Abstract
Pricing American-style options presents unique challenges due to an opportunity to exercise before maturity, which makes it an optimal stopping problem. Therefore, closed-form solutions have yet to be found. The thesis revises methodologies such as the binomial tree and finite difference methods, paying special attention to the Least- Squares Monte Carlo (LSMC) approach, which offers an elegant numerical solution to the optimal stopping problem for American options, enabling the estimation of continuation functions through cross-sectional regressions.A key contribution of this thesis lies in providing an extensive survey on the adaptation of the LSMC approach for estimating the Greeks, particularly Vega, in both constant and stochastic volatility environments. The study provides a mathematical framework for this extension and explores the nuances of sensitivity estimation in complex market environments, offering insights into the behavior of option prices and Greeks under varying conditions.
Simulation studies and real data analysis form an integral part of the thesis, al- lowing for the empirical validation of the methodologies. By analyzing simulated and real data, as well as real-market scenarios, the thesis evaluates the effectiveness of the proposed approach in practical settings, demonstrating its robustness and applicability in options pricing and hedging strategies.
We found that the algorithms work well for in-the-money (ITM) options, whereas for at-the-money (ATM) and out-of-the-money (OTM) options, which are highly sensitive to changes in volatility, the specification of the initial state in volatility is crucial. We also found that the GARCH Diffusion Stochastic Volatility Model balanced variance across the range of initial volatility states, providing a lower bound on the price of an option, which makes the value function significantly less sensitive toward volatility specification. Additionally, we found that the algorithms produce Vega with smaller bias and variance for larger initial volatilities. We tried to assess sensitivity of the algorithms using historical volatility, external aggregate implied volatilities, and data-induced implied volatilities. The latter has shown the best result with a slight systematic upward bias, enabling us to estimate the Greeks. The estimates of Delta are most robust across different algorithms and parameters, whereas Vega has a significantly larger bias and variance for longer-term (less liquid) options, which is exacerbated in the case of small initial volatilities.
Finally, we showcase how the algorithm can be applied to real data to create a hedge. With strategies like protective put, one can use this algorithm to gauge their risk exposure that comes with adding options to a portfolio. Creating a Vega-neutral portfolio has revealed that the hedge is very stable within a trading day. Nevertheless, it is a single example, and further confirmation will require a deeper investigation of the data.