Drift Approximation
Under terminal measure, the drifts of forward rate dynamics are state-dependent, which gives rise to sufficiently complicated non-lognormal distributions. This means that an explicit analytic solution to the forward rate stochastic differential equations cannot be obtained. Therefore, most work on the topic has focused on ways to approximate the drift, which is the fundamental trickiness in implementing the Market Model.
Our model works backwards recursively from forward rate N down to forward rate k . The N-th forward rate without drift can be determined exactly. By the time it takes to calculate the k-th forward rate , all forward rates from to at time t are already known. Therefore, the drift calculation (11b) is to estimate the integrals containing forward rate dynamics , for j=k+1,…,N , with known beginning and end points given by and . For completeness, we list all possible solutions below.
Frozen Drift (FD). Replace the random forward rates in the drift by their deterministic initial values, i.e.,