Research paperPrimal-dual active set method for pricing American better-of option on two assets
Introduction
An option is a financial derivative, which can dramatically amplify the changes of the underlying asset price with leverage, and thus is an important tool in hedging strategy [8], [10], [16]. Ever since the option price was modeled by Black and Scholes as free boundary problem (BS equation) in 1973 [3], this model was regarded as the standard model by researchers and traders, and has been expanded to various option pricing models [10], [16]. The vanilla options can be divided into European style and American style based on the exercise time, and can also be divided into call options and put options. European options allow the owner to exercise the option only on the maturity dates, while American options can be exercised on or before the maturity dates. The former has a closed-form solution, but the latter doesn’t have closed-form solution, which could only be approximated by numerical approaches.
With the development of financial market, the investors and corporations have found that the vanilla options on single asset are not efficient for hedging or diversification [6], [18], [19]. Therefore, various options on multi-asset are developed for portfolios. According to the payoff function, there are five general categories of American multi-asset options: rainbow options, basket options, quotient options, product options, and spread options [27]. Options on multi-assets are also called cross-asset options. In particular, examples of rainbow options include: best-of option, call-max option, call-min option, put-max option, and put-min option [22]. Although multi-asset options may cover any number of assets, the two-asset options are the most useful in practice. In particular, the rainbow options on two-asset, which payoff depends on two underlying risky assets, each of which is referred to as a color of the rainbow, is very useful. The better-of option is a special case of best-of options, which payoff depends only on two underlying risky assets. This option, known as “option on the maximum of two risky assets” [26], was first proposed by Stulz in 1982 and was developed by Jiang and Haug as better-of option [8], [13]. By buying a better-of option, an investor is guaranteed to receive a higher return according to the better of the two assets at maturity than buying the two assets at the same price. The applications include option-bonds issued by firms, foreign currency bons, incentive and/or risk-sharing contracts, and so on [4].
Using the formula for the pricing of an option to exchange one asset for another, Stulz provided closed-formula for European best-of option on two risky assets [26]. There is no analytical formula for an American best-of option up till now, which needs to be calculated using some numerical methods. Numerical methods are frequently resorted for solving vanilla American options, such as binomial tree methods [5], finite difference methods [7], finite element methods [1], [29] and Monte Carlo methods [23] have been developed and extensively studied in recent decades. To the best of our knowledge, there exists few work on the numerical methods for American best-of options. In this paper, a primal-dual active-set (PDAS) method is proposed for solving the discretized linear complementarity problem arising from pricing an American best-of option. It is well known that the PDAS method is extremely efficient for quadratic programming problems [2], [20], which arose as a special case of generalized Moreau-Yosida approximations to non-differentiable convex functions [11]. Based on a modified augmented Lagrangian function, Bergounioux et al. have shown the global convergence of PDAS [2], and Hintermuller et al. have shown that the PDAS is also a special case of the Newton-type method under mild assumptions [9]. The PDAS method has been expanded to many applications. For example, Ito and Kunisch have use it to solve optimal control of variational inequalities problems [12], while Kanzow shows that it is an efficient and accurate method for large-scale linear complementarity problems [15], and so on.
The main challenges for the numerical treatment of the pricing of an American best-of option are twofold: (1) The singularity at the maturity time T have to be carefully treated to improve the accuracy. (2) There are two optimal exercise boundaries, which lead to more nonlinear terms on the boundaries for the BS equation. For the first issue, by using the similarity reduction technique, we obtain the variational inequality form for the American best-of option, and deduce the corresponding one-dimensional LCP on a bounded domain. Moreover, we adopt the geometric partition instead of uniform partition in the spatial direction to guarantee the accuracy around the singularity at the maturity time T. Using the finite difference method with uniform partition in temporal direction, we derive a large scale discretized optimization system. For the second problem, we will take advantage of PDAS method for solving the LCP. Firstly, using this method we can obtain highly accurate solutions by explicitly identifying sets of constraints. Meanwhile, the primal-dual technique combines two complementary ideas that leads to rapid convergence. On one hand, as an active-set strategy, it requires a reduced amount of work at each iteration. On the other hand, as a semi-smooth Newton technique, it achieves superlinear local convergence. Finally, the PDAS method extract option values and the free boundaries simultaneously, which is two for one comparing to other numerical methods.
The paper is organized as follows. In the next section, we introduce some notations, and the governing LCP on bounded domain are derived based on the variational inequality, similarity reduction, and some known information on the free boundary. In Section 3, the full discretization scheme of LCP is constructed by finite difference with uniform partition and finite element method with geometric partition in temporal and spatial directions, respectively. Then the PDAS is introduced to solve the resulting discretized optimization system. We also systematically analyze the convergent results of our algorithm. Numerical tests are performed in Section 4 to verify the effectiveness and efficiency of our method. The last section is devoted to some concluding remarks.
Section snippets
The pricing model
The celebrated parabolic variational inequality model (VI) defined on a two-dimension unbounded domain and its associated linear complementarity problem (LCP) defined on a one-dimension bounded domain for American better-of options will be presented in this section.
The numerical strategy
In this section, the variational problem corresponding to the LCP (11) shall be presented firstly, which will be discretized by finite difference and finite element methods in temporal and spatial directions, respectively. For the resulting discretized system, the primal-dual active set method (PDAS) is applied to obtain the option price and the free boundary simultaneously.
Numerical simulations
In this section, numerical simulations are carried out to test the performance of our proposed method. To illustrate the efficiency of the primal-dual active set method (PDAS), we first verify its superiority in computing free boundaries by comparing it with the binomial method (BM, Jiang [13]). Then, we confirm the error estimates on options established in Section 3, and illustrate its superiority on the valuation of options by comparing our proposed PDAS method with the BM and the projection
Conclusions
In this paper, an efficient numerical method is proposed for the valuation of American better-of options. The two-dimensional free boundary problem satisfied by better-of options is transformed into a one-dimensional linear complementarity problem (LCP) on a bounded domain by the numeraire transformation and some known information about the free boundary. The bounded domain problem associated with American better-of options is discretized by finite difference and finite element methods in
Acknowledgments
The work of H. Song was supported by the NSF of China under the grant No.11701210, the science and technology department project of Jilin Province under the grant No. 20190103029JH. the education department project of Jilin Province under the grant No. JJKH20180113KJ, and the fundamental research funds for the Central Universities. The work of K. Zhang was supported by the NSF of China under the grant No. 11871245, 11771179, 11726102, and by the Key Laboratory of Symbolic Computation and
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