----------------------------------------------------- # # # Chapter 8: Stochastic Calculus # # and the Ito-Formula # # # ----------------------------------------------------- Summary: Recall the main result of the first chapter on trading strategies: Hold delta_k stocks at the end of day t_k and close the position on day t_N. Suppose V_0 is your initial money. Then at time t_N this stra- tegy has generated the (positive or negative) amount V_N = V_0 + sum_{k=1}^N delta_{k-1}*(S_k - S_{k-1}) (1) if S_k denotes the closing price of the traded stock on day t_k. Here we assumed zero interest rates. For non-zero interest rates, the quantities V and S in (1) above have to be substituted by discounted quantities. Now assume that the underlying price is modelled by a Black-Scholes model in continuous time. Then the sum on the right hand side of (1) is a sum over stochastic quantities. If we consider the continuous time limit of (1), then an obvious notation for (1) would be V_T = V_0 + int_0^T delta_t * dS_t (2) with a stochastic differential dS_t. Apparently, in equation (1) it makes a difference whether we have delta_{k-1} = delta_{k-1}(s_0,...,S_{k-1}) (3a) meaning we can use past or current information, or delta_{k-1} = delta_{k-1}(s_0,...,S_{k-1},S_k) (3b) which would mean that we can use future information: The case (3b) would mean that when we adjust our stock position at the end of day t_{k-1}, we already know which underlying price S_k will realize tomorrow and of course this information then can be used to make unlimited profit. Thus, mathematically, it is a crucial requirement that the delta's showing up in (1) or then in the continuous time version (2), fulfill the require- ment (3a), they cannot depend on future information. In the more mathematical framework of stochastic inte- grals, the condition (3a) leads to the notion of Ito- integrals, which are, at least in the context of option pricing, apparently the right ones to work with, whereas a more general condition (3b) may lead to so called Stratonovich integrals, and, in the continuous time limit, these objects have indeed different limits. From a prac- tical point of view this sounds quite obvious, but from a more mathematical perspective, for someone who is used to numerically calculate standard (that is, non-stocha- stic) integrals just by discretizing to Riemannian sums, this sounds a bit surprising. Thus, in this chapter these issues are considered more closely.

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