Continuous Time Finance by Kohn R.V.

By Kohn R.V.

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HJM considers the infinitestimal forward rate not the forward term rate, but one might hope it could also be lognormal. However there’s a problem. If we assume σ(t, T ) = σ0 f (t, T ), the HJM theory tells us the risk-neutral process has a nonconstant drift term T α(t, T ) = σ(t, T ) t σ(t, u) du = σ02 f (t, T ) T f (t, u) du. t Since the drift is random (it depends on f ) the solution will not be lognormal. But worse: the drift is quadratic in f . As a result, one can show that the associated HJM stochastic differential equation experiences finite-time blowup with positive probability.

In the present setting the money-market fund isn’t a natural object, because we’re working with term rates and a discrete set of maturities. Rather, the natural object is the rolling CD, which earns interest during the time interval (tk , tk+1 ) at term rate Fk (tk ). 4 Every numeraire has an associated martingale measure. The one associated to the rolling CD has the property that the value of any option divided by the value of the rolling CD is a martingale. Hull calls this the rolling forward risk-neutral measure.

The dependence of the skew/smile on maturity. What’s the goal here? Well, it’s natural to ask how the basic Black-Scholes model can be changed to make it consistent with the observed prices of calls and puts of all strikes and maturities. More practically: it’s natural to ask whether we can use information about the implied vol skew/smile for improved pricing and/or hedging. Concerning pricing: don’t expect too much – we use European option prices to deduce the implied volatility, so we can’t expect to get anything new about the prices of European options.

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