| This dissertation is concerned with the area of Mathematical Finance dealing with the pricing of American-style options. An option is a financial asset giving the owner the right, but not the obligation, to buy or sell another financial asset for a given price (the strike price). Options endowed with an American exercise feature allow its investors to exercise their rights at any time during the life of the options. As the majority of the exchange-traded stock and equity options are American-style, it is understandable that an enormous amount of attention has been drawn to this particular area during the past few decades.;The first chapter of the dissertation presents a simple yet powerful simulation-based approach for approximating the values of prices and Greeks (i.e. derivatives with respect to the underlying spot prices, such as delta, gamma, etc) for American-style options. This approach is primarily based upon the Least Squares Monte Carlo(LSM) algorithm and is thus termed the Modified LSM (MLSM) algorithm. The key to this approach is that with initial asset prices randomly generated from a carefully chosen distribution, we obtain a regression equation for the initial value function, which can be differentiated analytically to generate estimates for the Greeks. This approach is intuitive, easy to apply, computationally efficient and most importantly, provides a unified framework for estimating risk sensitivities of the option price to underlying spot prices. We demonstrate the effectiveness of this technique with a series of increasingly complex but realistic examples.;The second chapter of the dissertation presents another fast yet effective simulation-based approach for estimating the upper bounds of American-style option prices without nested simulations. We derive the analytical representation for the 'correct' martingale that is to be applied to the dual pricing formula to give the correct option price under a generic theoretical setting. This representation turns out to bear clear implication for hedging and is the main contribution of our paper. We further propose a simulation algorithm that directly builds on this analytical form for martingale construction and circumvents the time-consuming sub-simulation part that is usually required in other existing dual simulation procedures. This approach is shown to be capable of producing viable upper bounds as well as significantly reducing the computing time allocated to upper bound estimation. Likewise, we illustrate the effectiveness of our approach with a series of increasingly complex but realistic examples. |
