Continuous Time Finance by Kohn R.V.
By Kohn R.V.
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Additional info for Continuous Time Finance
Notice that (4) is simply a discrete analogue of the HJM equation dt f (t, T ) = α(t, T ) dt + σ(t, T ) dw with T α(t, T ) = σ(t, T ) σ(t, u) du t and σ(t, T ) = ζi (t)fi (t) when T ≈ ti . Indeed: formally as δ → 0 we expect Fk (t) ≈ f (t, tk ) since f is the infinitesimal forward rate. And formally as δ → 0 we have 1 + δFi (t) ≈ 1 while k T δFi (t)ζi (t) ≈ f (t, u)ζ(t, u) du. t i=m(t) Thus the Libor Market Model is essentially the discrete analogue of HJM with lognormal forward rates. Question: HJM with lognormal statistics was problematic due to blowup.
Explicit formulas are a bit tedious to obtain – but they’re not needed. From the structure of the calculation, we see quite easily (just as in the risk-neutral case) that the short rates and bond prices are lognormal. ***************** Aside from being too restrictive (not enough parameters), Vasicek has one other awkward feature: since r(t) is Gaussian, there is a positive probability that it is negative. This is of course unrealistic: interest rates must be positive. The Cox-Ingersoll-Ross model mentioned above was introduced to fix this problem: one can show that if θ ≥ σ 2 /2 then r stays positive (with probability one).
E. we commonly assume the underlying has lognormal dynamics. e. ) The situation with HJM is analogous. For one-factor HJM, the framework is dt f (t, T ) = α(t, T ) dt + σ(t, T ) dw. e. assuming w is a Brownian motion in the risk-neutral measure) the drift α is completely specified by σ. The choice of σ is again crucial. It is no longer so obvious how to choose it; we saw how to choose σ(t, T ) to get either Ho-Lee or Hull-White. But many other choices are possible. g. if it depends explicitly on f (t, T )) we have no exact pricing formulas.