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By Andreas J. Grau

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In the later Section, we will see Asian options, where pStT (s) is known analytically and EQ [P (S)|StT = s] has to be estimated numerically. Furthermore, we will see Parisian options, where pStT (s) has to be estimated numerically and EQ [P (S)|StT = s] is known at each time step of an induction in backwards time. Feature Extraction means to use as much analytical information in an option pricing process as possible. Suppose that pStT (s) is known and EQ [P (S)|StT = s] is not known. The intuition behind this method is that we interpret f (s) := EQ [P (S)|StT = s] as the payoff of a hypothetical European-style option, which is computed by Monte Carlo simulation.

906]): i i V˜j+1 = max(P (S i , t), Vj+1 ). This solver is sufficient for the purpose of this thesis. However, a better method is given by internal iterations using a penalty method [50]. 1 Overview A challenging problem in option pricing is the evaluation of path dependant options. This chapter presents a method which can increase the convergence of Monte Carlo pricing [21, 23, 53, 32, 81] significantly. The method is extended such that Monte Carlo simulation and numerical integration methods are combined in a consistent framework called Feature Extraction.

1. The figure shows a stock-price index and the corresponding moving average. The analysts claim that there is information about the future in such charts. However, we will not discuss whether this is true or not, we will use the moving average in a different setting, as a strike of stock options. This idea is simple and leads to a product which is easy to understand for investors. But, only a few options which have a moving average as a strike or as an underlying are actively traded [70]. More common is the moving average computation in issuer-call features of some fixed income securities [71].

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Applications of Least-Squares Regressions to Pricing and Hedging of Financial Derivatives by Andreas J. Grau

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