有限期美式封顶股票抵押贷款的定价
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摘要
本文研究有限期的美式封顶式股票抵押贷款的定价问题.股票抵押贷款是一种用股票作为抵押品的贷款产品,它们的定价问题是一种最优停时问题.带封顶的股票抵押贷款通过设定股票价格的上限,将"上限"功能纳入股票抵押贷款中,这样贷款人就可以避免由于股价上涨而造成巨大损失.本文利用随机分析方法推导出最优执行边界函数的积分方程,从而得到有限期的美式封顶式股票抵押贷款价格的解析公式.本文还进一步研究一类随机利率模型下美式封顶式股票抵押贷款的定价问题,通过数值算例分析最优执行边界的性质.
In this paper we study the valuation of finite-maturity American capped stock loan using the stochastic approach. The stock loans are loans with stocks as collateral and their valuation is a kind of optimal stopping problems. The capped stock loan is a kind of product incorporating a "cap" feature to the stock loans by setting a limit for the stock price, in which way the lender can avoid the possible large loss due to the rise of the stock price. This paper utilizes the stochastic approach to derive an integral equation for the optimal exercise boundary function and then obtain the analytical formulas for the price of the finite-maturity American capped stock loan.A stochastic interest rate model is also studied in this paper. Numerical examples are carried out to investigate the properties of the optimal exercise boundaries.
In this paper we study the valuation of finite-maturity American capped stock loan using the stochastic approach. The stock loans are loans with stocks as collateral and their valuation is a kind of optimal stopping problems. The capped stock loan is a kind of product incorporating a "cap" feature to the stock loans by setting a limit for the stock price, in which way the lender can avoid the possible large loss due to the rise of the stock price. This paper utilizes the stochastic approach to derive an integral equation for the optimal exercise boundary function and then obtain the analytical formulas for the price of the finite-maturity American capped stock loan.A stochastic interest rate model is also studied in this paper. Numerical examples are carried out to investigate the properties of the optimal exercise boundaries.
引文
1 Xia J,Zhou X Y.Stock loans.Math Finance,2007,17:307-317
2 Dai M,Xu Z Q.Optimal redeeming strategy of stock loans with finite maturity.Math Finance,2011,21:775-793
3 Pascucci A,Suárez-Taboada M,Vázquez C.Mathematical analysis and numerical methods for a PDE model of a stock loan pricing problem.J Math Anal Appl,2013,403:38-53
4 Lu X,Putri E R M.Semi-analytic valuation of stock loans with finite maturity.Commun Nonlinear Sci Numer Simul,2015,27:206-215
5 Liu G,Xu Y.Capped stock loans.Comput Math Appl,2010,59:3548-3558
6 Liang Z,Wu W,Jiang S.Stock loan with automatic termination clause,cap and margin.Comput Math Appl,2010,60:3160-3176
7 Broadie M,Detemple J.American capped call options on dividend-paying assets.Rev Financ Stud,1995,8:161-191
8 Chen W,Xu L,Zhu S P.Stock loan valuation under a stochastic interest rate model.Comput Math Appl,2015,70:1757-1771
9 Kim I J,Yu G G.An alternative approach to the valuation of American options and applications.Rev Deriv Res,1996,1:61-85
10 Bernard C,MacKay A,Muehlbeyer M.Optimal surrender policy for variable annuity guarantees.Insurance Math Econom,2014,55:116-128
11 Bj?rk T.Arbitrage Theory in Continuous Time.Oxford:Oxford University Press,2004
12 Karatzas I,Shreve S E.Brownain Motion and Stochastic Calculus.New York:Springer,1991
13 Jeanblanc M,Yor M,Chesney M.Mathematical Methods for Financial Markets.Berlin:Springer,2006
2 Dai M,Xu Z Q.Optimal redeeming strategy of stock loans with finite maturity.Math Finance,2011,21:775-793
3 Pascucci A,Suárez-Taboada M,Vázquez C.Mathematical analysis and numerical methods for a PDE model of a stock loan pricing problem.J Math Anal Appl,2013,403:38-53
4 Lu X,Putri E R M.Semi-analytic valuation of stock loans with finite maturity.Commun Nonlinear Sci Numer Simul,2015,27:206-215
5 Liu G,Xu Y.Capped stock loans.Comput Math Appl,2010,59:3548-3558
6 Liang Z,Wu W,Jiang S.Stock loan with automatic termination clause,cap and margin.Comput Math Appl,2010,60:3160-3176
7 Broadie M,Detemple J.American capped call options on dividend-paying assets.Rev Financ Stud,1995,8:161-191
8 Chen W,Xu L,Zhu S P.Stock loan valuation under a stochastic interest rate model.Comput Math Appl,2015,70:1757-1771
9 Kim I J,Yu G G.An alternative approach to the valuation of American options and applications.Rev Deriv Res,1996,1:61-85
10 Bernard C,MacKay A,Muehlbeyer M.Optimal surrender policy for variable annuity guarantees.Insurance Math Econom,2014,55:116-128
11 Bj?rk T.Arbitrage Theory in Continuous Time.Oxford:Oxford University Press,2004
12 Karatzas I,Shreve S E.Brownain Motion and Stochastic Calculus.New York:Springer,1991
13 Jeanblanc M,Yor M,Chesney M.Mathematical Methods for Financial Markets.Berlin:Springer,2006