----------------------------------------------------- # # # Chapter 7: The Black-Scholes Equation # # # ----------------------------------------------------- Summary: In the second chapter on the Binomial model we saw that an arbitrary option payoff can be replicated with a suitable trading strategy in the underlying. The value V_k of the replicating portfolio at time t_k was shown to fulfill a recursion relation which allowed us to calculate V_k from the portfolio values at time t_{k+1}, V_{k+1}. Now, at maturity T = t_N, the value V_N of the replicating strategy has to be equal to the option payoff H(S_N) which is a given function. Thus, using these recur- sion relations, we were able in chapter 2 to calculate V_0, the initial amount which is necessary to set up the replicating strategy, which is nothing else than the option price, in the Binomial model. Now, we take the Binomial model of chapter 5, which is an approximate model for the Black-Scholes model, and consider the continuous time limit of the recursion relations for the V_k, the value of the replicating portfolio at time t_k, which is the same as the option price at time t_k. If we denote these values in continuous time simply by V_t = V(t,S_t), where we also made the dependence of V with res- pect to the underlying price S_t explicit, then this con- sideration leads to a partial differential equation for the option price V = V(t,S_t). This partial differential equation is typically referred to as "the Black-Scholes equation".

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