Approximating stochastic volatility by recombinant trees

TL;DR

This paper introduces a method to approximate stochastic volatility models, specifically the Heston model, using recombinant trees, enabling efficient pricing of various options with proven convergence properties.

Contribution

It develops a novel recombinant tree construction for stochastic volatility models, providing a practical and convergent approach for option pricing.

Findings

Efficient numerical implementation for American and European options.

Proved weak and extended weak convergence of the approximation.

Applicable to diverse pay-off types including barrier, lookback, and Asian options.

Abstract

A general method to construct recombinant tree approximations for stochastic volatility models is developed and applied to the Heston model for stock price dynamics. In this application, the resulting approximation is a four tuple Markov process. The first two components are related to the stock and volatility processes and take values in a two-dimensional binomial tree. The other two components of the Markov process are the increments of random walks with simple values in $\{-1,+1\}$. The resulting efficient option pricing equations are numerically implemented for general American and European options including the standard put and calls, barrier, lookback and Asian-type pay-offs. The weak and extended weak convergences are also proved.