American options under stochastic volatility

Tese de mestrado em Matemática Financeira, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2016An option is a contract that gives the holder the right to buy, in the case of a call, or sell, in the case of a put, an underlying asset at a pre-determined strike price. A European option allows the holder to exercise the option only on a pre-determined expiration date, while with an American option the holder can exercise the option at any point in time until the maturity date. Options can incorporate dividends, which are a portion of a company's earning distributed to its shareholders, that can be issued as cash payments, as shares of stock or other property. Black and Scholes (1973) derived a closed form solution for the value of European options with constant volatility, while Heston (1993) provides a solution for European options with stochastic volatility. It was proved that assuming constant volatility leads to considerable mispricing. Bakshi, Cao and Chen (1997) did a series of tests comparing the Black and Scholes (1973) with three models which allow for stochastic volatility. They showed that incorporating stochastic volatility reduces the absolute pricing error by 20% to 70%. For example a call option with the price $1:68, under the Black and Scholes model has an error of $0:78, while with a model with stochastic volatility the error is reduced to $0:42. Hence, models that allow the volatility of the underlying asset to be stochastic better describe the market behavior. Unlike European options, American options do not have a closed form solution for its value with constant or stochastic volatility, due to the fact that the price depends on the optimal exercise policy. The models on American options under stochastic volatility can be separated in two approaches: the Partial Differential Equation, PDE, based and the non PDE based. There are various numerical methods to price American options. For example, Brennan and Schwartz (1977) introduced finite difference methods; the least squares Monte Carlo is a model developed by Longstaff and Schwartz (2001), where the model uses simulations of cash flows generated by the option and compare them to the value of immediate exercise to calculate the price. In Beliaeva and Nawalkha (2010) a bivariate tree is used where two independent trees are created for the stock price and for the variance. Broadie and Detemple (1996) developed a method for lower and upper bounds on the prices of American options based on regression coefficients. In the Clarke and Parrott (1999) model they use the Heston PDE, transformed into a non dimensional form, with a multigrid iteration method to solve the problem of option pricing. Detemple and Tian (2002) determine the exercise region by a single exercise boundary under general conditions on the interest rate and the dividend yield and derive a recursive integral equation for the exercise boundary. In this work, we will develop an implementation based on the Heston model with the explicit method. First, we will derive the Heston PDE, showing how it is used in the method described. Then we will test the accuracy of the results, randomly creating options and using the various methods to price them and calculate the errors of each method