An Asymptotic Formula for Pricing an American Exchange Option
In this paper, the American exchange option (AEO) valuation problem is modelled as a free boundary problem. The critical stock price for an AEO is satisfied an integral equation implicitly. When the remaining time is large enough, an asymptotic formula is provided for pricing an AEO. The numerical results reveal that our asymptotic pricing formula is robust and accurate for the long-term AEO.
[1] G. Barone-Adesi and R. Whaley, Efficient Analytic Approximation of
American Option Values, J. Finance 42 (1987) 301-320.
[2] M. Broadie and J. Detemple, The Valuation of American Options on
Multiple Assets, Math. Finance 7 (1997) 241-286.
[3] P. Carr, The Valuation of American exchange options with applications to
real options In L. Trigeorgis. Real options in Capital Investment Westport,
CT: Praeger, 1995, pp.109-120,
[4] X. Chen and J. Chadam, A Mathematical Analysis for the Optimal
Exercise Boundary of American Put Option, University of Pittsburgh,
2001.
[5] R. Carmona and V. Durrleman, Pricing and Hedging Spread Option, SIAM
Review 45 (2003) 627-685,
[6] J.C. Cox, S.A. Ross and M. Rubinstein, Option Pricing: A Simplified
Approach, J. Finan. Econ. 3 (1979) 145-166.
[7] J.D. Evans, R. Kuske and J.B. Keller, American Options with Dividends
Near Expiry, Math. Finance 12 (2002) 219-237.
[8] R. Gesker and H.E. Johnson, The American Put Option Valued Analytically,
J. Finance 39 (1984) 1511-1524.
[9] R.A. Kuske and J.B. Keller, Optimal Exercise Boundary for An American
Put, Appl. Math. Finance 5 (1998) 107-116.
[10] J. Lee and D.A. Paxson, Valuation of R&D Real American sequential
exchange options, R&D Manage. 31 (2001) 191-200.
[11] M.L. Liu and H.K. Liu, An Asymptotic Solution for an Integral Equation
Arising form a Finite Maturity American Exchange Option. submitted to
Japan Journal of Industrial and Appilied Mathematics, 2006.
[12] W. MacMillan, Analytic Approximation for the American Put Option,
Advance in Futures and Options Research 1 (1986) 119-141.
[13] W. Margrabe, The Pricing of an Option to Exchange One Asset for
Another, J. Finance 33 (1978) 177-186.
[14] R. Merton, The Theory of Rational Option Pricing, The Bell Journal of
Economics and Management science 4 (1973) 141-183.
[15] R. McDonald and D. Siegel, The Value of Waiting to Invest, Quart. J.
Economics 101 (1986) 707-727.
[16] H. Rhys, J. Song and I. Jindrichovska, The Time of Real Option
Exercise: Some Recent Developments, The Engineering Economist 47
(2002) 436-450.
[17] M. Rubinstein, Return to Oz, Risk Magazine 7 (1994) 67-70.
[18] W.P. Sharpe, Investments, Prentice-Hall, Englewood Cliffs, New Jersey,
1978.
[1] G. Barone-Adesi and R. Whaley, Efficient Analytic Approximation of
American Option Values, J. Finance 42 (1987) 301-320.
[2] M. Broadie and J. Detemple, The Valuation of American Options on
Multiple Assets, Math. Finance 7 (1997) 241-286.
[3] P. Carr, The Valuation of American exchange options with applications to
real options In L. Trigeorgis. Real options in Capital Investment Westport,
CT: Praeger, 1995, pp.109-120,
[4] X. Chen and J. Chadam, A Mathematical Analysis for the Optimal
Exercise Boundary of American Put Option, University of Pittsburgh,
2001.
[5] R. Carmona and V. Durrleman, Pricing and Hedging Spread Option, SIAM
Review 45 (2003) 627-685,
[6] J.C. Cox, S.A. Ross and M. Rubinstein, Option Pricing: A Simplified
Approach, J. Finan. Econ. 3 (1979) 145-166.
[7] J.D. Evans, R. Kuske and J.B. Keller, American Options with Dividends
Near Expiry, Math. Finance 12 (2002) 219-237.
[8] R. Gesker and H.E. Johnson, The American Put Option Valued Analytically,
J. Finance 39 (1984) 1511-1524.
[9] R.A. Kuske and J.B. Keller, Optimal Exercise Boundary for An American
Put, Appl. Math. Finance 5 (1998) 107-116.
[10] J. Lee and D.A. Paxson, Valuation of R&D Real American sequential
exchange options, R&D Manage. 31 (2001) 191-200.
[11] M.L. Liu and H.K. Liu, An Asymptotic Solution for an Integral Equation
Arising form a Finite Maturity American Exchange Option. submitted to
Japan Journal of Industrial and Appilied Mathematics, 2006.
[12] W. MacMillan, Analytic Approximation for the American Put Option,
Advance in Futures and Options Research 1 (1986) 119-141.
[13] W. Margrabe, The Pricing of an Option to Exchange One Asset for
Another, J. Finance 33 (1978) 177-186.
[14] R. Merton, The Theory of Rational Option Pricing, The Bell Journal of
Economics and Management science 4 (1973) 141-183.
[15] R. McDonald and D. Siegel, The Value of Waiting to Invest, Quart. J.
Economics 101 (1986) 707-727.
[16] H. Rhys, J. Song and I. Jindrichovska, The Time of Real Option
Exercise: Some Recent Developments, The Engineering Economist 47
(2002) 436-450.
[17] M. Rubinstein, Return to Oz, Risk Magazine 7 (1994) 67-70.
[18] W.P. Sharpe, Investments, Prentice-Hall, Englewood Cliffs, New Jersey,
1978.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:62200", author = "Hsuan-Ku Liu", title = "An Asymptotic Formula for Pricing an American Exchange Option", abstract = "In this paper, the American exchange option (AEO) valuation problem is modelled as a free boundary problem. The critical stock price for an AEO is satisfied an integral equation implicitly. When the remaining time is large enough, an asymptotic formula is provided for pricing an AEO. The numerical results reveal that our asymptotic pricing formula is robust and accurate for the long-term AEO.
", keywords = "Integral equation, asymptotic solution, free boundary problem, American exchange option.", volume = "5", number = "4", pages = "672-4", }
