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

By Andreas J. Grau

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Compute the probabilities Pi,i+1 (s) of survival for all values of s and i. g. by Monte Carlo simulations, starting at the asset value St0 . 2. 9). 3. 11) and go back to step 2 if i = 0. 4. The price of the path dependent option is given by Vt0 = e−rtT EQ [f (S)] = e−rtT EQ [f (S)|St0 = St0 , I(S, t0 ) = 1]. This algorithm can be extended to Parasian, lookback or similar options without large efforts. 1 Numerical Example: A Parisian Option In this section, the efficiency of the new method will be compared with the classical Monte Carlo method.

The resulting algorithm provides a convergent and practical method for pricing the moving window Asian options as well as other high-dimensional, exercisable securities, which to our knowledge have not yet been solved with reasonable accuracy. 2 Introduction Methods for pricing a large variety of exotic options have been developed in the past decades. Still, the pricing of high-dimensional American style options remains challenging. The price of this kind of option depends on the complete price path not only on the stock price at the final exercise date.

Monte Carlo methods, on the other hand, allow for complex path dependencies (high-dimensional problems) but their efficiency is limited by their relatively low convergence rate as presented in the previous chapter. In this chapter we suggest to combine elements of the Monte Carlo and the direct numerical integration in order to increase the accuracy of the former. We call this method Feature Extraction because the feature of the option’s payoff, which is hard to compute by other means is estimated from a Monte Carlo simulation.

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