1 Introduction
An option is a contract that gives the right to buy or sell the underlying asset at a specific price within a period of time, which gives the buyer the right, but not the obligation to perform the transaction. Option pricing is a crucial target in financial decisionmaking. In the practical side, however, it is also one of the most complicated problems in the mathematical domain of financial engineering. The breakthrough came in 1973, when Black and Scholes proposed the original BlackScholes equation [5], which is the first complete European option pricing model. With an alternative method, Merton derived the formula and generalized it in important ways [22, 23]. The BlackScholes model has a huge influence to the financial market and drives an unexpected prosperity in the trading of derivatives.
With the advent of parallel computers, many complicated problems can be solved efficiently by the improving processing power and new algorithms. An example is the asynchronous iterative scheme. The pioneering research of such approach dates back to 1969 by Chazan and Miranker [6] that arises in the face of communication delays. Their works have been extended to the nonlinear cases in [24, 3, 7] based on the notion of contraction. A well known systematic formalizations is contained in the book of Bertsekas and Tsitsiklis [4]. Furthermore, twostage theories were proposed in the 1990s by Frommer and Szyld [8, 9]. We refer the reader to [10, 12] as surveys on asynchronous iterative methods. Recently, asynchronous iterations have been extended to efficient domain decomposition methods in space [25], including substructuring methods [21], optimized Schwarz method [20]. Another issue is the extension to domain decomposition methods in time. In this paper, we are interested in the asynchronous parareal scheme, which will be explored in Section 2. In Section 3 we present the transformation from the original BlackScholes equation to an appropriate form which applies to our asynchronous solver. Section 4 is devoted to the experimental results. Finally, concluding remarks and future directions are presented in Section 5.
2 Asynchronous Parareal Algorithm
2.1 Classical Parareal Scheme
The parareal algorithm is a method to solve the timedependent partial differential equations in parallel, first presented by Lions, Maday and Turinici in 2001
[13]. The ideas of parareal are relevant to both multiple shooting method and predictorcorrector scheme, see the historical remarks in [11] whereby a comprehensive analysis of parareal has been mathematically studied.Given a secondorder linear elliptic operator , consider the following timedependent problem
(1) 
where the boundary conditions are unnecessary to be mentioned. By introducing a function , the above mathematical description can be decomposed into sequential problems, according to the rule with . We now rewrite our problem (1) as
(2) 
where , together with the condition . The parareal scheme is established by two propagators. Let be a fine time step corresponding to the problem (2). Suppose that the timedependent problem is approximated by some appropriate classical discretization schemes. Let be a precise integration method based on , often called a fine propagator. Similarly, let be a coarse propagator with respect to the broad time step . It is seen that the discretization schemes used for such two levels might be different. Finally, parareal replaces the accurate sequential method by an iteration
which is a predictorcorrector iterative scheme with and . Additionally, we are obliged to respect two more conditions and throughout the iterative process. We note that the timedependent problems might benefit from such scheme only if the number of iterations is less than . Following the mathematical model, we arrive at the iterative process below
Algorithm 1.
(Parareal Iterative Algorithm)
Notice that can be evaluated independently using parallel computers. Thus, parareal is often preferable than the serial setting for the timedependent problems.
2.2 Asynchronous Parareal Scheme
To derive the corresponding asynchronous scheme, now we present the asynchronous iterative model. The fundamental asynchronous model (see, e.g., [4]) is
(3) 
where is the number of processor, , and are integers satisfying . In addition, the model (6) is subject to the following conditions
(4) 
which indicates that each processor reads eventually the latest information for each component and no processor stops updating. Unfortunately, this model fails to be applied to the twostage problems. Therefore, we explore an alternative method which is often called asynchronous iterations with flexible communication, see [9]. To the end we will consider the model
with
where are integers satisfying and similarly we have
(5) 
In the context of the asynchronous twostage iterative model, we can eventually derive the asynchronous parareal method. Consider the following iteration
where , and , with the similar conditions
We supposed that the essential assumptions (7) and (8) are satisfied.
2.3 Asynchronous Parareal Scheme
To derive the corresponding asynchronous scheme, now we present the asynchronous iterative model. The fundamental asynchronous model (see, e.g., [4]) is
(6) 
where is the number of processor, , and are integers satisfying . In addition, the model (6) is subject to the following conditions
(7) 
which indicates that each processor reads eventually the latest information for each component and no processor stops updating. Unfortunately, this model fails to be applied to the twostage problems. Therefore, we explore an alternative method which is often called asynchronous iterations with flexible communication, see [9]. To the end we will consider the model
with
where are integers satisfying and similarly we have
(8) 
In the context of the asynchronous twostage iterative model, we can eventually derive the asynchronous parareal method. Consider the following iteration
where , and , with the similar conditions
We supposed that the essential assumptions (7) and (8) are satisfied.
3 Formalization of the Option Pricing Model
We now turn to the study of the option pricing problem that is expressed by appropriate evolution models. Accordingly, they apply to the above described scenario. To the end we will concentrate on the European call option, which can be exercised only on the expiration date, using the BlackScholes model as following
(9) 
where is the option price as a function of underlying asset price and time . The parameters are volatility and riskfree interest rate . Furthermore, the BlackScholes equation is backward parabolic satisfying
where the first formula is a final condition, is the time to maturity and is the exercise price. We are particularly interested in the solution of its discretization. However, the irregular form poses some difficulties for the solution. Therefore, we now try to transform the formula (9) to a simplest case.
A well known special case is the heat equation, which leads to a really simple parabola. Thus, the transformation from the original to the latter can simplify the solving process. Therefore, a change of variables is
substituting into the BlackScholes equation (9) gives
where . Setting
(10) 
yields
then gives the heat equation in an infinite interval
(11) 
heat equation is forward parabolic as opposed to the original model, with the corresponding conditions
We notice here that equation (11) corresponds to the initial problem (1) with and . Hence, the parareal schemes can be used to approximate the solution.
We assume that the equation (11) is discretized in time using a twostage scheme. Generally, the same discretization approach as coarse solver or a method of higher order can be applied to the fine grid propagator. Nevertheless, the spatial discretization is not straightforward enough. We give the precise boundaries as following instead of limits to infinity
such that
where and are defined by (10).
4 Experimental Results
In this section we present numerical results for the above asynchronous iterative scheme in order to solve the BlackScholes equation. Let us give the volatility and the riskfree interest rate , which can be considered as constants since they have little relation with the asynchronous properties. Another intuitive configuration is to set equals one day, while taking into account representing one year. However, such assignment seems inflexible and yields a somewhat complicated constant. Hence, we keep fixed and vary to simulate the different remaining life of the financial instrument. Finally, the spatial discretization is implemented by the finite difference method with suitable equal subintervals.
The asynchronous implementation requests some advanced capabilities. We conduct our tests based on JACK (an asynchronous communication kernel) library [19], which provides highlevel supports for both synchronous and asynchronous iterative algorithms, developed on the top of MPI library. Then, mathematical operations and linear systems solvers are implemented by Alinea (An Advanced Linear Algebra) library [14] The Alinea library is implemented in C++, MPI, CUDA and OpenCL. For both central processing unit and graphic processing unit devices, there are different matrix storage formats, and real and complex arithmetics in single and doubleprecision. It includes several linear algebra operations [1] and numerous algorithms for solving linear systems such as iterative methods [16], [2], [15], together with some energy consumption optimization [18], and domain decomposition methods in space [17]. We carry out two sets of experiments on a SGI ICE X cluster connected with InfiniBand (56 Gbit/s). Each computing node consists of two Intel Xeon E52670 v3 2.30 GHz CPUs. Furthermore, the version of the MPI library used is SGIMPT 2.14.
The first is performed by changing the length of on which we simulate the different time to maturity. The backward Euler method is chosen for both the coarse and the fine propagator. Given the initial stoke price and the strike price , table 1 illustrates the some average results for 16 cores and 250 subintervals, where is the approximate option prices, is the exact price, is the absolute error and is the relative error with .
Time  

0.1  0.4857  0.4853  0.0004  0.0008  0.958 
0.2  1.3950  1.3947  0.0003  0.0002  2.012 
0.3  2.3145  2.3140  0.0005  0.0002  3.014 
0.4  3.1932  3.1925  0.0007  0.0002  4.091 
0.5  4.0212  4.0203  0.0009  0.0002  5.041 
0.6  4.7972  4.7961  0.0011  0.0002  6.084 
0.7  5.5226  5.5213  0.0013  0.0002  6.873 
0.8  6.1995  6.1981  0.0014  0.0002  7.925 
0.9  6.8306  6.8291  0.0015  0.0002  8.526 
1.0  7.4184  7.4169  0.0015  0.0002  9.969 
Table 2 reports the similar results with and .
Time  

0.1  1.5204  1.5179  0.0025  0.0016  0.969 
0.2  3.3163  3.3141  0.0022  0.0007  1.972 
0.3  4.9525  4.9504  0.0021  0.0004  3.022 
0.4  6.4497  6.4476  0.0021  0.0003  3.817 
0.5  7.8264  7.8241  0.0023  0.0003  5.221 
0.6  9.0963  9.0939  0.0024  0.0003  6.108 
0.7  10.2699  10.2673  0.0026  0.0003  6.953 
0.8  11.3557  11.3530  0.0027  0.0002  7.731 
0.9  12.3611  12.3584  0.0027  0.0002  8.871 
1.0  13.2923  13.2898  0.0025  0.0002  10.396 
The absolute errors and the relative errors are small enough, which leads to the conclusion that the asynchronous iterations converge to the exact solution of the BlackScholes equation. Therefore, the fact above illustrates that the asynchronous parareal scheme is suitable to the option pricing problem, formalized by the BlackScholes model. Notice that when , which is beyond the scope of common requirements, even in a more general context.
The second is a comparison of synchronous and asynchronous parareal scheme. We choose , , and 150 subintervals. Table 3 reports the average experimental results, where is the number of processor cores.
Synchronous  Asynchronous  

Iter.  Time  Iter. (min, max, mean)  Time  
16  11  0.620  22  30  26  0.490 
32  11  0.781  30  47  40  0.677 
64  11  0.971  44  77  60  0.947 
The results illustrate that the number of iterations of the synchronous version remains the same throughout the test, whereas the asynchronous scheme requires further iterations with the increase of cores. Let us mention here that the context is not distributed, which rejects the possibility of large communication delays. We notice that the asynchronous version is faster.
5 Conclusions
In this paper we show the convergence performance of an original asynchronous extension of the parareal scheme applied to the BlackScholes equation, and illustrate some cases that the asynchronous method takes on superiority than its synchronous counterpart. The asynchronous iterative methods performs well even if there exists large communication delays or the computation bottlenecks in the systems, so long as all the processors keep updating using eventually the latest values. We illustrate the excellent precision and the conditional efficiency for the target approach. A theoretical and comprehensive analysis is under investigation by the authors to explain the results.
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