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By James Newbury

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Extra resources for Applications of Malliavin calculus to the pricing and hedging of Bermudan options

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We now analyse the results of the approximation of the option’s delta. 2 below shows that the approximations of the delta happen to be quite unstable for a number of paths less than 2000. 1) would enable us to achieve a greater level of convergence. Unfortunately, our own implementation involving 38 the control variate method did not yield any satisfactory results, and they have therefore not been included here. 3 precisely shows to what extent the non-localized version of the algorithm does not numerically work.

We take into consideration the associated empirical means: 1/2 (q) (q) N 2 2 q=1 . (tk , Xkδt )(δk (u)) k λ. = (q) N 2 q=1 . (tk , Xkδt ) In the particular case of λk1 , we generally have closed-form expressions, which we shall give explicitly in the case of geometric Brownian motion and the Ornstein-Uhlenbeck process. We now examine the application of the representation formula of the derivative of conditional expectation in order to be able to compute an approximation ˆ of the delta of the option.

Indeed, the results presented here seem to be reasonably in line with ”reference” values when the number of paths is relatively high, but this mechanically increases (and ”squares”) the required CPU time for the computations. This naturally leads us to question of the efficiency of the algorithm in terms of computational cost, especially in comparison with other methods such as the Longstaff-Schwartz algorithm. Bally et al. [1] state that the latter is undenaiably more competitive with regard to the computational speed, although it does not necessarily yield more accurate results.

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