A Stochastic Approach to the Valuation of Barrier Options in Heston’s Stochastic Volatility Model

Abstract

In the valuation of continuous barrier options the distribution of the first hitting time plays a substantial role. In general, the derivation of a hitting time distribution poses a mathematically challenging problem for continuous but otherwise arbitrary boundary curves. When considering barrier options in the Heston model the non-linearity of the variance process leads to the problem of a non-linear hitting boundary. Here, we choose a stochastic approach to solve this problem in the reduced Heston framework, when the correlation is zero and foreign and domestic interest rates are equal. In this context one of our main findings involves the proof of the reflection principle for a driftless Ito process with a time-dependent variance. Combining the two results, we derive a closed-form formula for the value of continuous barrier options. Compared to an existing pricing formula, our solution provides further insight into how the barrier option value in the Heston model is constructed. Extending the results to the general Heston framework with arbitrary correlation and drift, we obtain approximations for the joint random variables of the Ito process and its maximum in a weak sense. As a consequence, an approximate formula for pricing barrier options is established. A numerical case study is also performed which illustrates the agreement in results of our developed formulas with standard finite difference methods.