Ivancevic_Applied-Diff-Geom

1026 Applied Differential Geometry: A Modern Introductiona suitable model for the time evolution of the asset prices. The assumptionof the BSM model is that the price S of an asset is driven by a Brownianmotion and verifies the stochastic differential equation (SDE) [Hull (2000);Paul and Baschnagel (1999)]dS = µSdt + σSdw, (6.35)which, by means of the Itô lemma, can be cast in the form of an arithmeticBrownian motion for the logarithm of Sd(ln S) = Adt + σdw, (6.36)where σ is the volatility, A = ( µ − σ 2 /2 ) , µ is the drift parameter and wis the realization of a Wiener process. Due to the properties of a Wienerprocess, (6.36) may be written asd(ln S) = Adt + σɛ √ dt, (6.37)where ɛ follows from a standardized normal distribution with mean 0and variance 1. Thus, in terms of the logarithms of the asset pricesz ′ = ln S ′ , z = ln S, the conditional transition probability p(z ′ |z) tohave at the time t ′ a price S ′ under the hypothesis that the pricewas S at the time t < t ′ is given by [Paul and Baschnagel (1999);Bennati et. al. (1999)]p(z ′ |z) ={1√2π(t′ − t)σ exp − [z′ − (z + A(t ′ − t))] 2 }2 2σ 2 (t ′ , (6.38)− t)which is a gaussian distribution with mean z +A(t ′ −t) and variance σ 2 (t ′ −t). If we require the options to be exercised only at specific times t i , i =1, · · · , n, the asset price, between two consequent times t i−1 and t i , willfollow (6.37) and the related transition probability will be{1p(z i |z i−1 ) = √ exp − [z i − (z i−1 + A∆t)] 2 }2π∆tσ2 2σ 2 , (6.39)∆twith ∆t = t i − t i−1 .A time–evolution model for the asset price is strictly necessary in atheory of option pricing because the fair price at time t = 0 of an optionO, without possibility of anticipated exercise before the expiration date ormaturity T (a so–called European option), is given by the scaled expectation