By Kohn R.V.

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**Extra info for Continuous Time Finance**

**Example text**

No problem: just apply the preceding result to any pair of numeraires and subtract. We find that if M and N are two numeraires, then their martingale measures are related by the fact that dwN = (σM − σN ) dt + dwM is a Brownian motion under QN if wM is a Brownian motion under QM . Thus: when we change numeraire, we introduce a drift equal to the difference of the volatilities of the two numeraires. We want to apply this with M = P (t, tk ) and N equal to the rolling CD. Let’s write vk (t) for the volatility of P (t, tk ).

T This holds for every T , so we may differentiate with respect to T . This gives α(t, T ) = Σ(t, T )∂T Σ(t, T ). Recalling the definition of Σ, this amounts to the statement that T α(t, T ) = σ(t, T ) σ(t, u) du. t 3 ***************** How can we get the mean-reverting Hull-White model from HJM? Well, from our analysis of Hull-White, we know that its instantaneous forward rates satisfy dt f (t, T ) = (stuff) dt + σe−a(T −t) dw under the risk-neutral measure. So we can expect one-factor HJM to specialize to HullWhite when σ(t, T ) = σ0 e−a(T −t) with σ0 constant.

5 ***************** A different viewpoint, which I won’t pursue at any length, is the following. Rather than considering only constant σ (corresponding to geometric Brownian motion) or considering arbitrary σ(S, t) (as we have done above), one can consider simple functional forms for σ(S, t) with just a few free parameters. Exact option pricing formulas can be given for some classes of such σ. If we’re lucky, there will be a choice of the parameters for which the associated smile/skew resembles what is seen in the marketplace.