Sample Question #146 (mathematics – stochastics)

Show that exp(-*t*/2 + *W*_{t}) is a martingale, where *W*_{t} is a Wiener process and exp() is the exponential operator.

(Comment: an oft-used result in studying derivative pricing)

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ANSWER

The expression may look a little intimidating at first (it did to me!) but it’s actually pretty easy to show its conditional expectation at time t, given all information available only up till time s < t, is the same as the expectation at time s. This, of course, is the basic definition of a martingale.

Or apply Ito’s Lemma: Z(t)=f(t,X(t))=f(t,W(t))=exp(W(t)-0.5t) (the diffusion X(t) is standard Brownian motion only, sigma=1,mu=0).

f(t,x)=exp(x-0.5t). f_t(t,x)=-0.5f(t,x). f_x(t,x)=f_xx(t,x)=f(t,x). Then dZ(t)=(-0.5+0.5)Z(t)dt+Z(t)dW(t). The drift is zero so Z(t) is a martingale, the risk-neutral density process.