----------------------------------------------------- # # # Chapter 9: The Risk Neutral Pricing Measure # # for the Black-Scholes Model # # # ----------------------------------------------------- Summary: This is the continuous time version of the chapter 3 on real world and risk neutral probabilities in the binomial model. Let us quickly recall the pri- cing logic of chapters 2 and 3 on the binomial model: Arbitrary option payoffs could be replicated by a suitable trading strategy in the underlying. The money which is necessary to set up such a replicating stra- tegy is the option price. In chapter 3 we saw then that we can actually write this price as an expectation value with respect to some stochastics, where this stochastics is actually not given as a result of some statistical analysis of the underlying dynamics (that is, no "real world" pricing), but that stochastics merely came from the requirement that in our basic replication equation V_N = V_0 + sum_{k=1}^N delta_{k-1}*(S_k - S_{k-1}) (1) the sum on the right hand side of (1) should simply vanish under this expectation value. In continuous time, (1) is substituted by the Ito- integral V_T = V_0 + int_0^T delta_t * dS_t = H(S_T) (2) with H(S_T) being the option payoff (which also can be path-dependent) and again we are able to arrive at a compact pricing equation V_0 = E[ H(S_T) ] (3) if we define the stochastics with respect to which the expectation on the right hand side of (3) is taken, in a suitable way. It is a quite fundamental result of mathematical finance that if the real world dynamics of the underlying S_t is supposed to be a Black-Scholes dynamics with some drift mu, dS_t/S_t = mu*dt + sigma*dx_t (4) with dx_t a Brownian motion, then the stochastics which has to be used in (3) is given by a Black-Scholes dyna- mics with exactly the same volatility sigma, but with mu completely dropped out and substituted by the risk free interest rate r, dS_t/S_t = r*dt + sigma*dx_t (5) As a consequence, Black-Scholes option prices do not de- pend on the real world drift parameter mu, but only de- pend on the risk free rate r and the volatility sigma.

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