June 5th, 2003, 11:30 am
TREES:In the driftless model it is extremely easy to implement a recombining tree for the model.To do this we have to drop the number of factors.Suppose we decide that 4 factors are good enough.This means that four of the state variables determine all the others as deterministic functions of these four and it is easy to make this explicit.Set Y_j=log(U_j). Then the Y_j follow a dynamicsdY_j(t)=-0.5*sigma^2_j(t)dt + sigma_j(t)u_jdW(t)with deterministic drift. So we need to kow only the driftless "volatility parts" V_j(t)V_j(0)=Y_j(0), dV_j(t)=sigma_j(t)u_jdW(t).At any time t the vector V(t) is multinormal with covariance matrix C of rank four, that is,C=RR' where R is of dimension k by 4.Since the vector V(t) is multinormal we have V(t)=RZ, where Z is a 4 dimensional standard normal vector.If you want to determine the value of the variables V_j(t) from any four of them simply use these four to solve for Zand use the values of Z to compute all the others. Of course we could formulate this in terms of multiplication with a matrix inverse.We then add the deterministic drift to get Y_j(t) and from this U_j(t)=exp(Y_j(t)).Consequently we only have to evolve 4 variables, say V_{n-4},...,V_{n-1} and these form a Gaussian process.To construct a tree for that is even easier than constructing a tree for a log-Gaussian dynamicsdS(t)=S(t)sigma(t)dW(t).If we use a simple up-down model each of the four state variables has t+1 possible states at time step t.This gives us (t+1)^4 possible nodes at time t -- still manageable.Four factors cover the four largest eigenvalues of the covariance matrix and from what I have seen in mycovariance matrices this should get about 97% of the variability.I believe the availability of trees will tilt the scale in favor of the driftless Libor market model.I'll try to implement it before I release the C++ code.
Last edited by
trc on June 4th, 2003, 10:00 pm, edited 1 time in total.