All articles published by MDPI are made immediately available worldwide under an open access license. No special
permission is required to reuse all or part of the article published by MDPI, including figures and tables. For
articles published under an open access Creative Common CC BY license, any part of the article may be reused without
permission provided that the original article is clearly cited. For more information, please refer to
Feature papers represent the most advanced research with significant potential for high impact in the field. A Feature
Paper should be a substantial original Article that involves several techniques or approaches, provides an outlook for
future research directions and describes possible research applications.
Feature papers are submitted upon individual invitation or recommendation by the scientific editors and must receive
positive feedback from the reviewers.
Editor’s Choice articles are based on recommendations by the scientific editors of MDPI journals from around the world.
Editors select a small number of articles recently published in the journal that they believe will be particularly
interesting to readers, or important in the respective research area. The aim is to provide a snapshot of some of the
most exciting work published in the various research areas of the journal.
A new explicit form is provided for the solution of optimal stopping problems involving a multidimensional geometric Brownian motion. A free-boundary value approach is adopted and the value function is obtained via fundamental solution methods. There are many applications for the valuation of perpetual options of American style, which are of interest for finance and managerial decisions.
Despite the huge progress on the pricing methods for American options, an explicit solution for their value is known only for options with infinite maturity written on a single underlying asset. Under classical assumptions on the underlying assets, the problem can be formulated as a free-boundary value for a Black–Scholes-type PDE. When the underlying stochastic processes are geometric Brownian motions (hereafter GBM) the value-matching and smooth-fit conditions apply, but finding the exercise threshold in explicit form remains an elusive research question. Early exercise boundary approximation techniques have been developed in many articles to obtain the option price and hedge ratio (see the Refs. Ait-Sahlia and Lai (2001); Huang et al. (1996); Kim (1990), among others). It is almost impossible to summarize all approaches to approximate the price of American options, starting from the first attempts to replace the exercise boundary with a finite set of discrete points (Geske and Johnson 1984) or to resort to finite-difference schemes (Brennan and Schwartz 1977).
While multi-asset European options have been priced extensively in the literature even in a non-Gaussian Black–Merton–Scholes framework (see the Ref. Agliardi (2012), for a comprehensive approach), there are few studies on the American option written on several assets. For these options, an explicit solution is not available even in the perpetual case. Various payoff functions for American options on two assets have been considered in the Ref. Broadie and Detemple (1997), which remains a fundamental work for the multiasset case. On the other hand, the regularity of the free boundary for multi-asset American options has been studied in the Ref. Laurence and Salsa (2009).
Recently, some traditional numerical schemes and simulation-based approaches have been applied to price these American-style options. In particular, finite difference schemes are used to discretize the complementarity problem involving Black–Scholes-type equations (see, e.g., the Ref. Nielsen et al. 2008; O’Sullivan and O’Sullivan 2011, and a comonotonic finite difference method in the Ref. Hanbali and Linders 2019), radial basis techniques are used in the Ref. Egorova et al. (2018), while least-square Monte Carlo algorithms extending the Ref. Longstaff and Schwartz (2001) are another popular approach in this field (see the Ref. Chan et al. 2006; Samimi and Mehrdoust 2018). Overall, the pricing of multi-asset American options is recognized in the literature as a quite difficult issue from a computational point of view; on the other hand, the lack of analytical expressions for the solution renders the judgement of the accuracy of the approximation a hard task.
For single-asset models, there exist several articles studying the pricing problem under more general stochastic processes than GBM, such as Lévy processes (Boyarchenko and Levendorskiĭ 2002; Gukhal 2001; Levendorskiĭ 2004; Mordecki 2002). For multi-asset options the pricing problem has no explicit solution in general, even in the GBM case. In some special cases (e.g., two-dimensional exchange options) an explicit solution can be obtained in an elementary way. When the options are written on a basket of risky assets and the payoff function is non-homogeneous, only numerical methods are available (e.g., finite difference schemes). The most difficult task remains to identify the optimal exercise boundary. Thus, in this paper, we concentrate on this case, which still deserves investigation. We use an analytical approach to solve the pricing problem for perpetual American options in the multidimensional case. We obtain an explicit expression for the option value which is written as an integral involving modified Bessel functions of the second kind (see Section 3). A proof relating modified Bessel functions and the n-dimensional modified Helmoltz equation is provided in the Appendix A, which is of theoretical interest for other possible applications.
Although perpetual options are rare, they can be used in efficient numerical procedures to approximate the price of American options with finite maturity, for example, Carr’s randomization method (Carr 1998). More importantly, perpetual options provide a technical tool to attack several problems arising in real option theory (see the Ref. Boyarchenko and Levendorskiĭ 2007; Dixit and Pindyck 1994), and thus their scope of application spans far beyond financial derivatives.
2. Problem-Setting and Motivating Examples
In what follows, we assume that the value of the underlying asset, , follows an -dimensional geometric Brownian motion with respect to a given filtration, , representing the available information. (See Section 3 for the detailed notation).
Given a positive payoff function , usually a continuous function, and a risk-free interest rate , the rational price of a perpetual American option with instantaneous payoff is obtained as
where the supremum is taken over all stopping times w.r.t. . Here, denotes the expectation operator conditional on The optimal stopping time represents the optimal exercise time. In the Ref. Shiryaev (2008), existence of is obtained when and is the first entry time of into a stopping region, S (or exercise region) where equals G, that is, it is optimal to exercise the option immediately. On the contrary, is in the so-called continuation region, C. The boundary between the two regions is the optimal exercise boundary, The boundary-value formulation of this optimal stopping problem is as follows:
with in C and in S. Here, L is the differential operator associated with , which is defined in (6).
Some examples of payoff functions for multi-asset American options are reported below. Let and let denote the call/put attribute, with denoting a call option and a put option.
Basket (or index) options:
where the are the weights in the basket.
A special case is represented by the
Arithmetic average option:
The special case of exchange options is obtained setting .
Note that in (1) we can safely replace G with because it is not optimal to exercise the option when its payoff is negative. Moreover, arguing as in the Ref. Boyarchenko and Levendorskiĭ (2002), we will assume that can be written in terms of a cash flow , so that the valuation problem reduces to maximizing the following expected value:
where g represents the revenue flow which is acquired by exercising the option. In the case of regular payoff functions, this method can be easily motivated as follows. If is with absolutely continuous first derivatives (and other technical conditions), Ito’s formula implies that
and we denote by Note that g is a distribution rather than a function if is not sufficiently regular, so this method can be used also for less regular functions. If we assume that , then
Thus the expected value of equals and the optimal exercise problem reduces to maximizing (3). Alternatively, one can maximize
Then we can confine the analysis to the optimal stopping problem for (4). If V is the value function for the optimal stopping problem for (4), then is the option price. The boundary-value formulation of problem (4) is as follows:
A typical assumption on g is that g is measurable and satisfies an integrability condition of the form:
Note that if , then the optimal stopping time is ∞, while if it is optimal to stop immediately. To avoid such trivial situations one needs to assume that g changes sign.
Let denote the instantaneous dividend yield which is paid on the ith risky asset. Then, in view of the argument above, some examples of payoff streams, g, associated with American options are:
for a basket option;
for a spread option;
for a max-option in where is the Lebesgue measure on the line
For convex payoff streams, the following property can be proved.
Let g be a convex function. Then the stopping set for
Let us prove that V, the value function for this problem, is convex. Let , , and denote by Let denote the stochastic process started at . In view of convexity of g for any stopping time we have:
Then taking the of the left-hand side we get
which implies that V is convex. Let us show that the stopping set is convex arguing by contradiction. If and then which yields a contradiction. □
3. Valuation Formula
Assume that the option is written on n risky assets and let r denote the risk-free interest rate. Each risky asset pays a continuously compounded dividend yield, , per unit of time. The risk factors are modelled throughout an n-dimensional geometric Brownian motion, , where the dimension represents the number of components in the underlying portfolio. Its components are of the form
where is an n-dimensional Wiener process with respect to a given filtration, , and with where the non-negative definite matrix has Let denote for all Assume that is a Hermitian positive definite matrix. In what follows, will be denoted by
The differential operator associated with the process is the following differential operator in :
where , .
Let g denote the payoff stream associated with a perpetual option. The boundary-value formulation of the problem is as follows:
where denotes the continuation region, where it is not optimal to exercise the option, while denotes the early exercise region. The early exercise boundary, , separating the two regions should be determined as part of the problem. In the sequel we will consider explicitly the case where it can be represented as a function and we refer to Peskir (2019) and to Peskir and Shiryaev (2006) for the continuity property of the exercise boundary.
In order to solve problem (7), first we make the following change of variables:
where we choose such that
The continuation region is mapped into and problem (7) becomes:
Here and k is a real nonnegative number.
Let us now consider the Cholesky decomposition for the matrix , that is, where T is a lower triangular matrix (with positive diagonal entries). For example, in the two-dimensional case we can write:
Changing to variables and denoting the differential equation is transformed into:
where is the n-dimensional Laplace operator in the variable Z and The continuation region is transformed into . For example, if C is of the form
then takes the form
where are the entries of and denotes the first rows of .
Then we can write the solution for this modified Helmholtz-type equation in terms of its fundamental solution, , as follows:
The n-dimensional Green kernel can be computed in terms of modified Bessel functions of the second kind, . (See Appendix A). More precisely:
Changing back variables, we can write:
where, in the above-mentioned case,
If we insert the optimal boundary into the above-obtained expression for the value function then we get
that is, we get an equation for the unknown function . We need a guess for b and then we improve it through numerical iteration. For example, one can start with a hyperplane joining the points (if any) where the optimal boundary meets the Cartesian axes, as these points can be obtained solving one-dimensional problems in explicit form.
The advantage of having the equation for the unknown threshold derived above lies in the possibility of controlling for the approximated expression for b, which provides guidance for the numerical procedure. As Bessel functions are included in built-in-function packages in various numerical softwares, the formula above can be easily computed numerically and thus the procedure can be implemented in practice. In the next section we connect our analytical procedure to its probabilistic interpretation.
4. Probabilistic Interpretation
In this Section we provide a probabilistic interpretation for the expression of the value function obtained in Section 3. It is known (see the Ref. Revuz and Yor 1999) that the transition density of a Bessel process of order can be expressed throughout modified Bessel functions of order , that is:
On the other hand, Bessel processes can be connected to multi-dimensional Wiener processes. More precisely, in view of a relationship between n-dimensional Brownian motions and Bessel processes of order , one can compute the resolvent kernel for the operator in (10) in terms of modified Bessel functions of order . Consider an n-dimensional Wiener process, , where denotes the probability density function of . Then the Green kernel
can be computed as
using an identity for this integral transform which is found, for instance, in the Ref. Erdélyi et al. (1953).
As this Green kernel is the inverse of the operator , we obtain the expression for the fundamental solution which is derived in Section 3 through a different method. (See the Appendix A).
5. Discussion and Concluding Remarks
The problem of finding an exact solution for American options on a multiple underlying asset has never been solved, even in the case of a perpetual option. As Firth (2005) claims, “Pricing single asset American options is a hard problem in mathematical finance.…Pricing multi–asset (high-dimensional) American options is still more difficult”. As we discussed in the introduction, numerous simulation-based methods and numerical schemes have been proposed to find an approximate solution to this challenging pricing problem. The problem can be formulated starting from the stochastic differential equations modelling the underlying assets and then employing Monte Carlo simulations to derive approximate solutions; alternatively, one can consider the problem in its partial differential formulation and construct a discretization with good properties and accuracy. These approximation methods are based on traditional techniques which have been successfully employed with European-type derivatives, but need extra effort when applied to American-type derivatives, which require additional information about the future paths of all underlying assets.
One popular methodology employs the least-square Monte Carlo approach introduced by Longstaff and Schwartz (2001). While Monte Carlo methods are widely known and easy to implement in principle, they require simulation of the paths of each single stock; the combination with least-square estimations for American options further slows down the algorithm. Thus, the execution can become prohibitively time-consuming when the basket size underlying the option is large and various technical improvements have been developed to reduce the computational effort (e.g., multi-level Monte Carlo simulation along with proper numerical schemes, regression at the end of each time step, splitting of the simulation into several processes and parallel implementation, quantization methods).
Another class of approximation methodologies is based on finite difference approximations of the differential operator which also contains second-order mixed derivative terms. Thus, the classical finite difference methods lead to some off-diagonal entries in the matrix associated with the discretized operator, which determines oscillations in the computed solution and the difference approximation may become unstable. When the size of the underlying basket is large, the numerical scheme may become unstable, the results inaccurate, and sometimes pricing multi-dimensional derivatives may even be impossible. Some improvements have been devised to cope with these deficiencies related to multi-dimensionality (see Hanbali and Linders 2019; Wu 2013).
In any case, in the absence of an analytical solution, benchmarking the numerically computed solutions is a hard task and testing the accuracy of the approximation requires ad hoc theorems for each algorithm.
In this paper, instead of proposing another numerical procedure, we follow an analytical approach to solve the pricing problem for perpetual American options in the multidimensional case. We obtain an explicit expression for the option value which is written as an integral involving modified Bessel functions of the second kind. Although these functions are non-elementary ones, they are included in built-in-function packages in various numerical softwares, and thus can be computed numerically in practice.
At the end of Section 3 we provided an equation to determine the optimal exercise threshold. One limitation of this result is that this equation requires an n-dimensional integration, which makes effective evaluation difficult to perform for high dimensions. For low dimensions, the results is of practical use; for higher dimensions, it can be combined with a numerical solution and may serve as a benchmark to test the reliability of the numerically computed solution.
Finally, our analysis is limited to the case of perpetual options. Although they are of limited usage in the financial markets, the analysis of perpetual options suggests methods for pricing American options with finite maturity (Carr 1998). More importantly, our results are applicable to real options where usually no expiration date is prescribed to the decision process. For the real option theory, our result is valuable as it solves a long-standing open problem: how to express the value function of a multi-factor optimal decision problem in a mathematically rigorous form. This opens the way to further applications in management science.
Conceptualization, E.A. and R.A.; methodology, R.A.; writing—original draft preparation and review, E.A. and R.A. All authors have read and agreed to the published version of the manuscript.
This research received no external funding.
Informed Consent Statement
Data Availability Statement
No empirical data were used in this research.
Conflicts of Interest
There are no potential conflicts of interest to disclose.
Appendix A. Bessel Functions and Modified Helmoltz Equation
Let denote a modified Bessel function of the second kind, which satisfies
solves the differential equation:
Moreover, the following properties hold:
Now we show that is the fundamental solution of
Let . Then
where the integral is written in polar coordinates, , , and the contribution of is null as the function is radial. In view of Equation (A1) we have
and thus integration by parts in yields:
Using (A2) and (A3) the last integral can be written as
Ait-Sahlia, Farid, and Tze Leung Lai. 2001. Exercise boundaries and efficient approximation to American option prices and hedge parameters. Journal of Computational Finance 4: 85–104. [Google Scholar]
Agliardi, Rossella. 2012. A comprehensive mathematical approach to exotic option pricing. Mathematical Methods in the Applied Sciences 35: 1256–68. [Google Scholar] [CrossRef]
Boyarchenko, Svetlana I., and Sergei Z. Levendorskiĭ. 2002. Perpetual American options under Lévy processes. SIAM Journal on Control and Optimization 40: 1663–96. [Google Scholar] [CrossRef]
Boyarchenko, Svetlana I., and Sergei Z. Levendorskiĭ. 2007. Irreversible Decisions under Uncertainty: Optimal Stopping Made Easy. Studies in Economic Theory 27. Berlin/Heidelberg: Springer. [Google Scholar]
Brennan, Michael J., and Eduardo S. Schwartz. 1977. The valuation of the American put option. Journal of Finance 32: 449–62. [Google Scholar] [CrossRef]
Broadie, Mark, and Jerome Detemple. 1997. The valuation of American options on multiple assets. Mathematical Finance 17: 241–86. [Google Scholar] [CrossRef]
Chan, Raymond H., Chi-Yan Wong, and Kit-Ming Yeung. 2006. Pricing multi-asset American-style options by memory reduction Monte Carlo methods. Applied Mathematics and Computation 179: 535–44. [Google Scholar] [CrossRef]
Dixit, Robert K., and Robert S. Pindyck. 1994. Investment Under Uncertainty. Princeton: Princeton University Press. [Google Scholar]
Egorova, Vera N., Lucas Jodar, and Fazlollah Soleymani. 2018. A local radial basis function method for high-dimensional American option pricing problems. Mathematical Modelling and Analysis 23: 117–38. [Google Scholar]
Erdélyi, Arthur, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi. 1953. Higher Trascendental Functions. New York: McGraw-Hill. [Google Scholar]
Firth, Neil P. 2005. Pricing High-Dimensional American Options. Ph.D. thesis, University of Oxford, Oxford, UK. [Google Scholar]
Geske, Robert, and Herbert E. Johnson. 1984. The American put option valued analytically. Journal of Finance 39: 1511–24. [Google Scholar] [CrossRef]
Gukhal, Chandrasekhar Reddy. 2001. Analytical valuation of American options on jump-diffusion processes. Mathematical Finance 11: 97–115. [Google Scholar] [CrossRef]
Hanbali, Hamza, and Daniel Linders. 2019. American-type basket option pricing: A simple two-dimensional Partial Differential Equation. Quantitative Finance 19: 1689–704. [Google Scholar] [CrossRef]
Huang, Jing-zhi, Marti G. Subrahmanyam, and G. George Yu. 1996. Pricing and hedging American options: A recursive integration method. Review of Financial Studies 9: 277–300. [Google Scholar] [CrossRef][Green Version]
Kim, In Joon. 1990. The analytic valuation of American options. Review of Financial Studies 3: 547–72. [Google Scholar] [CrossRef]
Laurence, Peter, and Sandro Salsa. 2009. Regularity of the free-boundary of an American option on several assets. Communications on Pure and Applied Mathematics 62: 969–94. [Google Scholar] [CrossRef]
Levendorskiĭ, Sergei Z. 2004. Pricing of American put under Lévy processes. International Journal of Theoretical and Applied Finance 7: 303–35. [Google Scholar] [CrossRef][Green Version]
Longstaff, Francis A., and Eduardo S. Schwartz. 2001. Valuing American options by simulation: A simple least-square approach. Review of Financial Studies 14: 113–47. [Google Scholar] [CrossRef][Green Version]
Mordecki, Ernesto. 2002. Optimal stopping and perpetual options under Lévy processes. Finance and Stochastics 6: 473–93. [Google Scholar] [CrossRef]
Nielsen, Bjørn Fredrik, Ola Skavhaug, and Aslak Tveito. 2008. Penalty methods for the numerical solution pf American multi-asset problems. Journal of Computational and Applied Mathematics 222: 3–16. [Google Scholar] [CrossRef][Green Version]
O’Sullivan, Stephen, and Conall O’Sullivan. 2011. On the acceleration of explicit finite difference methods for option pricing. Quantitative Finance 11: 1177–91. [Google Scholar] [CrossRef]
Peskir, Goran. 2019. Continuity of the optimal stopping boundary for two-dimensional diffusions. Annals of Applied Probability 29: 505–30. [Google Scholar] [CrossRef][Green Version]
Peskir, Goran, and Albert Shiryaev. 2006. Optimal Stopping and Free-Boundary Problems. Basel: Birkhäuser. [Google Scholar]
Revuz, Daniel, and Marc Yor. 1999. Continuous Martingales and Browian Motion. Berlin: Springer. [Google Scholar]
Samimi, Oldouz, and Farshid Mehrdoust. 2018. Pricing multi-asset American option under Heston stochastic volatility model. International Journal of Financial Engineering 5: 1850026. [Google Scholar] [CrossRef]
Shiryaev, Albert N. 2008. Stochastic Modelling and Applied Probability 8. In Optimal Stopping Rules. Translated from the 1976 Russian second edition. Berlin: Springer. [Google Scholar]
Wu, Xianbin. 2013. Accurate numerical method for pricing two-asset American put options. Journal of Function Spaces and Applications 2013: 189235. [Google Scholar] [CrossRef][Green Version]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely
those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or
the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas,
methods, instructions or products referred to in the content.