PRICING STOCK OPTIONS UNDER THE HESTON STOCHASTIC-VOLATILITY MODEL VIA A TRINOMIAL TREE IN A REGIME-SWITCHING ECONOMY WITH ADDED TRANSACTION COSTS

Abstract

In this paper we will discuss the procedure on how to develop computationally efficient trinomial-tree method for pricing only for European call options under Heston’s stochastic-volatility model, incorporating both regime-switching approximations and proportional transaction costs. A decorrelation transform developed by Beliaeva and Nawalkha and discretization of the variance process via a continuous-time Markov chain is applied to the Heston model, yielding the reduction of the twodimensional dynamics to a family of coupled one-dimensional processes. On each regime, we build a recombining trinomial lattice whose jumpsizes match the local drift and variance, and we connect the lattices across regimes using the Markov-chain generator. This yields a five-point tree with only O(N) nodes at step N, in contrast to the exponential growth of na¨ıve multinomial schemes [11]. At each section the paper will be as detailed as possible when deriving the PDE variations starting from the famous Heston model [5] ending with the option’s PDE which is dependent on Markov Chain generator. Numerical approximations show that this method achieves less than 1% accuracy on benchmark European calls only for a few dozen variance states and a few thousands time-steps. We further include proportional transaction costs to our grid design of the trinomial tree by penalizing deltas (hedge) at each node, and we demonstrate the realizations on the same example as with plain trinomial tree