# zbMATH — the first resource for mathematics

Hedging options for a large investor and forward-backward SDE’s. (English) Zbl 0856.90011
Summary: In the classical continuous-time financial market model, stock prices have been understood as solutions to linear stochastic differential equations, and an important problem to solve is the problem of hedging options (functions of the stock price values at the expiration date). In this paper, we consider the hedging problem not only with a price model that is nonlinear, but also with coefficients of the price equations that can depend on the portfolio strategy and the wealth process of the hedger. In mathematical terminology, the problem translates to solving a forward-backward stochastic differential equation with the forward diffusion part being degenerate. We show that, under reasonable conditions, the four step scheme of Ma, Protter and Yong for solving forward-backward SDE’s still works in this case, and we extend the classical results of hedging contingent claims to this new model. Included in the examples is the case of the stock volatility increase caused by overpricing the option, as well as the case of different interest rates for borrowing and lending.

##### MSC:
 91G20 Derivative securities (option pricing, hedging, etc.) 60H30 Applications of stochastic analysis (to PDEs, etc.) 93A99 General systems theory
Full Text:
##### References:
 [1] ANSEL, J. P. and STRICKER, C. 1993. Lois de martingale, densites et decomposition de \' \' Follmer-Schweizer. Ann. Instit. H. Poincare 28 375 392. \" \' · Zbl 0772.60033 [2] BLACK, F. and SCHOLES, M. 1973. The pricing of options and corporate liabilities. J. Polit. Economy 81 637 659. · Zbl 1092.91524 [3] CVITANIC, J. 1996. Nonlinear financial markets: hedging and portfolio optimization. Proćeedings of the Isaac Newton Institute for Mathematical Sciences. [4] CVITANIC, J. and KARATZAS, I. 1993. Hedging contingent claims with constrained portfolios. Ánn. Appl. Probab. 3 652 681. · Zbl 0825.93958 [5] DELBAEN, F. and SCHACHERMAy ER, W. 1994. A general version of the fundamental theorem of asset pricing. Math. Ann. 300 463 520. · Zbl 0865.90014 [6] DUFFIE, D. and EPSTEIN, L. G. 1992. Stochastic differential utility. Econometrica 60 Z. 353 394. Appendix with Skiadas, C. JSTOR: · Zbl 0768.90006 [7] DUFFIE, D., MA, J. and YONG, J. 1995. Black’s console rate conjecture. Ann. Appl. Probab. 5 356 382. · Zbl 0830.60052 [8] FRIEDMAN, A. 1964. Partial Differential Equations of Parabolic Ty pe. Prentice-Hall, Englewood Cliffs, NJ. [9] GROSSMAN, S. J. 1988. An analysis of the implications for stock and futures price volatility of program trading and dy namic hedging strategies. Journal of Business 61 275 289. [10] HARRISON, J. M. and KREPS, D. M. 1979. Martingales and arbitrage in multiperiod security markets. J. Econom. Theory 20 381 408. · Zbl 0431.90019 [11] HARRISON, J. M. and PLISKA, S. R. 1981. Martingales and stochastic integrals in the theory of continuous trading. Stochastic Process. Appl. 11 215 260. · Zbl 0482.60097 [12] HARRISON, J. M. and PLISKA, S. R. 1983. A stochastic calculus model of continuous time trading: complete markets. Stochastic Process. Appl. 15 313 316. · Zbl 0511.60094 [13] KARATZAS, I. 1989. Optimization problems in the theory of continuous trading. SIAM J. Control Optim. 27 1221 1259. · Zbl 0701.90008 [14] KARATZAS, I., LEHOCZKY, J. P. and SHREVE, S. E. 1991. Equilibrium models with singular asset prices. Math. Finance 1 11 29. JSTOR: · Zbl 0900.90111 [15] KARATZAS, I. and SHREVE, S. E. 1991. Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York. · Zbl 0734.60060 [16] LADy ZENSKAJA, O. A., SOLONNIKOV, V. A. and URAL’CEVA, N. N. 1968. Linear and Quasilinear Equations of Parabolic Ty pe. Amer. Math. Soc., Providence, RI. [17] MA, J., PROTTER, P. and YONG, J. 1994. Solving forward backward stochastic differential equations explicitly a four step scheme. Probab. Theory Related Fields 98 339 359. · Zbl 0794.60056 [18] MERTON, R. C. 1969. Lifetime portfolio selection under uncertainty: the continuous time case. Rev. Econom. Statist. 51 247 257. [19] PARDOUX, E. and PENG, S. 1990. Adapted solution of a backward stochastic differential equation. Sy stems Control Lett. 14 55 61. · Zbl 0692.93064 [20] PARDOUX, E. and PENG, S. 1992. Backward stochastic differential equations and quasi-linear parabolic partial differential equations. Stochastic Partial Differential Equations and Their Applications. Lecture Notes in Control and Inform. Sci. 176 200 217. Springer, Berlin. · Zbl 0766.60079 [21] PENG, S. 1991. Probabilistic interpretation for sy stems of quasilinear parabolic partial differential equations. Stochastics Stochastics Rep. 37 61 74. · Zbl 0739.60060 [22] PLATEN, E. and SCHWEIZER, M. 1994. On smile and skewness. [23] PROTTER, P. 1990. Stochastic Integration and Differential Equations. A New Approach. Springer, Berlin. · Zbl 0694.60047 [24] ROGERS, L. C. G. and WILLIAMS, D. 1987. Diffusion, Markov Processes, and Martingales 2. Ito Calculus. Wiley, New York. · Zbl 0977.60005 [25] NEW YORK, NEW YORK 10027 WEST LAFAy ETTE, INDIANA 47907-1395 E-MAIL: cj@stat.columbia.edu E-MAIL: majin@math.purdue.edu
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.