Ivancevic_Applied-Diff-Geom

1028 Applied Differential Geometry: A Modern Introduction(usually large) number of expectation conditional probabilities.Standard Numerical ProceduresTo value derivatives when analytical formulae are not available, appropriatenumerical techniques have to be advocated. They involve the useof Monte Carlo (MC) simulation, binomial trees (and their improvements)and finite–difference methods [Hull (2000); Wilmott et. al. (1993)].A natural way to simulate price paths is to discretize (6.37) asln S(t + ∆t) − ln S(t) = A∆t + σɛ √ ∆t,or, equivalently,[S(t + ∆t) = S(t) exp A∆t + σɛ √ ]∆t , (6.42)which is correct for any ∆t > 0, even if finite. Given the spot price S 0 , i.e.,the price of the asset at time t = 0, one can extract from a standardizednormal distribution a value ɛ k , (k = 1, . . . , n) for the random variable ɛ tosimulate one possible path followed by the price by means of (6.42):√ ]S(k∆t) = S((k − 1)∆t) exp[A∆t + σɛ k ∆t .Iterating the procedure m times, one can simulate m price paths{(S 0 , S (j)1 , S(j) 2 ,. . . , S n(j) ≡ S (j)T) : j = 1, . . . , m} and evaluate the price of the option. Insuch a MC simulation of the stochastic dynamics of asset price (MonteCarlo random walk) the mean values E[O i |S i−1 ], i = 1, . . . , n are given byE[O i |S i−1 ] = O(1) i + O (2)i+ · · · + O (m)iwith no need to calculate transition probabilities because, through the extractionof the possible ɛ values, the paths are automatically weighted accordingto the probability distribution function of (6.39). Unfortunately,this method leads to an estimated value whose numerical error is proportionalto m −1/2 . Thus, even if it is powerful because of the possibilityto control the paths and to impose additional constrains (as it is usuallyrequired by exotic and path-dependent options), the MC random walk isextremely time consuming when precise predictions are required and appropriatevariance reduction procedures have to be used to save CPU time[Hull (2000)]. This difficulty can be overcome by means of the method of them,