Pricing Barrier Options Research Articles

Spectrally accurate option pricing under the time fractional Black-Scholes model

We propose a Legendre–Laguerre spectral approximation to price the European and double barrier options in the time-fractional framework. By choosing an appropriate basis function, the spectral discretization is used for the approximation of the spatial derivatives of the time-fractional Black–Scholes equation. For the time discretization, we consider the popular \(L1\) finite difference approximation, which converges with order \(\mathcal{O}((\Delta \tau)^{2-\alpha})\) for functions which are twice continuously differentiable. However, when using the \(L1\) scheme for problems with nonsmooth initial data, only the first-order accuracy in time is achieved. This low-order accuracy is also observed when solving the time-fractional Black–Scholes European and barrier option pricing problems for which the payoffs are all nonsmooth. To increase the temporal convergence rate, we therefore consider a Richardson extrapolation method, which when combined with the spectral approximation in space, exhibits higher order convergence such that high accuracies over the whole discretization grid are obtained. Compared with the traditional finite difference scheme, numerical examples clearly indicate that the spectral approximation converges exponentially over a small number of grid points. Also, as demonstrated, such high accuracies can be achieved in much fewer time steps using the extrapolation approach. doi:10.1017/S1446181121000286

Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit

<p style='text-indent:20px;'>We continue a series of papers devoted to construction of semi-analytic solutions for barrier options. These options are written on underlying following some simple one-factor diffusion model, but all the parameters of the model as well as the barriers are time-dependent. We managed to show that these solutions are systematically more efficient for pricing and calibration than, e.g., the corresponding finite-difference solvers. In this paper we extend this technique to pricing double barrier options and present two approaches to solving it: the General Integral transform method and the Heat Potential method. Our results confirm that for double barrier options these semi-analytic techniques are also more efficient than the traditional numerical methods used to solve this type of problems.</p>

Valuation of piecewise linear barrier options

This paper discusses the valuation of piecewise linear barrier options that generalize classical barrier options. We establish formulas for joint probabilities of the logarithmic returns of the underlying asset and its partial running maxima when the process has a piecewise constant drift. In particular, we show that our results embrace the famous reflection principle as a special case, and that our established proposition delivers useful scalability for computing desired probabilities related to various types of barriers. We derive the closed-form prices of piecewise linear barrier options under the Black–Scholes framework, which are obtainable with little effort by relying on the derived probabilities. In addition, we provide numerical examples and discuss how option prices respond to several types of piecewise linear barriers.

On pricing of vulnerable barrier options and vulnerable double barrier options

In this paper, we provide analytical pricing formulae of vulnerable barrier options and vulnerable double barrier options. To obtain the price of vulnerable double barrier options, we give the joint distribution of a special range of the two-dimensional correlated Brownian motions. This result is of independent interest. Based on the derived formula, we numerically analyze the impacts of counterparty risk on option prices.

American Barrier Option Pricing Formulas for Currency Model in Uncertain Environment.

Option pricing problem is one of the central issue in the theory of modern finance. Uncertain currency model has been put forward under the foundation of uncertainty theory as a tool to portray the foreign exchange rate in uncertain finance market. This paper uses uncertain differential equation involved by Liu process to dispose of the foreign exchange rate. Then an American barrier option of currency model in uncertain environment is investigated. Most important of all, the authors deduce the formulas to price four types of American barrier options for this currency model in uncertain environment by rigorous derivation.

Enhancing finite difference approximations for double barrier options: mesh optimization and repeated Richardson extrapolation

We show that the performances of the finite difference method for double barrier option pricing can be strongly enhanced by applying both a repeated Richardson extrapolation technique and a mesh optimization procedure. In particular, first we construct a space mesh that is uniform and aligned with the discontinuity points of the solution being sought. This is accomplished by means of a suitable transformation of coordinates, which involves some parameters that are implicitly defined and whose existence and uniqueness is theoretically established. Then, a finite difference scheme employing repeated Richardson extrapolation in both space and time is developed. The overall approach exhibits high efficacy: barrier option prices can be computed with accuracy close to the machine precision in less than one second. The numerical simulations also reveal that the improvement over existing methods is due to the combination of the mesh optimization and the repeated Richardson extrapolation.

SPECTRALLY ACCURATE OPTION PRICING UNDER THE TIME-FRACTIONAL BLACK–SCHOLES MODEL

AbstractWe propose a Legendre–Laguerre spectral approximation to price the European and double barrier options in the time-fractional framework. By choosing an appropriate basis function, the spectral discretization is used for the approximation of the spatial derivatives of the time-fractional Black–Scholes equation. For the time discretization, we consider the popular$L1$finite difference approximation, which converges with order$\mathcal {O}((\Delta \tau )^{2-\alpha })$for functions which are twice continuously differentiable. However, when using the$L1$scheme for problems with nonsmooth initial data, only the first-order accuracy in time is achieved. This low-order accuracy is also observed when solving the time-fractional Black–Scholes European and barrier option pricing problems for which the payoffs are all nonsmooth. To increase the temporal convergence rate, we therefore consider a Richardson extrapolation method, which when combined with the spectral approximation in space, exhibits higher order convergence such that high accuracies over the whole discretization grid are obtained. Compared with the traditional finite difference scheme, numerical examples clearly indicate that the spectral approximation converges exponentially over a small number of grid points. Also, as demonstrated, such high accuracies can be achieved in much fewer time steps using the extrapolation approach.

Pricing external barrier options under a stochastic volatility model

An external barrier option has a random variable which determines whether the option is knock-in or knock-out. In this paper, we deal with the pricing of the external barrier option under a stochastic volatility model incorporated by a fast mean-reverting process. By using a singular perturbation method (asymptotic analysis) on the given partial differential equation for the option price, and applying the double Mellin transform technique and the method of images, we derive the corrected option price, which is an explicit analytical approximated solution for the external barrier option. For numerical experiments, we verify the price accuracy of the external barrier option with a stochastic volatility model by comparing the approximated option price with the option price obtained by Monte Carlo simulation. Finally, we investigate the behavior and sensitivity of option prices to model parameters.

A computational weighted finite difference method for American and barrier options in subdiffusive Black–Scholes model

Subdiffusion is a well established phenomenon in physics. In this paper we apply the subdiffusive dynamics to analyze financial markets. We focus on the financial aspect of time fractional diffusion model with moving boundary i.e. American and barrier option pricing in the subdiffusive Black–Scholes (B–S) model. Two computational methods for valuing American options in the considered model are proposed - the weighted finite difference (FD) and the Longstaff–Schwartz method. In the article it is also shown how to valuate numerically wide range of barrier options using the FD approach.

Reliability index and Asian barrier option pricing formulas of the uncertain fractional first-hitting time model with Caputo type

Abstract This paper mainly studies the pricing problem of arithmetic average Asian barrier options in the continuous case and analyzes the corresponding reliability index. Owing to the fact that the return function of the option is closely related to the average price of the underlying asset in a certain period, which can effectively alleviate the market speculation, the Asian barrier option is widely active in the financial market as a derivative product. Firstly, considering that there exists difficult to predict and measure reliability based on historical data, traditional stock models relied on stochastic theory are abandoned, and further assumes that the underlying assets follow an uncertain process. Then, we introduce the uncertain fractional differential equations with Caputo-type to describe the dynamic changes of risky asset prices. Secondly, in order to analyze the impact on the first hitting time of the barrier being triggered and the option execution result theoretically, a novel first-hitting time model is established to measure the relationship between the option reliability index and the value of risky assets. Meanwhile, the pricing formulas of four Asian barrier options and the corresponding reliability index under the first-hitting time model are derived, respectively. Finally, numerical algorithms and corresponding methods are designed to verify the results of our model.

Wavelet-optimized compact finite difference method for convection–diffusion equations

Abstract In this article, compact finite difference approximations for first and second derivatives on the non-uniform grid are discussed. The construction of diffusion wavelets using compact finite difference approximation is presented. Adaptive grids are obtained for non-smooth functions in one and two dimensions using diffusion wavelets. High-order accurate wavelet-optimized compact finite difference (WOCFD) method is developed to solve convection–diffusion equations in one and two dimensions on an adaptive grid. As an application in option pricing, the solution of Black–Scholes partial differential equation (PDE) for pricing barrier options is obtained using the proposed WOCFD method. Numerical illustrations are presented to explain the nature of adaptive grids for each case.

A Fourier-Cosine Method for Pricing Discretely Monitored Barrier Options under Stochastic Volatility and Double Exponential Jump

In this paper, the valuation of the discrete barrier options on the condition that the underlying asset price process follows the GARCH volatility and double exponential jump is studied. We derived an analytical approximation of the characteristic function for the underlying log-asset price. Then, a quasianalytical approximate formula of the price of the discrete barrier option is obtained based the on Fourier-cosine method. Numerical examples show that the Fourier-cosine method is fast and efficient for pricing discrete barrier options compared with the Monte Carlo simulation method. Finally, the influences of some important parameters on the prices of discrete barrier options are studied to further illustrate the rationality of the model.

Geometric Asian barrier option pricing formulas of uncertain stock model

Abstract In the high-risk modern financial market, option is an effective tool to hedge the risks caused by uncertain demand, the fluctuation of price and foreign exchange rate, because the option can provide the holder with an entitlement to sell or purchase an asset with an exercise price. The acquisition of this entitlement requires the investor to pay option fee, which raises the option pricing issue. This article analyzes how to price Geometric Asian barrier option for uncertain stock model, where barrier option becomes activated or inactivated depending on whether a given barrier level is hit. Here, we suppose that stock price obeys an uncertain differential equation, based on which the pricing formulas of Geometric Asian barrier option are discovered. Furthermore, to express how to use the pricing formulas to calculate corresponding option prices, some numerical examples are given.

Pricing of Barrier Options on Underlying Assets with Jump-Diffusion Dynamics: A Mellin Transform Approach

A barrier option is an exotic path-dependent option contract where the right to buy or sell is activated or extinguished when the underlying asset reaches a certain barrier price during the lifetime of the contract. In this article we use a Mellin transform approach to derive exact pricing formulas for barrier options with general payoffs and exponential barriers on underlying assets that have jump-diffusion dynamics. With the same approach we also price barrier options on underlying futures contracts.

Barrier option pricing formulas of an uncertain stock model

As applications of the uncertainty theory to finance, uncertain stock models have been presented to describe the prices of stocks strongly influenced by human uncertainty. So far, large progress has been achieved on pricing problems of path-independent options of the uncertain stock models. This paper investigates a type of path-dependent exotic options of an uncertain stock model which are named barrier options. Pricing formulas are derived based on the structure of the solutions of uncertain differential equations, and numerical algorithms are designed to calculate the prices of the barrier options based on these formulas.

Semi-Closed Form Prices of Barrier Options in the Time-Dependent CEV and CIR Models

The authors continue a series of articles where prices of the barrier options written on the underlying, whose dynamics follow a one-factor stochastic model with time-dependent coefficients and the barrier, are obtained in semi-closed form; see Carr and Itkin (2020) and Itkin and Muravey (2020). This article extends this methodology to the Cox–Ingersoll–Ross model for zero-coupon bonds and to the constant elasticity of variance model for stocks, which are used as the corresponding underlying for the barrier options. The authors describe two approaches. One is a generalization of the method of heat potentials (for the heat equation) to the Bessel process, so it is named “the method of Bessel potentials.” The authors also propose a general scheme for constructing the potential method for any linear differential operator with time-independent coefficients. The second approach is the method of generalized integral transform, which is also extended to the Bessel process. In all cases, a semi-closed solution means that first, we need to numerically solve a linear Volterra equation of the second kind, and then the option price is represented as a one-dimensional integral. The authors demonstrate that their method is computationally more efficient than both the backward and forward finite difference methods, while providing better accuracy and stability. Also, it is shown that these methods do not duplicate, but instead they complement each other, as one provides very accurate results at shorter maturities and the other provides such results at longer maturities. TOPICS:Options, statistical methods Key Findings • This article extends this methodology to the CIR model for zero-coupon bonds and to the CEV model for stocks, which are used as the corresponding underlying for the barrier options. • One approach is a generalization of the method of heat potentials (for the heat equation) to the Bessel process, so we call it “the method of Bessel potentials.” The second approach is the method of generalized integral transform, which is also extended to the Bessel process. • These methods do not duplicate, but instead they complement each other, as one provides very accurate results at shorter maturities and the other provides such results at longer maturities.

The Wiener-Hopf Factorization for Pricing Options Made Easy

The paper suggest a new approach to pricing barrier options under pure non-Gaussian Levy processes with jumps of finite variation. The key idea behind the method to represent the process under consideration as a difference between subordinators (increasing Levy processes). Such splitting rule applied to the process at exponentially distributed randomized time points gives us the possibility to find the option price by analytically solving a sequence of simple Wiener-Hopf equations.

ANALYTICAL PATH-INTEGRAL PRICING OF DETERMINISTIC MOVING-BARRIER OPTIONS UNDER NON-GAUSSIAN DISTRIBUTIONS

In this work, we present an analytical model, based on the path-integral formalism of statistical mechanics, for pricing options using first-passage time problems involving both fixed and deterministically moving absorbing barriers under possibly non-Gaussian distributions of the underlying object. We adapt to our problem a model originally proposed by De Simone et al. (2011) to describe the formation of galaxies in the universe, which uses cumulant expansions in terms of the Gaussian distribution, and we generalize it to take into account drift and cumulants of orders higher than three. From the probability density function, we obtain an analytical pricing model, not only for vanilla options (thus removing the need of volatility smile inherent to the Black & Scholes (1973) model), but also for fixed or deterministically moving barrier options. Market prices of vanilla options are used to calibrate the model, and barrier option pricing arising from the model is compared to the price resulted from the relative entropy model.

Pricing Barrier Options with DeepBSDEs

This paper presents a novel and direct approach to price boundary and final-value problems, corresponding to barrier options, using forward deep learning to solve forward-backward stochastic differential equations (FBSDEs). Barrier instruments are instruments that expire or transform into another instrument if a barrier condition is satisfied before maturity; otherwise they perform like the instrument without the barrier condition. In the PDE formulation, this corresponds to adding boundary conditions to the final value problem. The deep BSDE methods developed so far have not addressed barrier/boundary conditions directly. We extend the forward deep BSDE to the barrier condition case by adding nodes to the computational graph to explicitly monitor the barrier conditions for each realization of the dynamics as well as nodes that preserve the time, state variables, and trading strategy value at barrier breach or at maturity otherwise. Given these additional nodes in the computational graph, the forward loss function quantifies the replication of the barrier or final payoff according to a chosen risk measure such as squared sum of differences. The proposed method can handle any barrier condition in the FBSDE set-up and any Dirichlet boundary conditions in the PDE set-up, both in low and high dimensions.

Barrier Option Pricing with Heavy Tailed Distribution

Barrier Option Pricing with Heavy Tailed Distribution