By Kohn R.V.
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More precisely: in the limit δt → 0 the lognormal stock models with different µ’s but the same σ all assign the same values to options. So we may choose µ any way we please – there’s no reason to require that it match the actual expected return of the stock under consideration. The two most common choices are 1. choose µ to be the expected return of the stock nevertheless; or 2. e. µ = r − 21 σ 2 . The latter choice has the advantage that it puts q even closer to 1/2. This is the selection favored by Jarrow–Turnbull and many other authors.
But is it legitimate for describing the value of the option, as determined by arbitrage? This is less clear, since a continuous-time hedging strategy is unattainable in practice. In what sense can we “approximately replicate” the option by trading at discrete times? The Black-Scholes differential equation will help us answer these questions. • The differential equation approach gives fresh insight and computational flexibility. Imagine trying to understand the implications of compound interest without using the differential equation df /dt = rf !
As we shall see in a moment, this strategy is no longer self-financing – but it is nearly so, in a suitable stochastic sense, in the limit δt → 0. ” The answer, of course, is that the hedge portfolio is held fixed from t to t + δt. The following discussion – in which δt is small but not infinitesimal – should help clarify this point. OK, let’s return to that investment bank. The question is: how much additional money will the bank have to spend over the life of the option as a result of its discrete-time (rather than continuous-time) hedging?