VISCOSITY SOLUTIONS OF HJB EQUATIONS ARISING FROM THE VALUATION OF EUROPEAN PASSPORT OPTIONS

doi:10.1016/S0252-9602(10)60036-7

Abstract

Abstract:

The passport option is a call option on the balance of a trading account. The option holder retains the gain from trading, while the issuer is liable for the net loss. In this article, the mathematical foundation for pricing the
European passport option is established. The pricing equation which is a fully nonlinear equation is derived using the dynamic programming principle. The comparison principle, uniqueness and convexity preserving of the viscosity solutions of related HJB equation are proved. A relationship between the passport and lookback options is discussed.

Key words: passport option, HJB equation, viscosity solution, uniqueness, convexity preserving

CLC Number:

91B02

BIAN Bao-Jun, WANG Yang, ZHANG Ji-Zhou. VISCOSITY SOLUTIONS OF HJB EQUATIONS ARISING FROM THE VALUATION OF EUROPEAN PASSPORT OPTIONS[J].Acta mathematica scientia,Series B, 2010, 30(1): 187-202.

Trendmd

References

[1] Hyer T, Lipton-Lifschitz A, Pugachevsky D. Passport to success. Risk, 1997, 10(9): 127--131

[2] Andersen L, Andreasen J, Brotherton-Ratcliffe R. The passport option. Journal of Computational Finance, 1998, 1(3): 15--36

[3] Ahn H, Penaud A, Wilmott P. Various passport options and their valuation. Applied Mathematical Finance, 1998, 6(4): 275--292

[4] Penaud A, Wilmott P, H. Ahn H. Exotic passport options. Asia-Pacific Financial Markets, 1999, 6(2): 171--182

[5] Nagayama I. Pricing of passport option. J Math Sci Univ Tokyo, 1998, 5(4): 747--785

[6] Henderson V, Hobson D. Local Time, coupling and the passport option. Finance and Stochastic, 2000, 4(1): 69--80

[7] Shreve S E, Vecer J. Options on a traded account: vacation calls vacation puts and passport options. Finance and Stochastic, 2000, 4(3): 255--274

[8] Chan S S. The Valuation of American Passport Options. University of Wisconsin-Madison Working Paper, 1999

[9] Hendersen V, Hobson D. Passport options with stochastic volatility. Applied Mathematical Finance, 2001, 8(2): 97--119

[10] Crandall M G, Lions P L. Viscosity solutions of Hamilton-Jacobi equations. Transactions of the American Mathematical Society, 1983, 277: 1--42

[11] Crandall M G, Ishii H, Lions P L. User's guide to viscosity solutions of second order partial differential equations. Bull Amer Math Soc, 1992, 27(1): 1--67

[12] Soner H M. Controlled Markov processes, viscosity solutions and applications to mathematical finance Dolcetta I C, Lions P L, eds. Lecture Notes in Mathematics, Vol 1660. Berlin: Springer, 1997:134--185

[13] Soner H M. Stochastic Optimal Control in Finance. Pisa: Scuola Normale Superiore, 2007

[14] Wang M. Höder continuity of viscosity solutions of fully nonlinear parabolic equation. Acta Mathematica Scientia, 1993, 13(4): 399--406 (in Chinese)

[15] Alvarez O, Lasry J M, Lions P L. Convexity viscosity solutions and state constraints. J Math Pures Appl, 1997, 76(3): 265--288

[16] Feldman M, McCuan J. Constructing convex solutions via Perron's method. Annali dell'Universita di Ferrara, 2007, 53(1): 65--94

[17] Giga Y, Goto S, Ishii H, et al. Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Indiana Univ Math J, 1991, 40: 443--470

[18] Jiang L. Mathematical Model and Methods of Option Pricing (in Chinese). Beijing: Higher Education Press, 2003

[19] Benth F E, Karlsen H K, Reikvam K. A semilinear Black and Scholes partial differential equation for valuing American options. Finance and Stochastic, 2003, 7(3): 277--298