Asymptotic expansion of solutions to the Black-Scholes equation arising from American option pricing near the expiry

详细信息    查看全文

• 作者：Seyed-Mohammad-Mahdi Kazemi^a ; ^{smm.kazemi@aut.ac.ir" class="auth_mail" title="E-mail the corresponding author} ; Mehdi Dehghan^a ; ^{mdehghan@aut.ac.ir" class="auth_mail" title="E-mail the corresponding author} ; ^{mdehghan.aut@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Ali Foroush Bastani^b ; ^{bastani@iasbs.ac.ir" class="auth_mail" title="E-mail the corresponding author}
• 关键词：91G60 ; 34E05 ; 35R35 ; 65N15
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：311
• 期：Complete
• 页码：11-37
• 全文大小：1924 K

文摘

Our aim in this paper is to approximate the price of an American call option written on a dividend-paying stock close to expiry using an asymptotic analytic approach. We use the heat equation equivalent of the Black–Scholes partial differential equation defined on an unbounded spatial domain and decompose it into inner and outer problems. We extend the idea presented in [H. Han and X. Wu, A fast numerical method for the Black-Scholes equation of American options, SIAM Journal on Numerical Analysis 41 (6) (2003) 2081-2095.] in which a weakly singular memory-type transparent boundary condition (TBC) is obtained for the special case that the initial condition is equal to zero. We first derive this TBC in the general case and then focus on the outer problem in conjunction with an equivalent non-singular version of the TBC (dubbed ETBC) which is more tractable for analytical purposes. We then obtain the general solution of the outer problem in series form based on “the repeated integrals of the complementary error function” which also satisfies the introduced ETBC. As the next step, using the machinery of Poincaré asymptotic expansion and taking “time-to-expiry” as the expansion parameter, we find the general term of this series in closed form when the risk-free interest rate (an id="mmlsi63" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716303053&_mathId=si63.gif&_user=111111111&_pii=S0377042716303053&_rdoc=1&_issn=03770427&md5=c62fa2aa6f0741d4da2bf7804c93f789" title="Click to view the MathML source">ran>an class="mathContainer hidden">an class="mathCode">ath altimg="si63.gif" overflow="scroll">rath>an>an>an>) is less than the dividend yield (an id="mmlsi64" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716303053&_mathId=si64.gif&_user=111111111&_pii=S0377042716303053&_rdoc=1&_issn=03770427&md5=e85c6c92bf65abce025a78a9dcbe3a22" title="Click to view the MathML source">δan>an class="mathContainer hidden">an class="mathCode">ath altimg="si64.gif" overflow="scroll">δath>an>an>an>). We also obtain the first five terms in the opposite case (an id="mmlsi2" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716303053&_mathId=si2.gif&_user=111111111&_pii=S0377042716303053&_rdoc=1&_issn=03770427&md5=d2ff20e922806090a94e07e9ecef70d5" title="Click to view the MathML source">r>δan>an class="mathContainer hidden">an class="mathCode">ath altimg="si2.gif" overflow="scroll">r>δath>an>an>an>) in a systematic manner. We also prove the convergence properties of the obtained series rigorously under some general conditions. Our numerical experiments based on the obtained asymptotic series, demonstrate the applicability and effectiveness of the results in valuation of a wide range of American option problems.