With the development of financial markets and increasing demand for managing risk exposure, researchers and practitioners have developed various financial instruments over the years. Options, Futures, Forwards, Swaps are few examples of such instruments. There are many financial models design to price such derivatives and they all have one thing in common: arbitrage free valuation of these derivative contracts.In this thesis we focus on pricing mechanism of one the widely traded derivatives: American option. We employ HJM forward modeling approach introduced by Heath, Jarrow and Morton (1992). HJM model is originally introduced as an alternative method to bond pricing. Traditional bond pricing is done via short rate modeling while HJM method attempt to price bonds via modeling the evolution of entire yield curve. In recent years, Schweizer and Wissel (2008) and Carmona and Nadtochiy (2009) extend the forward modeling idea to equity market by modeling forward volatility allowing researchers to look at a dynamic curve which relax the Black - Scholes constrain of constant volatility. This modeling paradigm also allows easy calibration to market data, which makes the HJM model popular among practitioners. Here we propose an alternative approach to value American type options in the spirit of HJM approach. Since American option is essentially an optimal stopping problem, it's value given by the Snell envelop of the value process. By adapting HJM method method using forward drift we formulate a new value process of American option. We propose a new value function, a new stopping criteria and a new stopping time. We investigate this new method in both additive and multiplicative model settings using the forward modeling approach. Then we give an example for the new proposed method under additive model to solve American option pricing problem theoretically.Numerical investigation of the additive and multiplicative models is carried out for Option Matrix data for August 2007 to August 2015 using three methods: principal component analysis, robust principal component analysis and Karhunen - Loeve transformation.
