Monte-Carlo based solution formulations.