A consistent stable numerical scheme for a nonlinear option pricing model in illiquid markets

Markets liquidity is an issue of very high concern in financial risk management. In a perfect liquid market the option pricing model becomes the well-known linear Black–Scholes problem. Nonlinear models appear when transaction costs or illiquid market effects are taken into account. This paper deals with the numerical analysis of nonlinear Black–Scholes equations modeling illiquid markets when price impact in the underlying asset market affects the replication of a European contingent claim. Numerical analysis of a nonlinear model is necessary because disregarded computations may waste a good mathematical model. In this paper we propose a finite-difference numerical scheme that guarantees positivity of the solution as well as stability and consistency.     

Option pricing under the Merton model of the short rate

Previous option pricing research typically assumes that the risk-free rate or the short rate is constant during the life of the option. In this study, we incorporate the stochastic nature of the short rate in our option valuation model and derive explicit formulas for European call and put options on a stock when the short rate follows the Merton model. Using our option model as a benchmark, our numerical analysis indicates that, in general, the Black–Scholes model overvalues out-of-the-money calls, moderately overvalues at-the-money calls, and slightly overvalues in-the-money calls. Our analysis is directly extensible to American calls on non-dividend-paying stocks and to European puts by virtue of put-call parity.     

Generalized trapezoidal formulas for the black–scholes equation of option pricing

For the celebrated Black–Scholes parabolic equation of option pricing, we present new time integration schemes based on the generalized trapezoidal formulas introduced by Chawla et al. [3]. The resulting GTF(α) schemes are unconditionally stable and second order in both space and time. Interestingly, since the Black–Scholes equation is linear, GTF (1/3) attains order three in time. The computational performance of the obtained schemes is compared with the Crank–Nicolson scheme for the case of European option valuation. Since the payoff is nondifferentiable having a “corner” on expiry at the exercise price, the classical trapezoidal formula used in the Crank–Nicolson scheme can experience oscillations at this corner. It is demonstrated that our present GTF (1/3) scheme can cope with this situation and performs consistently superior than the Crank–Nicolson scheme.     

Radial-basis-function-based finite difference operator splitting method for pricing American options

We present a new radial-basis-function (RBF)-based numerical method for pricing European and American option problems. The governing equation is time semi-discretized by a linear-implicit backward difference method. The spatial discretization is done by using the RBF-based finite difference method. The numerical scheme first derived for an European option is extended for American options by using an operator splitting method. Numerical experiments with multiquadric RBF for one- and two-asset option problems are carried out, and the results obtained are compared with the existing ones.     

A new radial basis functions method for pricing American options under Merton's jump-diffusion model

A new radial basis functions (RBFs) algorithm for pricing financial options under Merton's jump-diffusion model is described. The method is based on a differential quadrature approach, that allows the implementation of the boundary conditions in an efficient way. The semi-discrete equations obtained after approximation of the spatial derivatives, using RBFs based on differential quadrature are solved, using an exponential time integration scheme and we provide several numerical tests which show the superiority of this method over the popular Crank–Nicolson method. Various numerical results for the pricing of European, American and barrier options are given to illustrate the efficiency and accuracy of this new algorithm. We also show that the option Greeks such as the Delta and Gamma sensitivity measures are efficiently computed to high accuracy.     

Normalized particle swarm optimization for complex chooser option pricing on graphics processing unit

An option is a financial instrument that derives its value from an underlying asset, for example, a stock. There are a wide range of options traded today from simple and plain (European options) to exotic (chooser options) that are very difficult to evaluate. Both buyers and sellers continue to look for efficient algorithms and faster technology to price options for better profit and to beat the competition. There are mathematical models like the Black–Scholes–Merton model used to price options approximately for simple and plain options in the form of closed form solution. However, the market is flooded with various styles of options, which are difficult to price, and hence there are many numerical techniques proposed for pricing. The computational cost for pricing complex options using these numerical techniques is exorbitant for reasonable accuracy in pricing results. Heuristic approaches such as particle swarm optimization (PSO) have been proposed for option pricing, which provide same or better results for simple options than that of numerical techniques at much less computational cost. In this study, we first map the PSO parameters to option pricing parameters. Analyzing the characteristics of PSO and option pricing, we propose a strategy to normalize some of the parameters, which helps in better understanding of the sensitivity of these and other parameters on option pricing results. We then avail of the inherent concurrency of the PSO algorithm while searching for an optimum solution, and design an algorithm for implementation on a modern state-of-the-art graphics processor unit (GPU). Our implementation makes use of the architectural features of GPU in accelerating the computing performance while maintaining accuracy on the pricing results.     

An option pricing by eliminating observation noise when the model parameters are unknown

The stock price that is observed in the market contains irregular fluctuations that cannot be explained by the theory. We assume this kind of fluctuation to be observation noise. By eliminating observation noise from the underlying stock price, we can evaluate the option that better reflects the entity price. We define the stock price that eliminates observation noise as the entity stock price. The most important problem in eliminating the observation noise is the choice of unknown constant parameters of the market model. We develop a parameter estimation method by using the relationship between stock price and its option price or future price. We obtain satisfactory results in testing the algorithm by some simulations. We apply the parameters obtained by this method to estimate the stock price by eliminating observation noise from the real market data. We then calculate the call option price based on the estimated entity stock price.     

Modeling and Simulation of an Artificial Stock Option Market

Since their introduction in 1973, options have become an important and very popular financial instrument. However, despite much research performed on the subject, the effects of option trading on the underlying asset market are still debated. Both empirical and theoretical studies have failed to point out how price volatility and volumes of the underlying asset are affected. In this paper we present the first study on the effects of an option market related to an underlying stock market, using an artificial financial market based on heterogeneous agents. We modeled a realistic European option using two market models. The microstructure of the first model is kept as simple as possible, being composed only of random traders. The second model is more complex and realistic, involving the presence of various kinds of trading strategies (random, fundamentalist and chartist). We show that the introduction of options, in the proposed models, tends to decrease the volatility of the underlying stock price. Moreover, the traders’ wealth can be strongly affected by the use of option hedging.     

A closed-form approximation for the fractional Black–Scholes model with transaction costs

In this paper, we investigate option valuation problems under the fractional Black–Scholes model. The aim is to propose a pricing formula for the European option with transaction costs, where the costs structure contains fixed costs, a cost propositional to the volume traded, and a cost proportional to the value traded. Precisely, we provide an approximate solution of the nonlinear Hoggard–Whalley–Wilmott equation. The comparison results reveal that our approximate solutions are close to the numerical computations. Moreover, the comparison results demonstrate that the price of the European option decreases as the Hurst exponent increases.     

An efficient numerical technique based on the extended cubic B-spline functions for solving time fractional Black–Scholes model

Financial theory could introduce a fractional differential equation (FDE) that presents new theoretical research concepts, methods and practical implementations. Due to the memory factor of fractional derivatives, physical pathways with storage and inherited properties can be best represented by FDEs. For that purpose, reliable and effective techniques are required for solving FDEs. Our objective is to generalize the collocation method for solving time fractional Black–Scholes European option pricing model using the extended cubic B-spline. The key feature of the strategy is that it turns these type of problems into a system of algebraic equations which can be appropriate for computer programming. This is not only streamlines the problems but speed up the computations as well. The Fourier stability and convergence analysis of the scheme are examined. A proposed numerical scheme having second-order accuracy via spatial direction is also constructed. The numerical and graphical results indicate that the suggested approach for the European option prices agree well with the analytical solutions.

     

Application of the local radial basis function-based finite difference method for pricing American options

In this paper we discuss a local radial basis function-based finite difference (RBF-FD) scheme for numerical solution of multi-asset American option problems. The governing equation is discretized by the θ-method and the option price is approximated by the RBF-FD method. Numerical experiments are performed with the multiquadratic radial basis function for single and double asset problem and results obtained are compared with existing ones. We show numerically that the scheme is second-order accurate. Stability of the scheme is also discussed.     

A meshless method for Asian style options pricing under the Merton jump-diffusion model

In this paper, we consider the partial integro-differential equation arising when a stock follows a Poisson distributed jump process, for the pricing of Asian options. We make use of the meshless radial basis functions with differential quadrature for approximating the spatial derivatives and demonstrate that the algorithm performs effectively well as compared to the commonly employed finite difference approximations. We also employ Strang splitting with the exponential time integration technique to improve temporal efficiency. Throughout the numerical experiments covered in the paper, we show how the proposed scheme can be efficiently employed for the pricing of American style Asian options under both the Black–Scholes and the Merton jump-diffusion models.     

A numerical method to price discrete double Barrier options under a constant elasticity of variance model with jump diffusion

We develop a numerical method to price discrete barrier options on an underlying described by the constant elasticity of variance model with jump-diffusion (CEVJD). In particular, the partial integro differential equation associated to this model is discretized in time using an operator splitting scheme whose accuracy is enhanced by repeated Richardson extrapolation. Such an approach allows us to approximate the differential terms and the jump integral by means of two different numerical techniques. Precisely, the spatial derivatives, which exist only in the weak sense, are discretized using a finite element method based on piecewise quadratic polynomials, whereas the jump integral is directly collocated at the mesh points, so that it can be easily evaluated by Simpson numerical quadrature. As shown by extensive numerical simulation, the proposed approach is very efficient from the computational standpoint, and performs significantly better than the finite difference scheme developed in Wade et al. [On smoothing of the Crank–Nicolson scheme and higher order schemes for pricing barrier options, J. Comput. Appl. Math. 204 (2007), pp. 144–158].     

High-accuracy finite-difference methods for the valuation of options

A finite-difference scheme often employed for the valuation of options from the Black–Scholes equation is the Crank–Nicolson (CN) scheme. The CN scheme is second order in both time and asset. For a rapid valuation with a reasonable resolution of the option price curve, it requires extremely small steps in both time and asset. In this paper, we present high-accuracy finite-difference methods for the Black–Scholes equation in which we employ the fourth-order L-stable Simpson-type (LSIMP) time integration schemes developed earlier and the well-known Numerov method for discretization in the asset direction. The resulting schemes, called LSIMP–NUM, are fourth order in both time and asset. The LSIMP–NUM schemes obtained can provide a rapid, stable and accurate resolution of option prices, allowing for relatively large steps in both time and asset. We compare the computational efficiency of the LSIMP–NUM schemes with the CN and Douglas schemes by considering valuation of European options and American options via the linear complementarity approach.     

Valuation of option price in commodity markets described by a Markov-switching model: A case study of WTI crude oil market

This paper suggests a Markov-switching model to evaluate commodity futures and spot dynamics, such that the diffusion coefficients and jump size parameter are associated with a hidden Markov chain. We improve the current models in the literature of the commodity markets by modeling the sudden jumps in the commodity prices through the hidden Markov chain. From the crude oil spot price in West Texas Intermediate, we estimate the parameters of proposed Markov-switching model based on expectation–maximization algorithm. To perform this task, we apply this estimation algorithm to the model discretized by Euler scheme and provide some convergence analysis for this discretization method. There are options, such as European options, which are written on the commodity futures. In this study, we evaluate them under the regime-switching model with various economic states. In the following, we calibrate the option prices resulting from the proposed commodity model to a set of observed European call options written on crude oil futures. For this purpose, we first apply an inverse Fourier transform and obtain a semi-analytical option pricing formula. Then, we use the fast Fourier transform method to compute option prices. Since the investors need to calculate Greeks in order to understand the risk involved in option investments, the Greek formulas of Delta, Rho, Theta, and Gamma are derived.     

Calibrating parametric exponential Lévy models to option market data by incorporating statistical moments priors

We investigate a parametric method for calibrating European option pricing using the state-of-art exponential Lévy models. We propose a derivative-free calibration method constrained by four observable statistical moments (mean, variance, skewness and kurtosis) from underlying time series to conquer the ill-posed inverse problem and to incorporate priors on observable statistical moments. We present a numerical implementation scheme for calibrating the exponential Lévy models and show that it can resolve the instability of the inverse problems empirically and can produce good calibration results. In particular, we apply our approach to real market data sets of S&P 500 call options with significantly better performance.     

Finite Difference Method for the Black–Scholes Equation Without Boundary Conditions

We present an accurate and efficient finite difference method for solving the Black–Scholes (BS) equation without boundary conditions. The BS equation is a backward parabolic partial differential equation for financial option pricing and hedging. When we solve the BS equation numerically, we typically need an artificial far-field boundary condition such as the Dirichlet, Neumann, linearity, or partial differential equation boundary condition. However, in this paper, we propose an explicit finite difference scheme which does not use a far-field boundary condition to solve the BS equation numerically. The main idea of the proposed method is that we reduce one or two computational grid points and only compute the updated numerical solution on that new grid points at each time step. By using this approach, we do not need a boundary condition. This procedure works because option pricing and computation of the Greeks use the values at a couple of grid points neighboring an interesting spot. To demonstrate the efficiency and accuracy of the new algorithm, we perform the numerical experiments such as pricing and computation of the Greeks of the vanilla call, cash-or-nothing, power, and powered options. The computational results show excellent agreement with analytical solutions.     

On the Solution of the Black–Scholes Equation Using Feed-Forward Neural Networks

Computational Economics - This paper deals with a comparative numerical analysis of the Black–Scholes equation for the value of a European call option. Artificial neural networks are used for...     

A Multi-Level Mixed Element Method for the Eigenvalue Problem of Biharmonic Equation

In this paper, we discuss approximating the eigenvalue problem of biharmonic equation. We first present an equivalent mixed formulation which admits natural nested discretization. Then, we present multi-level finite element schemes by implementing the algorithm as in Lin and Xie (Math Comput 84:71–88, 2015) to the nested discretizations on a series of nested grids. The multi-level mixed scheme for the biharmonic eigenvalue problem possesses optimal convergence rate and optimal computational cost. Both theoretical analysis and numerical verifications are presented.     

A pseudo-spectral scheme for the approximate solution of a time-fractional diffusion equation

We consider an initial-boundary-value problem for a time-fractional diffusion equation with initial condition u₀(x) and homogeneous Dirichlet boundary conditions in a bounded interval [0, L]. We study a semidiscrete approximation scheme based on the pseudo-spectral method on Chebyshev–Gauss–Lobatto nodes. In order to preserve the high accuracy of the spectral approximation we use an approach based on the evaluation of the Mittag-Leffler function on matrix arguments for the integration along the time variable. Some examples are presented and numerical experiments illustrate the effectiveness of the proposed approach.