Laplace transforms and installment options. (English) Zbl 1102.91042
Summary: An installment option is a derivative financial security where the price is paid in installments instead of as a lump sum at the time of purchase. The valuation of these options involves a free boundary problem in that at each installment date, the holder of the derivative has the option of continuing to pay the premiums or allowing the contract to lapse, and the decision will depend upon whether the present value of the expected pay-off is greater or less than the present value of the remaining premiums. Using a model installment option where the premiums are paid continuously rather than on discrete dates, an integral equation is derived for the position of this free boundary by applying a partial Laplace transform to the underlying partial differential equation for the value of the security. Asymptotic analysis of this integral equation allows us to deduce the behavior of the free boundary close to expiry.
MSC:
| 91B28 | Finance etc. (MSC2000) |
| 44A10 | Laplace transform |
References:
| [1] | Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions, 55, 1964, US Government Printing Office · Zbl 0171.38503 |
| [2] | Alobaidi, G.Mallier, R., Int. J. Math. Math. Sci.27, 177 (2001), DOI: 10.1155/S0161171201005701. · Zbl 1018.91020 |
| [3] | Alobaidi, G.Mallier, R., J. Appl. Math.1, 39 (2001), DOI: 10.1155/S1110757X01000018. |
| [4] | Alobaidi, G.Mallier, R., J. Appl. Math.2, 121 (2002), DOI: 10.1155/S1110757X02110011. |
| [5] | Barles, G.et al., Math. Finance5, 77 (1995), DOI: 10.1111/j.1467-9965.1995.tb00103.x. |
| [6] | Black, F.Scholes, M., J. Political Econ.81, 637 (1973). · Zbl 1092.91524 |
| [7] | Bleistein, N.; Handelsman, R. A., Asymptotic Expansions of Integrals, 1986, Dover |
| [8] | Chance, D. M., Essays in Derivatives, 1998, Wiley |
| [9] | Crank, J., Free and Moving Boundary Problems, 1984, Clarendon · Zbl 0547.35001 |
| [10] | M. Davis, W. Schachermayer and R. Tompkins, Pricing, no-arbitrage bounds and robust hedging of installment options, preprint, Technische Universität Wien (2000). |
| [11] | Davis, M.Schachermayer, W.Tompkins, R., J. Risk Fin.3, 46 (2002), DOI: 10.1108/eb043487. |
| [12] | Dewynne, J. N.et al., Eur. J. Appl. Math.4, 381 (1993), DOI: 10.1017/S0956792500001194. |
| [13] | Dixit, A. R.; Pindyck, R. S., Investment Under Uncertainty, 1994, Princeton Univ. Press |
| [14] | Evans, G. W.Isaacson, E.MacDonald, J. K. L., Quart. Appl. Math.8, 312 (1950). · Zbl 0039.21902 |
| [15] | Evans, J. D.Kuske, R.Keller, J. B., Math. Fin.12, 219 (2002), DOI: 10.1111/1467-9965.02008. |
| [16] | Geske, R., J. Fin. Econ.7, 63 (1979), DOI: 10.1016/0304-405X(79)90022-9. |
| [17] | Hakala, J.; Wystup, U., Foreign Exchange Risk. Models, Instruments and Strategies, 2002, Risk Publications |
| [18] | Junkus, J., The Handbook of Equity Derivatives, eds. Francis, J. C.Toy, W. W.Whittaker, J. G. (Wiley, 2000) pp. 55-76. |
| [19] | Junkus, J., The Handbook of Equity Derivatives, eds. Francis, J. C.Toy, W. W.Whittaker, J. G. (Wiley, 2000) pp. 142-166. |
| [20] | Karsenty, F.Sikorav, J., Risk Mag.6, 36 (1993). |
| [21] | Knessl, C., Stud. Appl. Math.107, 157 (2001), DOI: 10.1111/1467-9590.00183. · Zbl 1152.91522 |
| [22] | Kuske, R. E.Keller, J. B., Appl. Math. Fin.5, 107 (1998). · Zbl 1009.91025 |
| [23] | Majd, S.Pindyck, R. S., J. Fin. Econ.18, 7 (1987), DOI: 10.1016/0304-405X(87)90059-6. |
| [24] | Mallier, R.Alobaidi, G., Appl. Math. Fin.7, 241 (2000), DOI: 10.1080/13504860110060384. |
| [25] | Mallier, R.Alobaidi, G., J. Comput. Appl. Math.164-5, 543 (2004), DOI: 10.1016/S0377-0427(03)00490-4. |
| [26] | McKean, H. P., Ind. Management Rev.6, 32 (1965). |
| [27] | Merton, R. C., Bell J. Econ. Management Sci.4, 141 (1973), DOI: 10.2307/3003143. |
| [28] | Ockendon, J. R.; Ockendon, J. R.; Hodgkins, W. R., Moving Boundary Problems in Heat Flow and Diffusion, 1975, Clarendon · Zbl 0295.76064 |
| [29] | Olver, F. W. J., Asymptotics and Special Functions, 1974, Academic Press · Zbl 0303.41035 |
| [30] | Polyanin, A. D.; Manzhirov, V., Handbook of Integral Equations, 1998, CRC Press · Zbl 0896.45001 |
| [31] | Samuelson, P. A., Ind. Management Rev.6, 13 (1965). |
| [32] | Selby, M. J. P.Hodges, S. D., Management Sci.33, 347 (1987), DOI: 10.1287/mnsc.33.3.347. |
| [33] | Stamicar, R.Sevcovic, D.Chadam, J., Can. Appl. Math. Quart.427 (1999). · Zbl 0979.91031 |
| [34] | Van Moerbeke, J., Arch. Rational Mech. Anal.60, 101 (1976). · Zbl 0336.35047 |
| [35] | Wilmott, P., Derivatives. The Theory and Practice of Financial Engineering, 1998, Wiley |
| [36] | Wilmott, P., Paul Wilmott on Quantitative Finance, 2000, Wiley |
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