Pricing and estimates of Greeks for passport option: A three time level approach

We consider the problem of pricing the European type passport option (contingent on the balance on a trading account), for which, a closed form solution exists in the symmetric case, when the riskfree rate is identical to the cost of carry. However, in absence of an explicit solution for the nonsymmetric case, we need to resort to numerical methods in order to solve the corresponding valuation partial differential equation (PDE), in addition to evaluating the first and second derivative, since the optimal holding strategy is dependent on the Greeks, Delta and Gamma, respectively. It has been observed that classical schemes like the two time level Crank Nicholson scheme, is not particularly appropriate for the passport option pricing problem, unless a large number of appropriate time steps are used. This can be attributed to the nonsmooth initial data resulting in small oscillatory errors in the solution. In this article, we address these issues by resorting to the three time level finite difference scheme which can be implemented with larger time steps, without compromising on the accuracy of the price and the Greeks, namely, Delta, Gamma and Theta. We derive these Greeks for the passport option in the symmetric case. The key result is the improvement in the evaluation of the price of the option as well as estimation of these Greeks, with significantly better results being observed near zero accumulated gain, by using the three time level scheme as compared to the Crank Nicholson scheme in the symmetric case. Finally, the three time level scheme is extended to estimate the price and the Greeks in the nonsymmetric case.