A SURVEY ON AMERICAN OPTIONS: OLD APPROACHES AND NEW TRENDS

- Journal title : Bulletin of the Korean Mathematical Society
- Volume 48, Issue 4, 2011, pp.791-812
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/BKMS.2011.48.4.791

Title & Authors

A SURVEY ON AMERICAN OPTIONS: OLD APPROACHES AND NEW TRENDS

Ahn, Se-Ryoong; Bae, Hyeong-Ohk; Koo, Hyeng-Keun; Lee, Ki-Jung;

Ahn, Se-Ryoong; Bae, Hyeong-Ohk; Koo, Hyeng-Keun; Lee, Ki-Jung;

Abstract

This is a survey on American options. An American option allows its owner the privilege of early exercise, whereas a European option can be exercised only at expiration. Because of this early exercise privilege American option pricing involves an optimal stopping problem; the price of an American option is given as a free boundary value problem associated with a Black-Scholes type partial differential equation. Up until now there is no simple closed-form solution to the problem, but there have been a variety of approaches which contribute to the understanding of the properties of the price and the early exercise boundary. These approaches typically provide numerical or approximate analytic methods to find the price and the boundary. Topics included in this survey are early approaches(trees, finite difference schemes, and quasi-analytic methods), an analytic method of lines and randomization, a homotopy method, analytic approximation of early exercise boundaries, Monte Carlo methods, and relatively recent topics such as model uncertainty, backward stochastic differential equations, and real options. We also provide open problems whose answers are expected to contribute to American option pricing.

Keywords

American option;American put;optimal exercise boundary;free boundary problem;Monte Carlo methods;

Language

English

Cited by

References

1.

C. Ahn, H. Choe, and K. Lee, A long time asymptotic behavior of the free boundary for an American put, Proc. Amer. Math. Soc. 137 (2009), no. 10, 3425-3436.

2.

K. Amin and A. Khanna, Convergence of American option values from discrete-to continuous-time financial models, Math. Finance 4 (1994), no. 4, 289-304.

3.

L. Andersen, A simple approach to the pricing of Bermudan swaptions in the multifactor LIBOR market model, J. Comput. Finance 3 (2000), 1-32.

4.

L. Andersen and M. Broadie, A primal-dual simulation algorithm for pricing multidimensional American options, Management Science 50 (2004), 1222-1234.

5.

G. Barone-Adesi and R. Whaley, Efficient analytic approximation of American option values, Journal of Finance 42 (1987), 301-320.

6.

A. Bensoussan, On the theory of option pricing, Acta Appl. Math. 2 (1984), no. 2, 139-158.

7.

F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy 81 (1973), 637-654.

8.

M. Brennan and E. Schwartz, The valuation of American put options, Journal of Finance 32 (1977), 449-462.

9.

M. Brennan and E. Schwartz, Finite difference methods and jump processes arising in the pricing of contingent claims: a synthesis, Journal of Financial and Quantitative Analysis 13 (1978), 461-474.

10.

M. Broadie and J. Detemple, American option valuation: new bounds, approximations, and a comparison of existing methods, Review of Financial Studies 9 (1996), 1211-1250.

11.

M. Broadie and J. Detemple, Option pricing: valuation models and applications, Management Science 50 (2004), 1145-1177.

12.

M. Broadie and P. Glasserman, Pricing American-style securities using simulation, J. Econom. Dynam. Control 21 (1997), no. 8-9, 1323-1352.

13.

P. Boyle, Options: a Monte Carlo approach, Journal of Financial Economics, 4 (1977), 328-338.

14.

D. A. Bunch and H. Johnson, The American put option and its critical stock price, Journal of Finance 55 (2000), 2333-2356.

15.

S. Byun, Numerical procedures for valuing american options, Ph. D. dissertation, KAIST, 1996.

16.

S. Byun and I. Kim, Relationship between American puts and calls on futures, J. Korean Soc. Ind. Appl. Math. 4 (2000), 11-20.

17.

P. Carr, Randomization and the American put, Review of Financial Studies 81 (1998), 597-626.

18.

P. Carr and D. Faguet, Valuing Finite-Lived Options as Perpetual, Working Paper, 1996.

19.

P. Carr, R. Jarrow, and R. Myneni, Alternative characterizations of American put options, Math. Finance 2 (1992), 87-106.

20.

X. Chen and J. Chadam, A mathematical analysis for the optimal exercise boundary of American put option, SIAM J. Math. Anal. 38 (2007), 1613-1641.

21.

N. Clarke and K. Parrot, The multigrid solution of two-factor American put options, Technical Report 96-16, (1996), Oxford Computing Laboratory, Oxford.

22.

J. Cox, S. Ross, and M. Rubinstein, Option pricing: a simplified approach, Journal of Financial Economics 7 (1979), 141-148.

23.

M. Davis and I. Karatzas, A deterministic approach to optimal stopping, with applications, in Probability, Statistics and Optimization: A Tribute to Peter Whittle, F. Kelley, ed., (1994), 455-466, Wiley.

24.

J. Detemple, American-Style Derivatives, Chapman & Hall/CRC Financial Mathematics Series, 2006.

25.

A. Dixit and R. Pindyck, Investment Under Uncertainty, Princeton University Press, 1994.

26.

H. Follmer, Financial uncertainty, risk measures and robust preferences, Aspects of Mathematical Finance, 3-13, Springer, Berlin, 2008.

27.

P. Forsyth and K. Vetzal, Quadratic convergence for valuing American options using a penalty method, SIAM J. Sci. Comput. 23 (2002), no. 6, 2095-2122.

28.

J. Fouque, G. Papanicolau, and K. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, 2000.

29.

B. Gao, J. Huang, and M. Subrhamanyam, The valuation of American barrier options using the decomposition technique, J. Econom. Dynam. Control 24 (2000), 1783-1827.

30.

R. Geske and H. Johnson, The American put options valued analytically, Journal of Finance 39 (1984), 229-263.

31.

I. Gilboa and Schmeidler, Maxmin expected utility with non-unique prior, J. Math. Econom. 18 (1989), 141-153.

32.

P. Glasserman, Monte Carlo Methods in Financial Engineering, Applications of Mathematics (New York), 53. Stoch. Model. Appl. Probab., Springer-Verlag, New York, 2004.

33.

O. Grabbe, The pricing of call and put options on foreign exchange, Journal of International Money and Finance 2 (1983), 239-253.

34.

S. J. Grossman, The Information Role of Prices, The MIT Press, 1989.

35.

S. Guo and Q. Zhang, Closed-form solutions for perpetual American put options with regime switching, SIAM J. Appl. Math. 64 (2004), no. 6, 2034-2049.

36.

S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies 6 (1993), 327-343.

37.

Y. Hu and S. Tang, Multi-dimensional BSDE with oblique reflection and optimal switching, forthcoming in Probab. Theory Related Fields, 2009.

38.

J. Huang, M. Subrahmanyam, and G. Yu, Pricing and hedging of American options: a recursive integration method, Review of Financial Studies 9 (1996), 277-300.

39.

J. Hull, Options, Futures, and Other Derivatives, 7th ed., Pearson Education, 2009.

40.

J. Hull and A. White, The pricing of options on assets with stochastic volatilities, Journal of Finance 42 (1987), 281-300.

41.

S. Ikonen and J. Toivanen, Operator splitting methods for American option pricing, Appl. Math. Lett. 17 (2004), no. 7, 809-814.

42.

S. Ikonen and J. Toivanen, Operator splitting methods for pricing American options with stochastic volatility, Technical Report B11/2004, University of Jyvaskyla, 2004.

43.

S. Ikonen and J. Toivanen, Componentwise splitting methods for pricing American options with stochastic volatility, Technical Report B7/2005, University of Jyvaskyla, 2005.

44.

S. Ikonen and J. Toivanen, Efficient numerical methods for pricing American options under stochastic volatility, Technical Report B12/2005, University of Jyvaskyla, 2005.

45.

S. D. Jacka, Optimal stopping and the American put, Math. Finance 1 (1991), 1-14.

46.

B. Jang and H. Koo, American Put Options with Regime-Switching Volatility, Working Paper, 2005.

47.

H. Johnson, An analytic approximation for the American put price, Journal of Financial and Quantitative Analysis 18 (1983), 141-148.

48.

N. Ju, Pricing an American option by approximating its early exercise boundary as a multi-piece exponential function, Review of Financial Studies 11 (1998), 627-646.

50.

I. Karatzas and S. Shreve, Methods of Mathematical Finance, New York, Springer-Verlag, 1998.

51.

52.

I. Kim and S. Byun, Optimal exercise boundary in a binomial option pricing model, Journal of Financial Engineering 3 (1994), 137-158.

53.

I. Kim and B. Jang, An Alternative Numerical Approach for Valuation of American Options: A Simple Iteration Method, Working Paper, 2008.

54.

R. A. Kuske and J. B. Keller, Optimal exercise boundary for an american put option, Appl. Math. Finance 5 (1998), 107-116.

55.

A. Longstaff and E. Schwartz, Valuing American options by simulation: a simple leastsquares approach, Review of Financial Studies 14 (2001), 113-147.

56.

R. MacDonald and M. Schroder, A parity result for American options, J. Comput. Finance 1 (1998), 5-13.

57.

L. MacMillan, An analytical approximation for the Amercian put prices, Advances in Futures and Options Research 1 (1986), 119-139.

58.

H. McKean, Appendix: a free boundary problem for the heat equation arising from a problem in mathematical economics, Industrial Management Review 6 (1965), 32-39.

59.

R. Merton, The theory of rational option pricing, Bell Journal of Economics and Management Science 4 (1973), 141-183.

60.

G. Meyer and J. van der Hoek, The Evaluation of American Options with the Method of Lines, Working Paper, 1994.

61.

Oosterlee, On multigrid for linear complementary problems with applications to American-style options, Electron. Trans. Numer. Anal. 16 (2003), 165-185.

62.

D. N. Ostrove and J. Goodman, On the early exercise boundary of the American put option, SIAM J. Appl. Math. 62 (2002), 1823-1835.

64.

S. Peng, Nonlinear expectations, nonlinear evaluations and risk measures, Stochastic methods in finance, 165-253, Lecture Notes in Math., 1856, Springer, Berlin, 2004.

68.

P. Samuelson, Rational theory of warrant pricing, Industrial Management Review 6 (1965), 13-31.

69.

M. Schroder, Changes of numeraire for pricing futures, forwards and options, Review of Financial Studies 12 (1989), 1143-1163.

70.

E. Schwartz, The valuation of warrants: implementing a new approach, Journal of Financial Economics 4 (1977), 79-93.

71.

R. Stamicar, D. Sevcovic, and J. Chadam, The early exercise boundary for the American put near expiry: numerical approximation, Can. Appl. Math. Q. 7 (1999), no. 4, 427-444.

72.

D. Tavella and C. Randall, Pricing Financial Instruments: The Finite difference Method, John Wiley & Sons, 2000.

73.

J. Tilley, Valuing American options in a path simulation model, Transactions of the Society of Actuaries 45 (1993), 83-104.

74.

L. Trigeorgis, Real Options in Capital Investment, Praeger, 1995.

75.

J. Tsitsiklis and B. Van Roy, Regression analysis for pricing complex American-style options, IEEE Transactions on Neural Networks 12 (2001), 694-703.

76.

P. van Moerbeke, On optimal stopping and free boundary problems, Arch. Ration. Mech. Anal. 60 (1975/76), no. 2, 101-148.

77.

P. Wilmott, J. Dewynne, and S. Howison, Option Pricing, Oxford University Press, 1993.

78.

L.Wu and Y. Kwok, A front-fixing finite difference method for the valuation of American options, Journal of Financial Engineering 6 (1997), 83-97.