Stochastic approximation methods for American type options

ABSTRACT

Stochastic approximation methods for rewards of American type options are studied. Pay-off functions are non random possibly discontinuous functions or random càdlàg functions. General conditions of convergence for binomial, trinomial, and skeleton reward approximations are formulated. Underlying log-price processes are assumed to be random walks. These processes are approximated by log-price processes given by random walks with discrete distributions of jumps. Backward recurrence algorithms for computing of reward functions for approximating log-price processes are given. These approximation algorithms and their rates of convergence are numerically tested for log-price processes represented by Gaussian and compound Gaussian random walks. Comparison of the above approximation methods is made.

Keywords:

• American type option
• Binomial approximation
• Compound Gaussian random walk
• Convergence of rewards
• Gaussian random walk
• Geometric random walk
• Markov log-price process
• Random pay-off
• Rate of convergence
• Reward function
• Skeleton approximation
• Trinomial approximation

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 91G20, 91G60, 60J22
• Secondary: 65C40, 62L15