×

A new analytical approximation formula for the optimal exercise boundary of American put options. (English) Zbl 1140.91415

Summary: A new analytical formula as an approximation to the value of American put options and their optimal exercise boundary is presented. A transform is first introduced to better deal with the terminal condition and, most importantly, the optimal exercise price which is an unknown moving boundary and the key reason that valuing American options is much harder than valuing its European counterparts. The pseudo-steady-state approximation is then used in the performance of the Laplace transform, to convert the systems of partial differential equations to systems of ordinary differential equations in the Laplace space. A simple and elegant formula is found for the optimal exercise boundary as well as the option price of the American put with constant interest rate and volatility. Other hedge parameters as the derivatives of this solution are also presented.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
35K20 Initial-boundary value problems for second-order parabolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Allegretto W., Discrete and Continuous Dynamical Systems, Series B., Applications and Algorithm 8 pp 127–
[2] DOI: 10.1080/07362999108809230 · Zbl 0729.60056
[3] DOI: 10.1111/j.1540-6261.1987.tb02569.x
[4] DOI: 10.1086/260062 · Zbl 1092.91524
[5] DOI: 10.1111/j.1467-9965.1995.tb00103.x · Zbl 0866.90029
[6] DOI: 10.2307/2326779
[7] DOI: 10.1093/rfs/9.4.1211
[8] J. W. Brown and R. V. Churchill, Complex Variables and Applications (McGraw-Hill, Inc., 1996) p. 235, 183 and 213.
[9] Bunch D. S., Journal of Finance 5 pp 2333– · Zbl 0030.06302
[10] DOI: 10.1093/rfs/11.3.597 · Zbl 1386.91134
[11] DOI: 10.1111/j.1467-9965.1992.tb00040.x · Zbl 0900.90004
[12] DOI: 10.1016/0304-405X(79)90015-1 · Zbl 1131.91333
[13] Fulford G. R., Industrial Mathematics: Case Studies in the Diffusion of Heat and Matter (2002) · Zbl 0982.00007
[14] DOI: 10.1111/j.1540-6261.1984.tb04921.x
[15] Grant D., Journal of Financial Engineering 5 pp 211– · JFM 02.0361.06
[16] S. C. Gupta, The Classical Stefan Problem: Basic Concepts, Modelling and Analysis (Elsevier, Amsterdam, Boston, 2003) pp. 18–23.
[17] F. B. Hildebrand, Advanced Calculus for Applications (Prentice-Hall, 1976) p. 57.
[18] Hill J. M., Pitman Monographs and Surveys in Pure and Applied Mathematics 31 (1987)
[19] Hon Y. C., Journal of Financial Engineering 8 pp 31–
[20] DOI: 10.1093/rfs/9.1.277
[21] DOI: 10.1111/j.1467-9965.1991.tb00007.x · Zbl 0900.90109
[22] DOI: 10.2307/2330809
[23] DOI: 10.1093/rfs/11.3.627
[24] DOI: 10.1093/rfs/3.4.547
[25] Kuske R. A., Applied Mathematical Studies 5 pp 107–
[26] DOI: 10.1007/BF01448358 · Zbl 0699.90010
[27] DOI: 10.1093/rfs/14.1.113 · Zbl 1386.91144
[28] MacMillan L., Advances in Futures and Options Research 1 pp 119–
[29] McKean H. P., Industrial Management Review 6 pp 32–
[30] DOI: 10.2307/3003143
[31] Papoulis A., Quart. Appl. Math. 14 pp 405–
[32] Samuelson P. A., Industrial Management Review 6 pp 13–
[33] DOI: 10.1016/0304-405X(77)90037-X
[34] Stefan N., S.-B. Wien. Akad. Mat. Natur. 98 pp 173–
[35] DOI: 10.1145/361953.361969
[36] Stamicar R., Canadian Applied Mathematics Quarterly 7 pp 427–
[37] D. Tavella and C. Randall, Pricing Financial Instruments, The Finite Difference Method (John Wiley and Sons, Inc, New York, 2000) p. 88.
[38] DOI: 10.1002/(SICI)1099-1204(199805/06)11:3<153::AID-JNM299>3.0.CO;2-C · Zbl 0924.65135
[39] M. van Dyke, Perturbation Methods in Fluid Mechanics (Academic Press, 1965) pp. 30–32.
[40] DOI: 10.1007/BF00250676 · Zbl 0336.35047
[41] DOI: 10.1017/CBO9780511812545 · Zbl 0842.90008
[42] Wu L., Journal of Financial Engineering 6 pp 83–
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.