January 3rd, 2005, 3:12 pm
QuoteOriginally posted by: CuchulainnFermion How would one apply binomial method to Merton jump model or even Levy processes? has this been done anywhere?You generally can't use binomial methods for jump-diffusion or arbitrary Levy processes, since with only two possible moves, you can model either two possible diffusive steps (like Black diffusion in the original CRR tree), or drift vs jump steps to model intesity of pure jump processes (I use these for credit derivatives sometimes).General trees/lattices/meshes/grids can handle almost any process you like. A simple extention of a binomial tree having a jump step coming off as a third node is probably the most straightforward way to handle jump diffusion. For example, from a node in the binomial tree with the usual +1%/-1% steps, you add a -10% step to model the jump, and solve for its probability. One less conventional but usefully clear trick about n-nomial trees is that they can measure arbitrary processes characterized by n-1 moments, by solving the linear system Ap = m where A = (for n = 4) [1,1,1,1;dd,d,u,uu;dd^2,d^2,u^2,uu^2;dd^3,d^3,u^3,uu^3] (that is, row vectors of powers of the possible returns), and m are the moments of those returns. It is not too hard to see that this is just solving for the distribution from the definition of the distributions' moments, and that it is easy to ensure all these probabilities are positive.