Derivative Securities – Fall 2012
نویسنده
چکیده
(a) The link between risk-neutral expectations and PDE’s. We have discussed two apparently different approaches to the valuation of a European option: (i) take the discounted risk-neutral expected payoff, or (ii) solve the Black-Scholes PDE. Let’s show now that these two approaches are equivalent. First, consider options on a forward price. We saw long ago that in the discrete time setting, the forward process satisfies Ft = ERN[FT ] for any t < T . (In the terminology we’ll introduce soon, Ft is a martingale when we calculate expectations using the risk-neutral measure.) In the continuous-time setting, this means the SDE for F under the risk-neutral measure has no dt term. If in addition F is lognormal then its SDE under the risk-neutral measure must be
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