Ivancevic_Applied-Diff-Geom

1034 Applied Differential Geometry: A Modern Introductionconstraints, as in the case of interesting path–dependent options, such asAsian and barrier options [Hull (2000)].Integration PointsThe above illustrated method represents a powerful and fast tool tocalculate the transition probability in the path integral framework and itcan be employed if we need to value a generic option with maturity Tand with possibility of anticipated exercise at times t i = i∆t (n∆t = T )[Montagna et. al. (2002)]. As a consequence of this time slicing, one mustnumerically evaluate n − 1 mean values of the type (9), in order to checkat any time t i , and for any value of the stock price, whether early exerciseis more convenient with respect to holding the option for a future time. Tokeep under control the computational complexity and the time of execution,it is mandatory to limit as far as possible the number of points for theintegral evaluation. This means that we would like to have a linear growth ofthe number of integration points with the time. Let us suppose to evaluateeach mean value∫E[O i |S i−1 ] = dz i p(z i |z i−1 )O i (e zi ),with p integration points, i.e., considering only p fixed values for z i . To thisend, we can create a grid of possible prices, according to the dynamics ofthe stochastic process as given by (6.37)z(t + ∆t) − z(t) = ln S(t + ∆t) − ln S(t) = A∆t + ɛσ √ ∆t. (6.57)Starting from z 0 , we thus evaluate the expectation value E[O 1 |S 0 ] withp = 2m+1, m ∈ N values of z 1 centered on the mean value E[z 1 ] = z 0 +A∆tand which differ from each other of a quantity of the order of σ √ ∆tz j 1 = z 0 + A∆t + jσ √ ∆t,(j = −m, . . . , +m).Going on like this, we can evaluate each expectation value E[O 2 |z j 1 ] get fromeach one of the z 1 ’s created above with p values for z 2 centered around themean valueE[z 2 |z j 1 ] = zj 1 + A∆t = z 0 + 2A∆t + jσ √ ∆t.Iterating the procedure until the maturity, we create a deterministicgrid of points such that, at a given time t i , there are (p − 1)i + 1 values of