By Kohn R.V.
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Extra resources for Continuous Time Finance
12 Therefore the error terms in our trinomial tree (finite difference) numerical scheme are, at leading order, σ2 −(1/2)utt ∆t + uxxxx (∆x)2 . 24 But differentiating the PDE we have utt = (1/4)σ 4 uxxxx so the preceding expression is equal to uxxxx times σ4 σ2 − ∆t + (∆x)2 . 8 24 2 2 The special choice σ ∆t = (1/3)(∆x) makes this vanish. e. it is of order (∆x)4 ∼ (∆t)2 rather than of order (∆x)2 ∼ ∆t. In the trinomial tree version of Hull-White we have a nonzero drift. I doubt that p = 1/6 is any longer so special, because the preceding argument should be disturbed by the drift.
Recall that in continuous time, the solution of dr = (θ(t) − ar) dt + σ dw, r(0) = r0 was expressed as a sum of two terms: r = α(t) + x(t), where dx = −ax dt + σ dw, x(0) = 0 is independent of the initial term structure, and α(t) = f (0, t) + σ2 1 − e−at 2a2 2 . This decomposition suggests one way of proceeding: we may model r(t) by the trinomial tree obtained by adding α(tj ) to each nodal value of x at time tj = j∆t. This is reasonable – but at finite ∆t it only gets the initial term structure approximately correct.
There are three main approaches to modeling the volatility skew/smile quantitatively: (a) local vol models, (b) jump-diffusion models, and (c) stochastic vol models. We have time for just a very brief introduction to each; you’ll learn much more in the course Case Studies. Today we focus on local vol models, explaining why it is “in principle” easy but in practice quite difficult to extract a local volatility function from market data on calls. ” ***************** The implied volatility skew/smile.