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Pricing and estimates of Greeks for passport option: A three time level approach. (English) Zbl 1414.91413
Summary: 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.

MSC:
91G60 Numerical methods (including Monte Carlo methods)
91G20 Derivative securities (option pricing, hedging, etc.)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
Software:
PATH Solver
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References:
[1] Hyer, T.; Lipton-Lifschitz, A.; Pugachevsky, D., Passport to success, Risk, 10, 9, 127-131, (1997)
[2] Andersen, L.; Andreasen, J.; Brotherton-Ratcliffe, R., The passport option, J. Comput. Finance, 1, 3, 15-36, (1998)
[3] S.-S. Chan, The valuation of American passport options, University of Wisconsin-Madison, Working Paper, 1999.
[4] Dirkse, S. P.; Ferris, M. C., The PATH solver: A non-monotone stabilization scheme for mixed complementarity problems, Optim. Methods Softw., 5, 123-156, (1995)
[5] Topper, J., A finite element implementation of passport options, (2003), University of Oxford, (M. Sc. thesis)
[6] Ahn, H.; Penaud, A.; Wilmott, P., Various passport options and their valuation, Appl. Math. Finance, 6, 4, 275-292, (1999) · Zbl 1009.91038
[7] Penaud, A.; Wilmott, P.; Ahn, H., Exotic passport options, Asia-Pac. Financ. Mark., 6, 2, 171-182, (1999) · Zbl 1153.91553
[8] B. Baojun, W. Yang, A viscosity solution approach to valuation of passport options in a jump-diffusion model, in: Proceedings of the 27th Chinese Control Conference, Kunming, Yunnan, China, 2008, pp. 606-608.
[9] Baojun, B.; Yang, W.; Jizhou, Z., Viscosity solutions of HJB equations arising from the valuation of European passport options, Acta Math. Sci., 30, 1, 187-202, (2010) · Zbl 1224.91148
[10] Henderson, V.; Hobson, D., Local time, coupling and the passport option, Finance Stoch., 4, 1, 69-80, (2000) · Zbl 0944.60046
[11] Henderson, V.; Hobson, D., Passport options with stochastic volatility, Appl. Math. Finance, 8, 2, 97-118, (2001) · Zbl 1013.91046
[12] H. Malloch, P.W. Buchen, Passport option: continuous and binomial models, in: Finance and Corporate Governance Conference, 2011.
[13] Delbaen, F.; Yor, M., Passport options, Math. Finance, 12, 4, 299-328, (2002) · Zbl 1048.91063
[14] Nagayama, I., Pricing of passport option, J. Math. Sci. Univ. Tokyo, 5, 4, 747-785, (1998) · Zbl 0927.91011
[15] Shreve, S. E.; Vecer, J., Options on a traded account: vacation calls, vacation puts and passport options, Finance Stoch., 4, 3, 255-274, (2000) · Zbl 0997.91020
[16] Pooley, D., Numerical methods for nonlinear equations in option pricing, (2003), University of Waterloo, (Ph.D. thesis)
[17] Kampen, J., Optimal strategies of passport options, Math. Ind., 12, 666-670, (2008) · Zbl 1308.91164
[18] Richtmyer, R. D.; Morton, K. W., (Difference Method for Initial-Value Problem, Interscience Tracts in Pure and Applied Mathematics, (1967)) · Zbl 0155.47502
[19] Smith, G. D., (Numerical Solution of Partial Differential Equations: Finite Difference Methods, Oxford Applied Mathematics and Computing Science Series, (1985)) · Zbl 0576.65089
[20] Shaw, W. T., Modelling financial derivatives with Mathematica, (1998), Cambridge University Press · Zbl 0939.91002
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