Arbitrage in Stock Index Futures One and Two Dimensional Problems

Abstract

Stock indexes, unlike stocks, options, cannot be trader directly, so futures based on stock indexes are primary way of trading stock indexes. There are three type of investors in various financial markets, namely, speculator, hedger and arbitrager. In this thesis, we are interested in the arbitrage profit in stock index futures. This thesis mainly focus on on pricing options whose payoff is based on simple arbitrage profit in stock index futures and plotting their early exercise boundaries. We consider both one dimensional and two dimensional problems, for each we sub-divide as 'no position limits' case and 'with position limits' case.In one dimensional problem, we use Brownian Bridge process to model simple arbitrage profit. A one dimensional PDE for the options is derived. In two dimensional problem, we add one mean-reverting stochastic differential equation to model order imbalance. A two dimensional PDE for the options is derived. We also take into account of transaction costs and position limits and form complete models.For numerical experiement, we use fully implicit and Crank-Nicolson scheme to solve the variational inequality numerically. To handle American option type, we adopt projected SOR method. Numerical Results of the early exercise boundaries and option values are given and analyzed. These early exercise boundaries give us the optimal arbitrage strategy. We discuss various parameter effects on option values and early exercise boundary, for one dimensional problem, while we also examine the order imbalance impacts on early exercise boundary, for two dimensional problem. We also compare the numerical results between the 'no position limits' and 'with position limits' models, and find the optimal trading strategy is exactly the same for both cases.