Optimal hedging of asian options

Abstract

The Black-Scholes option pricing model (1973) illustrates the modern theories of option valuation and hedging strategy. Black and Scholes used geometric Brownian motion to model stock price dynamics and proposed a delta-neutral hedging portfolio. The Black-Sholes model is based on the concepts of risk-neutral measure, stochastic calculus and no arbitrage principle. Solving the Black-Scholes partial differential equation gives rise to the Black-Scholes model for pricing European-style options.

The delta-neutral hedging in the Black-Scholes model assumes ‘perfect markets’ and requires continuous recalibration of the pricing model. This project analyzes the delta-neutral portfolio and the model assumptions.

The influences of various factors on option price are discussed, based on the Black-Scholes formula. However, there is a mathematical error in the Black-Scholes model and the inconsistency in the derivation is discussed.

This project compares alternative option pricing models in which different features of the stock dynamics are captured. Measuring the hedging performances of pricing models is discussed. Also, another hedging strategy – minimum variance hedging – and its approach in obtaining the hedge ratio in the hedging portfolio are explored.

The limitation of the Black-Scholes option pricing model is that it can be only used for pricing path-independent options. This project introduces an optimal hedging strategy for path-dependent Asian options, which takes into account the historical data.

This project proposes a new concept – variational hedging – for hedging path-dependent Asian options, based on Hamilton’s principle and variational method. Variational hedging suggests a functional based on a newly defined Lagrangian for the dynamics of the hedging portfolio. The functional represents the total variance of the portfolio value over the specified period. Variational hedging is actually a variational problem that seeks the option price function which minimizes the hedging functional. Thus, the total fluctuations in the portfolio value over the specified period are minimized.