Option Pricing Problems Research Articles

Option pricing in time-changed Lévy models with compound Poisson jumps

The problem of European-style option pricing in time-changed Lévy models in the presence of compound Poisson jumps is considered. These jumps relate to sudden large drops in stock prices induced by political or economical hits. As the time-changed Lévy models, the variance-gamma and the normal-inverse Gaussian models are discussed. Exact formulas are given for the price of digital asset-or-nothing call option on extra asset in foreign currency. The prices of simpler options can be derived as corollaries of our results and examples are presented. Various types of dependencies between stock prices are mentioned.

Perpetual American Defaultable Options in Models with Random Dividends and Partial Information

We present closed-form solutions to the perpetual American dividend-paying put and call option pricing problems in two extensions of the Black–Merton–Scholes model with random dividends under full and partial information. We assume that the dividend rate of the underlying asset price changes its value at a certain random time which has an exponential distribution and is independent of the standard Brownian motion driving the price of the underlying risky asset. In the full information version of the model, it is assumed that this time is observable to the option holder, while in the partial information version of the model, it is assumed that this time is unobservable to the option holder. The optimal exercise times are shown to be the first times at which the underlying risky asset price process hits certain constant levels. The proof is based on the solutions of the associated free-boundary problems and the applications of the change-of-variable formula.

GENERALIZED JACOBI REPRODUCING KERNEL METHOD IN HILBERT SPACES FOR SOLVING THE BLACK-SCHOLES OPTION PRICING PROBLEM ARISING IN FINANCIAL MODELLING

Based on the reproducing kernel Hilbert space method, a new approach is proposed to approximate the solution of the Black-Scholes equation with Dirichlet boundary conditions and introduce the reproducing kernel properties in which the initial conditions of the problem are satisfied. Based on reproducing kernel theory, reproducing kernel functions with a polynomial form will be constructed in the reproducing kernel spaces spanned by the generalized Jacobi basis polynomials. Some new error estimates for application of the method are established. The convergence analysis is established theoretically. The proposed method is successfully used for solving an option pricing problem arising in financial modelling. The ideas and techniques presented in this paper will be useful for solving many other problems.

Barycentric spectral domain decomposition methods for valuing a class of infinite activity Lévy models

A new barycentric spectral domain decomposition methods algorithm for solving partial integro-differential models is described. The method is applied to European and butterfly call option pricing problems under a class of infinite activity Levy models. It is based on the barycentric spectral domain decomposition methods which allows the implementation of the boundary conditions in an efficient way. After the approximation of the spatial derivatives, we obtained the semi-discrete equations. The computation of these equations is performed by using the barycentric spectral domain decomposition method. This is achieved with the implementation of an exponential time integration scheme. Several numerical tests for the pricing of European and butterfly options are given to illustrate the efficiency and accuracy of this new algorithm. We also show that Greek options, such as Delta and Gamma sensitivity, are computed with no spurious oscillation.

Pricing multi-asset option problems: a Chebyshev pseudo-spectral method

The aim of this paper is to contribute a new second-order pseudo-spectral method via a non-uniform distribution of the computational nodes for solving multi-asset option pricing problems. In such problems, the prices are required to be as accurately as possible around the strike price. Accordingly, the proposed modification of the Chebyshev–Gauss–Lobatto points would concentrate on this area. This adaptation is also fruitful for the non-smooth payoffs which cause discontinuities in the strike price. The proposed scheme competes well with the existing methods such as finite difference, meshfree, and adaptive finite difference methods on several benchmark problems.

A class of intrinsic parallel difference methods for time-space fractional Black\u2013Scholes equation

To quickly solve the fractional Black–Scholes (B–S) equation in the option pricing problems, in this paper, we construct pure alternative segment explicit–implicit (PASE-I) and pure alternative segment implicit–explicit (PASI-E) difference schemes for time-space fractional B–S equation. It is a kind of intrinsic parallel difference schemes constructed on the basis of classic explicit scheme and classic implicit scheme combined with alternate segmentation technique. PASE-I and PASI-E schemes are analyzed to be unconditionally stable, convergent with second-order spatial accuracy and (2-alpha)th-order time accuracy, and they have a unique solution. The numerical experiments show that the two schemes have obvious parallel computing properties, and the computation time is greatly improved compared to Crank–Nicolson (C–N) scheme. The PASE-I and PASI-E intrinsic parallel difference methods are efficient to solve the time-space fractional B–S equation.

Efficient control variate methods with applications to exotic options pricing under subordinated Brownian motion models

In this paper we apply both biased and multivariate control variate methods to some exotic option pricing problems with exponential subordinated Brownian motion models for the underlying asset prices. For both arithmetic Asian and basket options, control variates conditional on geometric means of asset prices are constructed. To reduce biases, we derive expressions of expectations in terms of expectations of subordinators for relevant random variables. Numerical results show that the constructed control variates are more efficient than the classical control variates in reducing variances when pricing Asian and basket options under the normal inverse Gaussian and variance gamma models. We believe that the great variance reduction gains are due to the high correlations between the option playoffs and the constructed control variates. The efficiencies realized by the control variates are even much more significant when they are combined with quasi-Monte Carlo methods. Details are given in Section 6.

Pathwise Dynamic Programming

We present a novel method for deriving tight Monte Carlo confidence intervals for solutions of stochastic dynamic programming equations. Taking some approximate solution to the equation as an input, we construct pathwise recursions with a known bias. Suitably coupling the recursions for lower and upper bounds ensures that the method is applicable even when the dynamic program does not satisfy a comparison principle. We apply our method to three nonlinear option pricing problems, pricing under bilateral counterparty risk, under uncertain volatility, and under negotiated collateralization.

Localized kernel-based approximation for pricing financial options under regime switching jump diffusion model

In this paper, we consider European and American option pricing problems under regime switching jump diffusion models which are formulated as a system of partial integro-differential equations (PIDEs) with fixed and free boundaries. For free boundary problem arising in pricing American option, we use operator splitting method to deal with early exercise feature of American option. For developing a numerical technique we employ localized radial basis function generated finite difference (RBF-FD) approximation to overcome the ill-conditioning and high density issues of discretized matrices. The proposed method leads to linear systems with tridiagonal and diagonal dominant matrices. Also, in this paper the convergence and consistency of the proposed method are discussed. Numerical examples presented in the last section illustrate the robustness and practical performance of the proposed algorithm for pricing European and American options.

A Stock Model with Varying Stock Diffusion for Uncertain Market

The option-pricing problem is an important topic in modern finance. In this paper, we propose a stock model with varying stock diffusion based on uncertainty theory. The European option pricing formulas are derived from the proposed uncertain stock model, and some mathematical properties of these formulas are investigated. Moreover, extended uncertain stock models are introduced and discussed. Finally, numerical examples are given to illustrate the proposed model.

Pricing European vanilla options under a jump-to-default threshold diffusion model

In this paper, we study option prices under a feasible threshold diffusion model subject to jump-to-default risk where the default intensity takes a negative power of the underlying stock price. The model incorporates the regime switches endogenously by assuming the volatility to shift from one regime to another when the stock price crosses the pre-specified threshold level. This threshold can be understood as the psychological price barrier. Using the probabilistic approach, we obtain the Laplace-transform-based analytical solutions to the pricing problem of European vanilla options. Numerical analysis in the end examines the option-related quantities using the derived results and shows the impact of jump-to-default risk and threshold effect.

High dimensional integration of kinks and jumps—Smoothing by preintegration

We show how simple kinks and jumps of otherwise smooth integrands over R d can be dealt with by a preliminary integration with respect to a single well chosen variable. It is assumed that this preintegration, or conditional sampling, can be carried out with negligible error, which is the case in particular for option pricing problems. It is proven that under appropriate conditions the preintegrated function of d − 1 variables belongs to appropriate mixed Sobolev spaces, so potentially allowing high efficiency of Quasi Monte Carlo and Sparse Grid Methods applied to the preintegrated problem. The efficiency of applying Quasi Monte Carlo to the preintegrated function are demonstrated on a digital Asian option using the Principal Component Analysis factorization of the covariance matrix.

Lookback options pricing for uncertain financial market

Lookback options are among the most popular path-dependent options in financial market. In this paper, the option pricing problem of lookback options is investigated under the assumption that the underlying stock price follows an uncertain differential equation driven by Liu process instead of stochastic differential equation, and the lookback options pricing formulae are derived under this assumption. Several numerical examples are also discussed to illustrate the pricing formula.

Efficient Semi-Discretization Techniques for Pricing European and American Basket Options

This paper studies the valuation of option pricing problem in the presence of several assets. European- and American-style options are dealt with using high order semi-discretization techniques. We integrate the resulting system of semi-discretized ODEs in time with several time-stepping schemes such as explicit Euler method, explicit Runge–Kutta method, and the Runge–Kutta method with adaptive variable step sizes. Numerical results demonstrate that employing higher order semi-discretizations improves the computational performance of the multi-asset option pricing problems.

RBF-PU method for pricing options under the jump–diffusion model with local volatility

Meshfree methods based on radial basis functions (RBFs) are of general interest for solving partial differential equations (PDEs) because they can provide high order or spectral convergence for smooth solutions in complex geometries. For global RBF methods, one of the major disadvantages is the computational cost associated with the dense linear systems that arise. Therefore, this paper is currently directed toward localized RBF approximations known as the RBF partition of unity (RBF-PU) method for partial integro-differential equation (PIDE) arisen in option pricing problems in jump–diffusion model. RBF-PU method produces algebraic systems with sparse matrices which have small condition number. Also, for comparison, some stable time discretization schemes are combined with the operator splitting method to get a fully discrete problem. Numerical examples are presented to illustrate the convergence and stability of the proposed algorithms for pricing European and American options with Merton and Kou models.

A different approach to the European option pricing model with new fractional operator

In this work, we have derived an approximate solution of the fractional Black-Scholes models using an iterative method. The fractional differentiation operator used in this paper is the well-known conformable derivative. Firstly, we redefine the fractional Black-Scholes equation, conformable fractional Adomian decomposition method (CFADM) and conformable fractional modified homotopy perturbation method (CFMHPM). Then, we have solved the fractional Black-Scholes (FBS) and generalized fractional Black-Scholes (GFBS) equations by using the proposed methods, which can analytically solve the fractional partial differential equations (FPDE). In order to show the efficiencies of these methods, we have compared the numerical and exact solutions of these two option pricing problems by using in pricing the actual market data. Also, we have found out that the proposed models are very efficient and powerful techniques in finding approximate solutions of the fractional Black-Scholes models which are considered in conformable sense.

Mixed Fractional Merton Model to Evaluate European Options with Transaction Costs

This paper deals with the problem of discrete-time option pricing by the mixed fractional version of Merton model with transaction costs. By a mean-self-financing delta hedging argument in a discrete-time setting, a European call option pricing formula is obtained. We also investigate the effect of the time-step δt and the Hurst parameter H on our pricing option model, which reveals that these parameters have high impact on option pricing. The properties of this model are also explained.

The QLBS Q-Learner Goes NuQLear: Fitted Q Iteration, Inverse RL, and Option Portfolios

The QLBS model is a discrete-time option hedging and pricing model that is based on Dynamic Programming (DP) and Reinforcement Learning (RL). It combines the famous Q-Learning method for RL with the Black-Scholes (-Merton) model's idea of reducing the problem of option pricing and hedging to the problem of optimal rebalancing of a dynamic replicating portfolio for the option, which is made of a stock and cash. Here we expand on several NuQLear (Numerical Q-Learning) topics with the QLBS model. First, we investigate the performance of Fitted Q Iteration for a RL (data-driven) solution to the model, and benchmark it versus a DP (model-based) solution, as well as versus the BSM model. Second, we develop an Inverse Reinforcement Learning (IRL) setting for the model, where we only observe prices and actions (re-hedges) taken by a trader, but not rewards. Third, we outline how the QLBS model can be used for pricing portfolios of options, rather than a single option in isolation, thus providing its own, data-driven and model independent solution to the (in)famous volatility smile problem of the Black-Scholes model.

Interpretable Optimal Stopping

Optimal stopping is the problem of deciding when to stop a stochastic system to obtain the greatest reward, arising in numerous application areas such as finance, healthcare and marketing. State-of-theart methods for high-dimensional optimal stopping involve approximating the value function or the continuation value, and then using that approximation within a greedy policy. Although such policies can perform very well, they are generally not guaranteed to be interpretable; that is, a decision maker may not be able to easily see the link between the current system state and the policy's action. In this paper, we propose a new approach to optimal stopping, wherein the policy is represented as a binary tree, in the spirit of naturally interpretable tree models commonly used in machine learning. We show that the class of tree policies is rich enough to approximate the optimal policy. We formulate the problem of learning such policies from observed trajectories of the stochastic system as a sample average approximation (SAA) problem. We prove that the SAA problem converges under mild conditions as the sample size increases, but that computationally even immediate simplifications of the SAA problem are theoretically intractable. We thus propose a tractable heuristic for approximately solving the SAA problem, by greedily constructing the tree from the top down. We demonstrate the value of our approach by applying it to the canonical problem of option pricing, using both synthetic instances and instances using real S&P-500 data. Our method obtains policies that (1) outperform state-of-the-art noninterpretable methods, based on simulation-regression and martingale duality, and (2) possess a remarkably simple and intuitive structure.

Comparison: Binomial model and Black Scholes model

The Binomial Model and the Black Scholes Model are the popular methods that are used to solve the option pricing problems. Binomial Model is a simple statistical method and Black Scholes model requires a solution of a stochastic differential equation. Pricing of European call and a put option is a very difficult method used by actuaries. The main goal of this study is to differentiate the Binominal model and the Black Scholes model by using two statistical model -t-test and Tukey model at one period. Finally, the result showed that there is no significant difference between the means of the European options by using the above two models.