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.,