By Kohn R.V.
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3 These probabilities must be positive. Writing ξ = aj∆t, the conditions for positivity are 1 + ξ 2 − ξ > 0, 3 1 + ξ 2 + ξ > 0, 3 2 ξ2 < . 816. 816. ***************** Why does it work to truncate the tree? Fixing this problem is surprisingly easy. We truncate the tree at suitably chosen values of j, and use a different trinomial branching scheme at the top and bottom spatial nodes (see the figure, which however truncates the tree much earlier than would normally be done). To see that this works, consider the situation at the top spatial node jmax .
Some comments on the HJM approach to interest rate modelling: HJM is a “framework” not a “model”. When modeling equity-based derivatives, our usual framework is to assume the underlying solves a diffusion process ds = µ(s, t) dt + σ(s, t) dw. The market is complete for any choice of the functions µ(s, t) and σ(s, t) (with some minor restrictions, for example σ(s, t) > 0). But what to choose for µ and σ? The choice of µ is of course irrelevant for option pricing, because under the risk-neutral measure the underlying satisfies ds = r dt + σ(s, t) dw.
OK, the goal is clear. But what changes of Black-Scholes should we consider? There are basically three different approaches: (a) Local volatility models. This approach relaxes the hypothesis that σ be constant, permitting it instead to be a (deterministic) function of S and t. It has the advantage of staying very close to the original Black-Scholes framework: the market is still complete, options can still be hedged, etc. One disadvantage: the function σ(S, t) is difficult to extract from market data in a stable way.