In this chapter, we present methods to value American options in the Heston model. We first present the simulation-based algorithm of Longstaff and Schwartz (2001). Next, we present a finite difference method, the explicit method, for American options. This method will be presented in its entirety in Chapter 10. We present the bivariate tree of Beliaeva and Nawalkha (2010), an adaptation of the trinomial tree in which the stock price and the volatility evolve along separate trees. We also cover the method of Medvedev and Scaillet (2010), which approximates the American option price using an analytic expansion and is thus able to produce option prices very quickly. Finally, we present the method of Chiarella and Ziogas (2006) for the valuation of American calls.
LEAST-SQUARES MONTE CARLO
The Least-Squares Monte Carlo (LSM) algorithm was developed by Longstaff and Schwartz (2001) as a way to price American options using simulation. The algorithm can be applied to any stock price stochastic process that lends itself to simulation. It is especially useful for multi-dimensional processes for which high-dimension trees are difficult to construct. In this section, we implement the method for the Heston model.
Denote by C(ω, s; t, T) the set of cash flows generated by the option along the stock price path ω, conditional on the option not being exercised prior to ...