A quasi-closed-form solution for the valuation of American put options

Viegas, Christina; Azevedo-Pereira, José Article A quasi-closed-form solution for the valuation of American put options International Journal of Financial Studies Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Viegas, Christina; Azevedo-Pereira, José (2020) : A quasi-closed-form solution for the valuation of American put options, International Journal of Financial Studies, ISSN 2227-7072, MDPI, Basel, Vol. 8, Iss. 4, pp. 1-10, https://doi.org/10.3390/ijfs8040062 This Version is available at: https://hdl.handle.net/10419/257729 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ International Journal of Financial Studies Article A Quasi-Closed-Form Solution for the Valuation of American Put Options Cristina Viegas 1,* and JoséAzevedo-Pereira 2 1Faculty of Economics, University of the Algarve, CEFAGE (UAlg) Center for Advanced Studies in Management and Economics and Research Centre for Tourism, Sustainability and Well-Being (CinTurs); Campus de Gambelas, Edifício 9, 8005-139 Faro, Portugal 2ISEG—Lisbon School of Economics and Management, Universidade de Lisboa, Rua Miguel Lupi, 20, 1249-078 Lisboa, Portugal; [email protected] *Correspondence: [email protected] Received: 18 September 2020; Accepted: 12 October 2020; Published: 16 October 2020   Abstract: This study develops a quasi-closed-form solution for the valuation of an American put option and the critical price of the underlying asset. This is an important area of research both because of a large number of transactions for American put options on different underlying assets (stocks, currencies, commodities, etc.) and because this type of evaluation plays a role in determining the value of other financial assets such as mortgages, convertible bonds or life insurance policies. The procedure used is commonly known as the method of lines, which is considered to be a formulation in which time is discrete rather than continuous. To improve the quality of the results obtained, the Richardson extrapolation is applied, which allows the convergence of the outputs to be accelerated to values close to reality. The model developed in this paper derives an explicit formula of the finite-maturity American put option. The results obtained, besides allowing us to quickly determine the option value and the critical price, enable the graphical representation—in two and three dimensions—of the option value as a function of the other components of the model. Keywords: option valuation; American put option; quasi-closed-form solution JEL Classification: G13 1. Introduction The value of a European option is easily determined by applying the formula derived by Black and Scholes (1973) . In turn, the valuation of an American put option seems to be a much more complex task. The possibility of early exercise by the holder of such options is the main obstacle to determining a closed-form solution. It is thus necessary to determine the critical price of the underlying asset, below which the option must be exercised immediately. This calculation represents a major problem in obtaining a closed-form solution. Zhu et al. (2018) report that a closed-form solution is only possible in some special cases. For example, works such as those by Zhu (2006a) and Zhao and Wong (2012) have developed a closed form solution for the optimal exercise boundary and the option price of American put options by using the homotopy analysis method. However, the implementation of the infinite series solution of Zhu (2006a) is a very complex and time-consuming problem, and it requires a recursive procedure of solving differential equations. On the other hand, the application of the homotopy solution by Zhao and Wong (2012) requires a complex iterative integration. Recently, Leipold and Vasiljevic (2017) obtained the prices of American options in closed form in a hyper-exponential jump-diffusion framework. So, almost all the recent work uses a numerical method to solve the problem of pricing American options. For example, Ballestra (2018) develops Int. J. Financial Stud. 2020,8, 62; doi:10.3390/ijfs8040062 www.mdpi.com/journal/ijfs Int. J. Financial Stud. 2020,8, 62 2 of 16 a numerical method for American option pricing that combines the front-fixing approach and the repeated Richardson extrapolation. The paper shows that the method used yields very accurate and fast results. Much of the work that has presented analytical solutions for the option price use numerical methods to determine the critical price of the underlying asset. For example, studies such as Kim (1990) , Jacka (1991) and Carr et al. (1992), consider that the value of the American option is the value of the European option plus a premium that is given by the discounted value of the potential gains inherent to early exercise. This premium is given by an integral equation, while the exercise boundary is determined by a reversible numerical procedure. In turn, studies such as Geske and Johnson (1984), Bunch and Johnson (1992), Ho et al. (1994), Huang et al. (1996) and Lee and Paxson (2003) develop analytical approximations for the American put option price by calculating the values of sets of options with discrete exercise dates. Based on the properties discussed in Geske and Johnson (1984) and Kim (1990), Fabozzi et al. (2016) present the extension method. In this method, they develop a quasi-analytic approach that aims to improve the performance of existing methods in calculating the value of American options with long maturities. In parallel, there are works that, by presenting analytical formulations for the price of American put options, explicitly determine the critical price of the underlying asset. Particularly prominent in this area is the work of Carr and Faguet (1996), Carr (1998) and Bunch and Johnson (2000), which offer significant advances in the calculation of the value of American put options using quasi-explicit solutions. For example, the derivation of an expression for the so-called critical price of the underlying asset of an American put option is one of the main objectives of Bunch and Johnson (2000). Once the value of this variable has been determined, the authors use it to determine the value of the American put option, using the method proposed in Huang et al. (1996). It should, however, be noted that the Bunch and Johnson (2000) solution is only valid for American put options whose underlying asset does not distribute dividends. Another reference in the field of analytical approximations for the critical stock prices of American options is Zhu (2006b). This article proposed a method based on the Laplace transform of the fundamental valuation partial differential equation. However, Li (2010) shows that the technique presented in Zhu (2006b) gives very inaccurate results to the critical stock price, being that the Average Percentage Error is comprised between 8 and 9%. Two other reference works are those of Carr and Faguet (1996) and Carr (1998). These two studies transform the general partial differential equation (the equation given by Black and Scholes (1973) , which was the basis for determining the value of the European option on the stock) into a non-homogeneous ordinary differential equation. This change implies the replacement of the derivative of price function with respect to time by a finite difference. The basic assumptions for the application of this procedure are different in the two articles. Thus, Carr (1998) justifies his procedure by classifying time as random and considering that it follows an Erlang or gamma distribution. Carr and Faguet (1996) , by contrast, report that obtaining an analytical solution requires the implementation of the method of lines, which means that the time component of the model ceases to be continuous and becomes discrete. This article focuses on the development of an analytical solution capable of evaluating an American put option, assuming that the underlying asset distributes dividends. This is an important area of research both because of a large number of transactions for American put options on different underlying assets (stocks, currencies, commodities, etc.) and because this type of evaluation plays a role in determining the value of other financial assets such as mortgages, convertible bonds or life insurance policies. In this study, the method of lines is followed; this is considered to be a formulation in which time is discrete rather than continuous. This method was used in the evaluation of options by Carr and Faguet (1996) and Meyer and Hoek (1997). The former study used this method to derive an exact valuation formula for perpetual options subject to a finite series of credit downgrades, which can be used to approximate the value of finite-lived American options. The later takes a numerical Int. J. Financial Stud. 2020,8, 62 3 of 16 approximation to calculate the value of an American option. The model developed in this paper represents a contribution to the existing literature on the application of the method of lines to calculate the value of the American put option, once it is derived an explicit formula of the finite-maturity American put option. The remainder of this paper is organized as follows. Section 2develops a model to evaluate American put options in which the underlying asset pays dividends, a closed-form solution being reached for the case of constant dividends and a quasi-closed-form solution where the dividends are proportional to the value of the underlying asset. In Section 3, the results obtained by applying the formulas derived in the previous section are presented numerically and graphically. They are also compared with other values obtained by different authors who have presented work in this area of research. Finally, the conclusion is presented, including suggestions for future developments. 2. The American Put Option Valuation Model 2.1. Analytical Solution for the Value of an American Put Option The model presents two variants: constant dividends, φ , and dividends in proportion to the price of the underlying asset, δ . In the case of constant dividends, the model developed in this work allows obtaining a closed form solution to the American put option price, while for dividends in proportion to the value of the underlying asset, the model allows only to reach a quasi-explicit solution, since the critical price is only liable to be found numerically. In the case of constant dividends, it is assumed that the initial value of the underlying asset is liable to be decomposed by the following expression, following Roll (1977): S=s+φ r(1−e−rτ)(1) where the second term on the right hand side corresponds to the updated value, in continuous time, of the fixed component of the future flow of dividends receivable in the period τ ,rrepresents the interest rate, and sthe sum of the updated value of the variable component of dividends receivable in the period τand the updated value of all dividends expected to be received after that period. In the formulation developed in this study, time is considered to be discrete, whereby (1) is replaced by the following configuration: S=s+D where D=φτ n 1−1 (1+rτ n)n rτ n (2) Drepresents the updated value of an income with n constant terms, which means that the continuing value of the fixed component of the dividends has been replaced by ndiscrete values equal to φτ n each one, with a time period between two consecutive terms equal to τ n . The total period of time of the possession of the instrument is equal to τ . The expression (2) may also present a simpler configuration: S=s+φ r 1−1 1+rτ nn (3) After these initial considerations relating to the form in which the fixed component of the future flow of dividends receivable in the period τ should be considered in the valuation process, the development of a methodology to determine the value of an American put option on an asset that pays dividends begins. From here, the formulas to present consider, alternatively, constant dividends, φ , or dividends in proportion to the underlying asset, δ . In other words, for steady dividends, we have δ= 0 and φ, 0, while for proportional dividends, φ= 0 and δ, 0 are considered. In contrast, Int. J. Financial Stud. 2020,8, 62 4 of 16 the whole formulation used for this purpose is a function of s, the underlying asset price minus the updated value of the fixed component of the dividends expected to be received during the period τ. The process to be developed is based on the technique used for the valuation of contingent claims. It is considered that the option value depends on a single stochastic variable: the price of the underlying asset. The evolution of this variable is defined according to the following stochastic process: ds =(r−δ)sdt +σssdzs(4) where r represents the short-term risk-free interest rate, δ the rate of dividend distribution, σs the instantaneous standard deviation and zsthe standard Wiener process. Considering that the option value, P(s,τ) , depends on the stochastic variable—the price of the underlying asset—and that it follows the process defined in (4), the partial differential equation may be deduced: −∂P(τ,s) ∂τ +∂P(τ,s) ∂s(r−δ)s+1 2σS2s2∂2P(τ,s) ∂s2−rP(τ,s)=0 ifs≥sm(5) where sm corresponds to the critical value, the underlying asset price, less the updated value of the fixed component of the dividends expected to be received during the period τ. Note that, as with the underlying asset price, the following equality is also checked within the critical price: Sm=sm+φ r 1−1 1+rτ nn (6) The method to be developed lies in the transformation of the general partial differential equation into a non-homogeneous ordinary differential equation. This change is made by applying the method of lines. It involves replacing the derivative of the price function, with respect to time, by a finite difference, while the derivatives in relation to the prices of the underlying asset remain unchanged. This process results in a sequence of non-homogeneous ordinary differential equations that must be solved. The shorter the time interval considered in the finite difference, the higher the quality of the results reached with this change. So, to attain a good estimation of the value of the option, it would be convenient to consider time as being divided into a number of intervals tending towards infinity. However, a big number of time intervals means a significant increase in mathematical computation. Thus, there is a tendency to use a relatively small number of time intervals, and then the Richardson extrapolation is applied (the Richardson extrapolation has been applied to studies in the area with success; see, for example, Carr (1998)) in order to accelerate the convergence of the results obtained to values close to reality. For example, Carr and Faguet (1996) and Carr (1998) show that the subdivision of the total time of ownership of the option into four intervals leads to very acceptable estimates. So, in this paper, time is subdivided into four intervals. Thus, Equation (5) takes on the following general configuration for s≥sm: dP(m)(s) ds (r−δ)s+1 2σS2s2d2P(m)(s) ds2−rP(m)(s)=P(m)(s)−P(m−1)(s) τ n (7) P(m)(s) corresponds to the value of the option and mcorresponds to the number of sub-periods in relation to the total number of time intervals, n= 4, considered under the model. Thus, the expression of the value of the American put option is determined, relative to mτ n periods of time, with m= 1, 2, 3, 4 and m≤n. So, (7) is a non-homogeneous ordinary differential equation in which the option price becomes a function of a single variable. To determine P(m)(s) , we need to know the expression of P(m−1)(s) , whereby this equation must be solved for m=1, 2, 3, 4. Int. J. Financial Stud. 2020,8, 62 5 of 16 The value of the American put option should confirm (7) in conjunction with the following boundary conditions: lim s→∞P(m)(s)=0 (8) lim s→sm P(m)(s)=K−Sm(9) lim s→sm dP(m)(s) ds =−1 (10) where Kcorresponds to the strike price. To achieve the objective of determining the value of the American put option, by solving (7), subject to the restrictions (8–10), the procedure used in studies such as Huang et al. (1996) applies, where the value of the American option is the value of the European put plus a term that represents the early exercise premium of the American put. Thus: P(m)(s)=p(m)(s)+pr(m)(s)ifs≥sm(11) where p(m)(s) is the value of the European put and pr(m)(s) corresponds to the value of the early exercise premium of the American put. In turn, the European put option should confirm (7) and the following boundary conditions: p(m)(s)=max(K−S, 0)for τ=0 (12) lim s→0p(m)(s)=K 1+rτ nn(13) lim s→∞p(m)(s)=0 (14) The respective early exercise premium should confirm (7) and the following condition: lim s→∞pr(m)(s)=0 (15) By using this procedure, the general closed-form solution for the value of the American put option is given by different expressions, according to the range of variation in the price of the underlying asset. Thus, for s≥K , m= 1, 2, 3, 4, n= 1, 2, 3, 4 and m≤n , the solution of (7), subject to the restrictions (12), (13), and (14), leads to the value of the European put option: p[s≥K](m)(s)=s 2τδ+β−√αn 2τσ2am+2nam−1τσ2 αn+ln(s) √αn +2n2am−2ln(s) √αn+2τσ2 αn2 +n4am−3 330τ3σ6 αn3+30τ2σ4ln(s) αn 5 2 +9σ2τln(s)2 αn2+ln(s)3 αn 3 2!# (16) where: αn=2rτ+τσ22+4δτ2−2r+δ+σ2+8nσ2for n=1, 2, 3, 4 β=τσ2−2rτ am−iis a parameter with am−i=0 for m−i≤0 and i=1,2,3. Int. J. Financial Stud. 2020,8, 62 6 of 16 The same reasoning is followed to determine the value of the early exercise premium, which must confirm the boundary condition defined in (15). Thus, for s≥s1 , m= 1, 2, 3, 4, n= 1, 2, 3, 4 and m≤n , the value of the premium is: pr(m)(s)=s 2τδ+β−√αn 2τσ2dm+2ndm−1τσ2 αn+ln(s) √αn +2n2dm−2ln(s) √αn+2τσ2 αn2 +n4dm−3 330τ3σ6 αn3+30τ2σ4ln(s) αn 5 2 +9σ2τln(s)2 αn2+ln(s)3 αn 3 2!# (17) where: dm−iis a parameter with dm−i=0 for m−i≤0 and i=1, 2, 3. Thus, it is possible to present the general solution in relation to the value of the American put option, when s≥Kand m=1, 2, 3, 4: P[s≥K](m)(s)=p[s≥K](m)(s)+pr(m)(s)(18) In the region s1≤s≤K , the value of the American put option also corresponds to the European option plus the premium, with the peculiarity that, in this area, the value of the European option to be determined is the call. The first step in solving (7) comprises of determining the option value at the time of the termination of the contract. This value corresponds to the payoffof the option, which is equal to zero in the out-of-money area of a European option. Thus, in the area s1≤s≤K and for τ= 0, the value of a European call is equal to zero. The put-call parity being subsequently applied to obtain the expression of the European put. In turn , the value of the premium is configured as defined in (17). Accordingly, the value of the European put, in the region s1≤s≤K , with m= 1, 2, 3, 4, n= 1, 2, 3, 4 and m≤n , is given by the following expression (again, the value of a European put must confirm the boundary conditions defined in (12)–(14)): p[s1≤s≤K](m)(s)=s 2τδ+β+√αn 2τσ2bm+2nbm−1τσ2 αn−ln(s) √αn +2n2bm−2ln(s) √αn−2τσ2 αn2 +n4bm−3 330τ3σ6 αn3−30τ2σ4ln(s) αn 5 2 +9σ2τln(s)2 αn2−ln(s)3 αn 3 2!# +K (1+rτ n)m−s (1+δτ n)m+em (19) where bm−iis a parameter with bm−i=0 for m−i≤0 and i=1,2,3. Whereby, the general expression for the amount of the American put option, when s1≤s≤K and m=1, 2, 3, 4, is given by: P[s1≤s≤K](m)(s)=p[s1≤s≤K](m)(s)+pr(m)(s)(20) Int. J. Financial Stud. 2020,8, 62 7 of 16 For s≤s1 , the option value is given by different expressions according to the range of variation of the price of the underlying asset. Thus, for s2≤s≤s1,m=2, 3, 4, n=2, 3, 4 and m≤n, we have: P[s2≤s≤s1](m)(s)=s 2τδ+β+√αn 2τσ2zm−1+2nzm−2τσ2 αn−ln(s) √αn +2n2zm−3ln(s) √αn−2τσ2 αn2#+K (1+rτ n)m−1 −s (1+δτ n)m−1− φ 1−1 (1+rτ n)n! r(1+rτ n)m−1 +s 2τδ+β−√αn 2τσ2ym−1+2nym−2τσ2 αn+ln(s) √αn +2n2ym−3ln(s) √αn+2τσ2 αn2# (21) where ym−iand zm−iare parameters with ym−iand zm−ifor m−i≤nand i=2,3. In turn, for s3≤s≤s2 , m= 3, 4, n= 3, 4 and m≤n , the formula for the value of the American put option is as follows: P[s3≤s≤s2](m)(s)=s 2τδ+β+√αn 2τσ2wm−2+2nwm−3τσ2 αn−ln(s) √αn +K (1+rτ n)m−2−s (1+δτ n)m−2− φ 1−1 (1+rτ n)n! r(1+rτ n)m−2 +s 2τδ+β−√αn 2τσ2xm−2+2nxm−3τσ2 αn+ln(s) √αn (22) where wm−iand xm−iare parameters with wm−i=0 and xm−i=0 for m−i≤0 and i=3. For the area s4≤s≤s3,m=4 and n=4, the option value corresponds to: P[s4≤s≤s3](m)(s)=u1s 2τδ+β+√αn 2τσ2+K 1+rτ n−s 1+δτ n − φ 1−1 (1+rτ n)n! r(1+rτ n)+v1s 2τδ+β−√αn 2τσ2 (23) where u1 and v1 are parameters. Finally, for s≤sm , with m= 1, 2, 3, 4 and m=n , the American put option is exercised immediately, whereby its value corresponds to: P[s≤sm](m)(s)=K−S(24) These different expressions for the value of the American put option contain a set of parameters ( am−i , dm−i , bm−i , ym−i , zm−i , wm−i , xm−i , u1 , v1 ) whose value is determined by solving a system of equations. For m=1, 2, 3, 4, n=1, 2, 3, 4, m≤nand s=K: p[s≥K](m)(s)=p[s1≤s≤K](m)(s) dp[s≥K](m)(s) ds = dp[s1≤s≤K](m)(s) ds (25) d2p[s≥K](m)(s) ds2= d2p[s1≤s≤K](m)(s) ds2 Int. J. Financial Stud. 2020,8, 62 8 of 16 For m=1, n=1, 2, 3, 4 and s=s1: P[s1≤s≤K](m)(s)=P[s≤s1](m)(s)(26) For m=2, 3, 4, n=2, 3, 4, m≤nand s=s1: P[s2≤s≤s1](m)(s)=P[s1≤s≤K](m)(s) dP[s2≤s≤s1](m)(s) ds = dP[s1≤s≤K](m)(s) ds (27) For m=2, 3, 4, n=2, 3, 4, m≤nand s=sn P[s≤sn](m)(s)=P[sn≤s≤sn−1](m)(s)(28) For m=3, 4, n=3, 4, m≤nand s=s2 P[s2≤s≤s1](m)(s)=P[s3≤s≤s2](m)(s) dP[s2≤s≤s1](m)(s) ds = dP[s3≤s≤s2](m)(s) ds (29) For s=s4: P[s4≤s≤s3](4)(s)=K−s4−φ r 1−1 1+rτ nn dP[s4≤s≤s3](4)(s) ds =−1 (30) To determine the value s1, the following equation must be solved for s=s1and n=1, 2, 3, 4: dP[s1≤s≤K](1)(s) ds =−1 (31) For δ, 0, the value s1 is given implicitly and must be found numerically (for numerical resolution, Maple software with “fsolve” was used). For δ=0 and φ,0, the value s1is calculated as follows: s1=exp σ2τln 4b12r2(n+rτ)2αn (−β+√αn)2φn−φ(n+rτ n)nn+φrτ+Kr2(n+rτ n)nτ−φ(n+rτ n)nrτ2+2nlnn+rτ n −β−√αn  (32) Whereby the critical value is equal to: S1=s1+φ r 1−1 1+rτ nn (33) In turn, for the value s=sm , m= 1, 2, 3, 4 and n= 2, 3, 4, the value of sm is determined by solving the following equation: dP[sm≤s≤sm−1](m)(s) dS =−1 (34) Int. J. Financial Stud. 2020,8, 62 15 of 16 Author Contributions: Conceptualization, C.V. and J.A.-P.; methodology, C.V.; software, C.V.; validation, C.V. and J.A.-P.; formal analysis, C.V. and J.A.-P.; investigation, C.V. and J.A.-P.; resources, C.V. and J.A.-P.; writing—original draft preparation, C.V.; writing—review and editing, C.V. and J.A.-P.; visualization, C.V. and J.A.-P.; funding acquisition, C.V. All authors have read and agreed to the published version of the manuscript. 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