A general continuous time Markov chain approximation for multi-asset option pricing with systems of correlated diffusions
Introduction
Option pricing for multi-asset systems is an especially challenging problem in quantitative finance, with numerous practical applications. Contracts such as spread options [1] are very common in energy and power markets, where crack spreads (oil) and spark spreads (electricity) are actively traded on major financial exchanges. In energy markets, various forms of averages are traded, often on multiple underlying futures, usually a set of consecutive maturities. Basket options, mountain range options, and best-of options are other common examples.
While the literature on single-asset option pricing is quite dense, the problem of multi-asset pricing remains relatively underdeveloped. The most commonly used approach is Monte Carlo and its many variations [2], [3], [4], [5], [6]. Techniques such as GPU acceleration [7], grid bundling [8], and quasi-Monte Carlo [9] have been proposed to enhance the efficiency of Monte Carlo. Memory reduction methods [10] have also been proposed for high dimensionality. For lower dimensional problems, partial differential equation (PDE) methods such as operator splitting are often applied, see Ikonen and Toivanen [11], and Clift and Forsyth [12] for an implicit method in two dimensions. Other PDE approaches include [13], [14]. Decorrelation techniques for removing the mixed derivatives in PDEs has been used by Company et al. [15], Egorova et al. [16]. Coordinate transformations with sparse grids are applied in [17] to price basket options by PDE, for up to five underlying assets. Radial basis techniques have also been applied [18], [19] in higher dimensional problems, as well as hexaonomial lattices [20], and finite element methods [21].
In two dimensions, Fourier transform techniques offer highly efficient alternatives, such as the CONV method [22], the COS method, [23], the SWIFT method [24], [25], and the PROJ method [26], [27]. Efficient specialized methods exist for certain payoffs, such as spread options [28], [29]. Closed form approximations have been provided for spread options [30], and basket options [31].
A more recent option pricing approach is provided by Markov chain approximation. In an early work, [32] consider the problem of barrier option pricing in one dimension. Later extensions include [33], [34], [35]. For a review of recent work in the area of CTMC approximation techniques for financial modeling, see [36], and rigorous error analysis can be found in [35], [37] for problems in one dimension.
The application of CTMC approximation in higher dimensions is still in its early stages of developement. For example, the case of stochastic volatility with two correlated diffusions is developed in [34], [38], [39], [40] using a combination of Markov chain approximation, regime-switching, and Fourier transform techniques. They then provided a two-layer CTMC approximation method which extends to the case of stochastic local volatility in [41], as well as one and two-layer approximations for the case of time-changed Markov processes in [42]. The use of hybrid Markov models, such as CTMC-Heston, is proposed in [43]. More recently, Xi et al. [44] provides an approach which can handle general two dimensional systems, based on matching correlations of the driving processes. However, the construction of the CTMC process proposed in [44] is implicit and it is not clear to extend their method to higher dimensional cases. In higher dimensions, the most important feature of the model is the correlation between various assets. In this work, we provide a CTMC approximation framework which can handle n-dimensional systems of diffusions with constant coefficients, allowing for arbitrary correlation between assets. The method relies on a novel decoupling transformation, proposed in [15], which allows us to de-correlate the diffusion system and approximate each dimension with a univariate CTMC. The final step is to re-correlate the CTMCs, and conduct pricing. Most importantly, the CTMC scheme is proved to weakly converge to the continuous diffusion in the second order. The methodology is quite general, and allows for European and exotic option pricing through simple matrix products, and is thus well-suited for practical applications. Extensive numerical examples not only confirm that our method is stable, fast as compared to the existing approach such as [23] but also validate the second order convergence of the CTMC scheme.
The rest of the paper is organized as follows: Section 2 introduces the correlated model and a general method to remove the correlations among the governing Brownian motions. In Section 3, we propose a general continuous time Markov chain framework to approximate the model introduced in Section 2. The weak convergence of the continuous time Markov chain approximation is considered in Section 3.2.1. Numerical examples in two and three dimensions are provided in Section 4. Section 5 concludes the paper with some future research directions.
Section snippets
Diffusion system
The multi-asset model we consider is the system of diffusionswhere Si is the ith underlying asset having expected return and continuous dividend of and volatility of σi > 0. The Brownian processes are correlated with where . Note that the model (1) was considered in [28] and [23].
Let be the state space of (Si(t))1 ≤ i ≤ n, and denote the set of functions on that vanish at
Continuous time Markov chain approximation
After de-correlating the diffusion system, the next step in our methodology is to obtain a Markov chain approximation to the system (11), which will result in closed-form value approximations. For notational simplification, let and then the system in (11) is reduced toFor each let . We consider a general n-dimensional continuous time Markov chain (CTMC)
Numerical examples
In this section we conduct numerical experiments demonstrating the versatility of the proposed N-Dimensional CTMC method (N-CTMC), and we verify the theoretical second order convergence proved in Theorem 3. Experiments are provided for European, Bermudan, and barrier options, with a variety of different payoffs and parameter settings. All experiments are conducted in Matlab 8.5 on a personal computer with Intel(R) Core(TM) i7-6700 CPU 3.40GHz.
Conclusion
We consider a general Markov chain approximation framework for correlated diffusion systems with constant coefficients. We prove that the continuous time Markov chain approximation leads to the second order convergence in n-dimensions, which is confirmed by extensive numerical examples. European, Bermudan, and barrier contracts are all considered in this work. The proposed scheme is stable, accurate, and fast as compared to the other existing methods. It would be interesting to extend the
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