Option Pricing Problems Research Articles

Numerical Methods for Pricing American Options with Time-Fractional PDE Models

In this paper we develop a Laplace transform method and a finite difference method for solving American option pricing problem when the change of the option price with time is considered as a fractal transmission system. In this scenario, the option price is governed by a time-fractional partial differential equation (PDE) with free boundary. The Laplace transform method is applied to the time-fractional PDE. It then leads to a nonlinear equation for the free boundary (i.e., optimal early exercise boundary) function in Laplace space. After numerically finding the solution of the nonlinear equation, the Laplace inversion is used to transform the approximate early exercise boundary into the time space. Finally the approximate price of the American option is obtained. A boundary-searching finite difference method is also proposed to solve the free-boundary time-fractional PDEs for pricing the American options. Numerical examples are carried out to compare the Laplace approach with the finite difference method and it is confirmed that the former approach is much faster than the latter one.

Option Pricing with Threshold Diffusion Processes

The threshold diffusion (TD) model assumes a piecewise linear drift term and piecewise smooth diffusion term, which can capture many nonlinear features and volatility clustering often observed in financial time series data. We solve the problem of option pricing with a TD asset pricing process by deriving the minimum entropy martingale measure, which is the risk-neutral measure closest to the underlying TD probability measure in terms of Kullback-Leibler divergence, given the historical regime-switching pattern. The proposed valuation model is illustrated with a numerical example.

Numerical schemes for pricing Asian options under state-dependent regime-switching jump–diffusion models

We study the pricing problem of Asian options when the underlying asset price follows a very general state-dependent regime-switching jump–diffusion process via a partial differential equation approach. Under this model, the price of the option can be obtained by solving a highly complex system of coupled two-dimensional parabolic partial integro-differential equations (PIDEs). We prove existence of the solution to this system of PIDEs by the method of upper and lower solutions via constructing a monotonic sequence of approximating solutions whose limit is a strong solution of the PIDE system. We then propose several numerical schemes for solving the system of PIDEs. One of the proposed schemes is built upon the constructive proof, hence its results are provably convergent to the solution of the system of PIDEs. We illustrate the accuracy of the proposed methods by several numerical examples.

Correlated continuous time random walk and option pricing

In this paper, we study a correlated continuous time random walk (CCTRW) with averaged waiting time, whose probability density function (PDF) is proved to follow stretched Gaussian distribution. Then, we apply this process into option pricing problem. Supposing the price of the underlying is driven by this CCTRW, we find this model captures the subdiffusive characteristic of financial markets. By using the mean self-financing hedging strategy, we obtain the closed-form pricing formulas for a European option with and without transaction costs, respectively. At last, comparing the obtained model with the classical Black–Scholes model, we find the price obtained in this paper is higher than that obtained from the Black–Scholes model. A empirical analysis is also introduced to confirm the obtained results can fit the real data well.

Radial basis function partition of unity methods for pricing vanilla basket options

Meshfree methods based on radial basis function (RBF) approximation are becoming widely used for solving PDE problems. They are flexible with respect to the problem geometry and highly accurate. A disadvantage of these methods is that the linear system to be solved becomes dense for globally supported RBFs. A remedy is to introduce localisation techniques such as partition of unity. RBF partition of unity methods (RBF–PUM) allow for a significant sparsification of the linear system and lower the computational effort. In this work we apply a global RBF method as well as RBF–PUM to problems in option pricing. We consider one- and two-dimensional vanilla options. In order to price American options we employ a penalty approach. A penalty term, suitable for American multi-asset call options, has been designed. RBF–PUM is shown to be competitive compared with a finite difference method and a global RBF method. It is as accurate as the global RBF method, but significantly faster. The results for RBF–PUM look promising for extension to higher-dimensional problems.

Solving partial integro-differential option pricing problems for a wide class of infinite activity Lévy processes

In this paper, numerical analysis of finite difference schemes for partial integro-differential models related to European and American option pricing problems under a wide class of Lévy models is studied. Apart from computational and accuracy issues, qualitative properties such as positivity are treated. Consistency of the proposed numerical scheme and stability in the von Neumann sense are included. Gauss–Laguerre quadrature formula is used for the discretization of the integral part. Numerical examples illustrating the potential advantages of the presented results are included.

Limitations and improvements of standard spectral methods for pricing standard options

We discuss the issue of how to improve the efficiency of standard spectral methods for pricing options. As a prototype example, we consider American options which are frequently used in the market as they enjoyed the early exercise opportunities. Such options are often modeled by nonlinear partial differential equations, solutions of which are not obtainable in a closed form and thus necessitates robust numerical techniques. For the discretisation in space (asset) direction, we use a rational spectral methods. For time integration involved in the process, we looked at comparing the efficiency of the method by using contour integral method based on the inversion of the Laplace transform, and the Exponential Time Differencing Runge–Kutta method of order 4. It is known that when these spectral methods are applied to option pricing problems, they no longer retain their higher order convergence. This order reduction is attributed due to the non-smooth initial conditions. To overcome this, we propose a domain decomposition method based on our rational spectral method. For the problem under consideration, we divide the domain into two sub-domains, with the transition point as the strike price. To represent the solution in each sub-domains, we choose to approximate the solution by linear rational interpolants due to their improved stability properties as compared to the polynomial interpolant. Comparative results are also presented.

Spatial low-discrepancy sequences, spherical cone discrepancy, and applications in financial modeling

In this paper we introduce a reproducing kernel Hilbert space defined on Rd+1 as the tensor product of a reproducing kernel defined on the unit sphere Sd in Rd+1 and a reproducing kernel defined on [0,∞). We extend Stolarsky’s invariance principle to this case and prove upper and lower bounds for numerical integration in the corresponding reproducing kernel Hilbert space.The idea of separating the direction from the distance from the origin can also be applied to the construction of quadrature methods. An extension of the area-preserving Lambert transform is used to generate points on Sd−1 via lifting Sobol’ points in [0,1)d to the sphere. The dth component of each Sobol’ point, suitably transformed, provides the distance information so that the resulting point set is normally distributed in Rd.Numerical tests provide evidence of the usefulness of constructing Quasi-Monte Carlo type methods for integration in such spaces. We also test this method on examples from financial applications (option pricing problems) and compare the results with traditional methods for numerical integration in Rd.

Nearest Neighbor Based Estimation Technique for Pricing Bermudan Options

Bermudan option is an option which allows the holder to exercise at pre-specified time instants where the aim is to maximize expected payoff upon exercise. In most practical cases, the underlying dimensionality of Bermudan options is high and the numerical methods for solving partial differential equations as satisfied by the price process become inapplicable. In the absence of analytical formula a popular approach is to solve the Bermudan option pricing problem approximately using dynamic programming via estimation of the so-called continuation value function. In this paper we develop a nearest neighbor estimator based technique which gives biased estimators for the true option price. We provide algorithms for calculating lower and upper biased estimators which can be used to construct valid confidence intervals. The computation of lower biased estimator is straightforward and relies on suboptimal exercise policy generated using the nearest neighbor estimate of the continuation value function. The upper biased estimator is similarly obtained using likelihood ratio weighted nearest neighbors. We analyze the convergence properties of mean square error of the lower biased estimator. We develop order of magnitude relationship between the simulation parameters and computational budget in an asymptotic regime as the computational budget increases to infinity.

The Application of Fractional Brownian Motion in Option Pricing

In this text, Fractional Brown Motion theory during random process is applied to research the option pricing problem. Firstly, Fractional Brown Motion theory and actuarial pricing method of option are utilized to derive Black-Scholes formula under Fractional Brown Motion and form corresponding mathematical model to describe option pricing. Secondly, based on BYD stock, estimation model on volatility of this stock is given to analyze and calculate the stock price volatility. Finally, make instance analysis for BYD’s option. Based on market data of BYD’s stock and option, calculate the actual option price and theoretical price of BYD by BlackScholes formula under Fractional Brown Motion. Compare the forecast price of this stock option given by model with actual price, relatively good effect is obtained, and then conclude that the model has relatively strong applicability.

Alternative Approach for the Derivation of Black-Scholes Partial Differential Equation in the Theory of Options Pricing Using Risk Neutral Binomial Process

This paper presents a risk neutral binomial process as an alternative approach for the derivation of analytic pricing equation called “Black - Scholes Partial differential Equation” in the theory of option pricing. Binomial option pricing is a powerful technique that can be used to solve many complex option pricing problems. In contra st to the Black - Scholes model and other option pricing models that require solutions to stochastic differential equations, the binomial model is mathematically simple. Binomial model is based on the assumption of no arbitrage. The assumption of no arbitra ge implies that all risk - free investments earn the risk - free rate of return and no investment opportunity exists that requires zero amounts of investment but yield positive returns.

Complete Analytical Solution of the Asian Option Pricing and Asian Option Value-at-Risk Problems: A Probability Density Function Approach

•The first ever explicit formulation of the concept of an option’s probability density functions has been introduced in our publication “Breakthrough in Understanding Derivatives and Option Based Hedging - Marginal and Joint Probability Density Functions of Vanilla Options - True Value-at-Risk and Option Based Hedging Strategies” (see link http://ssrn.com/abstract=2489601). •In this paper we report similar unique results for Asian Options, enabling complete analytical resolution of all problems associated with Asian Options. •Our discovery of the Asian Option probability density function enables exact closed-form analytical results for its expected value (price) for the first time without depending on Inverse Laplace or Fourier transforms that only abbreviate complex numerical integration procedures. •Expected value is the first moment. All higher moments are as easily represented in closed form based on our probability density function, but are not calculable by extensions of other numerical methods, such as Inverse Laplace or Fourier transforms, now used to represent the first moment. •Our formulation of the Asian Option probability density function is general enough to cover interesting and practical cases that are not addressed in the literature at all: for example, cases for which the averaging period is a terminal subset of the contract period, e.g. the last 3 months of the option lifetime. •Our formulation of the Asian Option probability density function is expressive enough to enable derivation for the first time ever of corollary closed-form analytical results for such Value-At-Risk characteristics as the probabilities that an Asian Option will be below or above any set of thresholds at any future time before or at termination. Such assessments are absolutely out of reach of current published methods for treating Asian Options. •Resolution of the Asian Option Pricing problem enables knowledge of their linear correlation with Vanilla Options and, therefore, hedging in terms of Vanilla Options. Moreover, an opportunity to construct “synthetic” Asian Options from Vanilla Options for a given termination becomes a trivial exercise. •All numerical evaluations based on our analytical results are practically instantaneous and absolutely accurate.

A finite volume method for pricing the American lookback option

Due to the characteristic of risk aversion, option has become one of the most fashionable derivatives in the financial field. More and more investigators are attracted to devote themselves to exploring the option pricing problem. In this paper, we are concerned with the valuation of American lookback options in terms of the Black-Scholes model. It is well known that the American lookback option satisfies a two-dimensional nonlinear partial differential equation in an unbounded domain, which couldn't be numerically solved directly. Based on the analysis of the issues for solving this problem, this paper introduces an approach to settle it. First, we transform the problem into a one-dimensional form by the numeraire transformation. And then, the Landau's transformation is applied to normalize the defined domain. For the nonlinear feature of the resulting problem, we propose a finite volume method coupled with Newton iterative method to obtain the optional value and the optimal exercise boundary simultaneously. We also give a proof on the nonnegativity of the numerical solutions under some appropriate assumptions. Finally, some numerical simulations are presented using the proposed method in this paper. Comparing with the binomial method, we can conclude that the proposed method is an effective one, which provides a theoretical basis for practical applications.

The Barone-Adesi Whaley Formula to Price American Options Revisited

This paper presents a method to solve the American option pricing problem in the Black Scholes framework that generalizes the Barone-Adesi, Whaley method [1]. An auxiliary parameter is introduced in the American option pricing problem. Power series expansions in this parameter of the option price and of the corresponding free boundary are derived. These series expansions have the Baroni-Adesi, Whaley solution of the American option pricing problem as zero-th order term. The coefficients of the option price series are explicit formulae. The partial sums of the free boundary series are determined solving numerically nonlinear equations that depend from the time variable as a parameter. Numerical experiments suggest that the series expansions derived are convergent. The evaluation of the truncated series expansions on a grid of values of the independent variables is easily parallelizable. The cost of computing the n-th order truncated series expansions is approximately proportional to n as n goes to infinity. The results obtained on a set of test problems with the first and second order approximations deduced from the previous series expansions outperform in accuracy and/or in computational cost the results obtained with several alternative methods to solve the American option pricing problem [1]-[3]. For example when we consider options with maturity time between three and ten years and positive cost of carrying parameter (i.e. when the continuous dividend yield is smaller than the risk free interest rate) the second order approximation of the free boundary obtained truncating the series expansions improves substantially the Barone-Adesi, Whaley free boundary [1]. The website: http://www.econ.univpm.it/recchioni/finance/w20 contains material including animations, an interactive application and an app that helps the understanding of the paper. A general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.

Multivariate Option Pricing with Pair-Copulas

We propose a copula-based approach to solve the option pricing problem in the risk-neutral setting and with respect to a structured derivative written on several underlying assets. Our analysis generalizes similar results already present in the literature but limited to the trivariate case. The main difficulty of such a generalization consists in selecting the appropriate vine structure which turns to be of D-vine type, contrary to what happens in the trivariate setting where the canonical vine is sufficient. We first define the general procedure for multivariate options and then we will give a concrete example for the case of an option written on four indexes of stocks, namely, the S&P 500 Index, the Nasdaq 100 Index, the Nasdaq Composite Index, and the Nyse Composite Index. Moreover, we calibrate the proposed model, also providing a comparison analysis between real prices and simulated data to show the goodness of obtained estimates. We underline that our pair-copula decomposition method produces excellent numerical results, without restrictive assumptions on the assets dynamics or on their dependence structure, so that our copula-based approach can be used to model heterogeneous dependence structure existing between market assets of interest in a rigorous and effective way.

A Radial Basis Function Partition of Unity Collocation Method for Convection–Diffusion Equations Arising in Financial Applications

Meshfree methods based on radial basis function (RBF) approximation are of interest for numerical solution of partial differential equations (PDEs) because they are flexible with respect to geometry, they can provide high order convergence, they allow for local refinement, and they are easy to implement in higher dimensions. For global RBF methods, one of the major disadvantages is the computational cost associated with the dense linear systems that arise. Therefore, research is currently directed towards localized RBF approximations such as the RBF partition of unity collocation method (RBF–PUM) proposed here. The objective of this paper is to establish that RBF–PUM is viable for parabolic PDEs of convection–diffusion type. The stability and accuracy of RBF–PUM is investigated partly theoretically and partly numerically. Numerical experiments show that high-order algebraic convergence can be achieved for convection–diffusion problems. Numerical comparisons with finite difference and pseudospectral methods have been performed, showing that RBF–PUM is competitive with respect to accuracy, and in some cases also with respect to computational time. As an application, RBF–PUM is employed for a two-dimensional American option pricing problem. It is shown that using a node layout that captures the solution features improves the accuracy significantly compared with a uniform node distribution.

Constructing positive reliable numerical solution for American call options: A new front-fixing approach

A new front-fixing transformation is applied to the Black–Scholes equation for the American call option pricing problem. The transformed non-linear problem involves homogeneous boundary conditions independent of the free boundary. The numerical solution by an explicit finite-difference method is positive and monotone. Stability and consistency of the scheme are studied. The explicit proposed method is compared with other competitive implicit ones from the points of view accuracy and computational cost.

Robust option pricing

In this paper, we combine robust optimization and the idea of ∊-arbitrage to propose a tractable approach to price a wide variety of options. Rather than assuming a probabilistic model for the stock price dynamics, we assume that the conclusions of probability theory, such as the central limit theorem, hold deterministically on the underlying returns. This gives rise to an uncertainty set that the underlying asset returns satisfy. We then formulate the option pricing problem as a robust optimization problem that identifies the portfolio which minimizes the worst case replication error for a given uncertainty set defined on the underlying asset returns. The most significant benefits of our approach are (a) computational tractability illustrated by our ability to price multi-asset, American and Asian options using linear optimization; and thus the computational complexity of our approach scales polynomially with the number of assets and with time to expiry and (b) modeling flexibility illustrated by our ability to model different kinds of options, various levels of risk aversion among investors, transaction costs, shorting constraints and replication via option portfolios.

A New Stock Model for Option Pricing in Uncertain Environment

The option-pricing problem is always an important part in modern nance. Assuming that the stock diusion is a constant, some literature has introduced many stock models and given corresponding option pricing formulas within the framework of the uncertainty theory. In this paper, we propose a new stock model with uncertain stock diusion for uncertain markets. Some option pricing formulas on the proposed uncertain stock model are derived and a numerical calculation is illustrated.

Research on Water Option Pricing Problem Based on the Binary Tree Model

Along with the progress of the water-saving technology, the water users who have excess water resources by using water saving technology may transfer the spare water rights to whom with water shortage. As a new attempt to water rights transaction, water options transaction has fully showed its efficiency and flexibility, and the water options pricing problem is the key problem in the process of trading. Using market mechanism enables water users participate in water rights transaction independently, water users who have excess water resources hold American put options based on the water rights, can use the binary tree pricing model to analyze the value of the American put option at various stages to help them deciding whether to perform rights or give up rights.