Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1035z i , in agreement with the request of linear growth. This procedure of selectionof integration points, together with the calculation of the transitionprobability previously described, is the basis of the path integral simulationof the price of a generic option.By applying the results derived above, we have at disposal an efficientpath integral algorithm both for the calculation of transition probabilitiesand the evaluation of option prices. In [Montagna et. al. (2002)] the applicationof the above path–integral method to European and Americanoptions in the BSM model was illustrated and comparisons with the resultswere get with the standard procedures known in the literature were shown.First, the path integral simulation of the probability distribution of thelogarithm of the stock prices, p(lnS), as a function of the logarithm of thestock price, for a BSM–like stochastic model, was given by (6.36). Once thetransition probability has been computed, the price of an option could becomputed in a path integral approach as the conditional expectation valueof a given functional of the stochastic process. For example, the price of anEuropean call option was given by∫ +∞C = e −r(T −t) dz f p(z f , T |z i , t) max[e z f− X, 0], (6.58)−∞while for an European put it will be∫ +∞P = e −r(T −t) dz f p(z f , T |z i , t) max[X − e z f, 0], (6.59)−∞where r is the risk–free interest rate. Therefore just 1D integrals need to beevaluated and they can be precisely computed with standard quadraturerules.6.3.6.3 Continuum Limit and American OptionsIn the specific case of an American option, the possibility of exercise at anytime up to the expiration date allows to develop, within the path integralformalism, a specific algorithm, which, as shown in the following, is preciseand very quick [Montagna et. al. (2002)].Given the time slicing considered above, the case of American optionsrequires the limit ∆t −→ 0 which, putting σ −→ 0, leads to a delta–liketransition probabilityp(z, t + ∆t|z t , t) ≈ δ(z − z t − A∆t).