连续型美式分期付款看跌期权
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摘要
本文讨论了连续型美式分期付款看跌期权.一方面,期权持有人拥有美式看跌期权的权利:在到期日之前实施合同以敲定价格卖出股票;另一方面,期权持有人拥有分期付款期权的权利:分期支付期权金以保证合同有效,也可以随时停止给付期权金以终止合同.因此,期权持有人在交易期间可行使的权利有三种:继续持有,实施合同或终止合同.这种期权的定价模型可表示为抛物型变分不等式,它同时是一个自由边界问题.该问题解的存在唯一性可利用惩罚函数法和常规的偏微分方程方法进行求解证明.不同于标准美式看跌期权,不管有没有分红,连续型美式分期付款看跌期权均有两条自由边界.本文将集中讨论该期权自由边界的性质,如单调性、正则性以及自由边界的位置.
Installment options are designed for an investor who is willing to pay a little extra for the opportunity to terminate a contract and reduce losses caused by a void investment position. Because of this extra privilege, installment options are weakly path-dependent. American continuous-installment put options are discussed in this paper.In addition to the right to terminate the contract by stopping the payments at any time,the holder also has the right to exercise the option at any time until maturity; this leads to three potential choices during the holding period: cancel, exercise, or hold on. The mathematical pricing model of this option can be formulated as a parabolic variational inequality, which is a free boundary problem. The existence and uniqueness of the solution can be solved using the penalty method and regular PDE arguments. Different from the standard American put option, this option has two free boundaries irrespective of dividends. Attention is focused on properties of the free boundaries, such as monotonicity,smoothness, and location.
Installment options are designed for an investor who is willing to pay a little extra for the opportunity to terminate a contract and reduce losses caused by a void investment position. Because of this extra privilege, installment options are weakly path-dependent. American continuous-installment put options are discussed in this paper.In addition to the right to terminate the contract by stopping the payments at any time,the holder also has the right to exercise the option at any time until maturity; this leads to three potential choices during the holding period: cancel, exercise, or hold on. The mathematical pricing model of this option can be formulated as a parabolic variational inequality, which is a free boundary problem. The existence and uniqueness of the solution can be solved using the penalty method and regular PDE arguments. Different from the standard American put option, this option has two free boundaries irrespective of dividends. Attention is focused on properties of the free boundaries, such as monotonicity,smoothness, and location.
引文
[1] BEN-AMEUR H, BRETON M, FRANCOIS P. A dynamic programming approach to price installment options[J]. European Journal of Operational Research, 2006, 169(2):667-676.
[2] DAVIS M, SCHACHERMAYER W, TOMPKINS R. Pricing, no-arbitrage bounds and robust hedging of installment options[J]. Quantitative Finance, 2001(1):597-610.
[3] DAVIS M, SCHACHERMAYER W, TOMPKINS R. Installment options and static hedging[J]. Journal of Risk Finance, 2002(3):46-52.
[4] ALOBAIDI G, MALLIERAND R, DEAKIN S. Laplace transforms and installment options[J]. Mathematical Models and Methods in Applied Sciences, 2004, 14(8):1167-1189.
[5] CIURLIA P, ROKO I. Valuation of American continuous-installment options[J]. Computational Economics,2005, 25(1):143-165.
[6]YANG Z, YI F H. Valuation of the European installment put options:Variational inequality approach[J].Communications in Contemporary Mathematics, 2009, 11(2):279-307.
[7] YANG Z, YI F H. A variational inequality arising from American installment call options pricing[J]. Journal of Mathematical Analysis and Applications, 2009, 357(1):54-68.
[8] CHEN X F, CHADAM J. A mathematical analysis for the optimal exercise boundary of American put option[J]. Siam Journal on Mathematical Analysis, 2007, 38:1613-1614.
[9] JIANG L S. Mathematical Modeling and Methods of Option Pricing[M]. New Jersey:World Scientific, 2005.
[10] KIM J. The analytic valuation of American options[J]. Review of Financial Studies, 1990, 3(4):542-572.
[11] KUSKE A, KELLER B. Optimal exercise boundary for an American put option[J]. Applied Mathematical Finance, 1998, 5(2):107-116.
[12] FRIEDMAN A. Parabolic variational inequalities in one space dimension and smoothness of the free boundary[J]. Journal of Functional Analysis, 1975, 18(2):151-176.
[2] DAVIS M, SCHACHERMAYER W, TOMPKINS R. Pricing, no-arbitrage bounds and robust hedging of installment options[J]. Quantitative Finance, 2001(1):597-610.
[3] DAVIS M, SCHACHERMAYER W, TOMPKINS R. Installment options and static hedging[J]. Journal of Risk Finance, 2002(3):46-52.
[4] ALOBAIDI G, MALLIERAND R, DEAKIN S. Laplace transforms and installment options[J]. Mathematical Models and Methods in Applied Sciences, 2004, 14(8):1167-1189.
[5] CIURLIA P, ROKO I. Valuation of American continuous-installment options[J]. Computational Economics,2005, 25(1):143-165.
[6]YANG Z, YI F H. Valuation of the European installment put options:Variational inequality approach[J].Communications in Contemporary Mathematics, 2009, 11(2):279-307.
[7] YANG Z, YI F H. A variational inequality arising from American installment call options pricing[J]. Journal of Mathematical Analysis and Applications, 2009, 357(1):54-68.
[8] CHEN X F, CHADAM J. A mathematical analysis for the optimal exercise boundary of American put option[J]. Siam Journal on Mathematical Analysis, 2007, 38:1613-1614.
[9] JIANG L S. Mathematical Modeling and Methods of Option Pricing[M]. New Jersey:World Scientific, 2005.
[10] KIM J. The analytic valuation of American options[J]. Review of Financial Studies, 1990, 3(4):542-572.
[11] KUSKE A, KELLER B. Optimal exercise boundary for an American put option[J]. Applied Mathematical Finance, 1998, 5(2):107-116.
[12] FRIEDMAN A. Parabolic variational inequalities in one space dimension and smoothness of the free boundary[J]. Journal of Functional Analysis, 1975, 18(2):151-176.