Discrete Barrier Options Research Articles

Min–max multi-step barrier options and their variants

This paper studies a new type of barrier option, min–max multi-step barrier options with diverse multiple up or down barrier levels placed in the sub-periods of the option’s lifetime. We develop the explicit pricing formula of this type of option under the Black–Scholes model and explore its applications and possible extensions. In particular, the min–max multi-step barrier option pricing formula can be used to approximate double barrier option prices and compute prices of complex barrier options such as discrete geometric Asian barrier options. As a practical example of directly applying the pricing formula, we introduce and evaluate a re-bouncing equity-linked security. The main theorem of this work is capable of handling the general payoff function, from which we obtain the pricing formulas of various min–max multi-step barrier options. The min–max multi-step reflection principle, the boundary-crossing probability of min–max multi-step barriers with icicles, is also derived.

The pricing and static hedging of multi-step double barrier options

As a sequel to Lee et al. (2022b), this paper explores the pricing of multi-step double barrier options with arbitrary European-type payoffs from a static hedging perspective. Using the reflection principle of Brownian motion, we develop how to construct an exact static hedging portfolio consisting of simple discrete barrier options under the Black–Scholes model. This equivalent conversion from continuous monitoring to discrete ones provides an efficient way of evaluating multi-step double barrier options, while overcoming the drawbacks of dynamic hedging. We illustrate our result with numerical examples, and extend it to other asset price dynamics such as jump diffusion.

Continuity correction: on the pricing of discrete double barrier options

This article deals with the pricing of double-barrier options monitored discretely. A continuity correction method is established to provide an analytical approximation for the price of such discrete options under the Black–Scholes model. We achieve this by applying the smooth-fit principle simultaneously to the two flat boundaries (barriers) associated. The resulting correction form still involves adjustments in the levels of barriers, but the amounts adjusted can be different for different boundaries. More interestingly, the shift for each boundary can also be in different directions, which depends largely on the position of the current level relative to the two boundaries. Numerical examples are provided as well which support our theoretical achievements.

A numerical method for pricing discrete double barrier option by Lagrange interpolation on Jacobi nodes

In this paper, a rapid and high accurate numerical method for pricing discrete single and double barrier knock‐out call options is presented. With regard to the well‐known Black‐Scholes model, the price of an option in each monitoring date could be calculated by computing a recursive integral formula that is based on the heat equation solution. We have approximated these recursive solutions with the aid of Lagrange interpolation on Jacobi polynomial nodes. After that, an operational matrix, which makes our computation significantly fast, has been derived. In some theorems, the convergence of the presented method has been shown and the rate of convergence has been derived. The most important benefit of this method is that its complexity is very low and does not depend on the number of monitoring dates. The numerical results confirm the accuracy and efficiency of the presented numerical algorithm.

Pricing Discrete Barrier Options Under the Jump-Diffusion Model with Stochastic Volatility and Stochastic Intensity

Pricing Discrete Barrier Options Under the Jump-Diffusion Model with Stochastic Volatility and Stochastic Intensity

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.

A Spectral Element Method for Option Pricing Under Regime-Switching with Jumps

In this paper, we propose the spectral element method to price European, digital, butterfly, American, discrete and continuous barrier options in a Markovian jump-diffusion regime-switching economy. The spectral element method discretisation is considered for the approximation of the spatial derivatives in a system of partial integro-differential equations and is chosen because it possesses spectral accuracy such that highly accurate option prices can be obtained using a small number of grid discretisation nodes. Essentially, the spectral element method consists of splitting the computational domain into as many elements as needed and approximating the basis functions by high-order orthogonal polynomials within each element. In order to sustain the high-order convergence in time, we also use an exponential time integration scheme to solve the semi-discrete system. Our numerical examples support our error analysis and indicate that the spectral element method converges exponentially for the values and the hedging parameters of the regime-dependent options. Therefore, the proposed scheme provides a viable alternative to the finite difference or finite element methods which usually converge only algebraically.

Pricing European-type, early-exercise and discrete barrier options using an algorithm for the convolution of Legendre series

This paper applies an algorithm for the convolution of compactly supported Legendre series (the CONLeg method) (cf. Hale and Townsend, An algorithm for the convolution of Legendre series. SIAM J. Sci. Comput., 2014, 36, A1207–A1220), to pricing European-type, early-exercise and discrete-monitored barrier options under a Lévy process. The paper employs Chebfun (cf. Trefethen et al., Chebfun Guide, 2014 (Pafnuty Publications: Oxford), Available online at: http://www.chebfun.org/) in computational finance and provides a quadrature-free approach by applying the Chebyshev series in financial modelling. A significant advantage of using the CONLeg method is to formulate option pricing and option Greek curves rather than individual prices/values. Moreover, the CONLeg method can yield high accuracy in option pricing when the risk-free smooth probability density function (PDF) is smooth/non-smooth. Finally, we show that our method can accurately price options deep in/out of the money and with very long/short maturities. Compared with existing techniques, the CONLeg method performs either favourably or comparably in numerical experiments.

A numerical method for pricing discrete double barrier option by Chebyshev polynomials

In this article, a fast numerical method based on orthogonal Chebyshev polynomials for pricing discrete double barrier option is illustrated. At first, a recursive formula for computing price of discrete double barrier option is obtained. Then, these recursive formulas are estimated by Chebyshev polynomials and expressed in operational matrix form that reduce CPU time of algorithm. Finally, the effectiveness and validity of the presented method is demonstrated by comparison with the obtained numerical results with some other algorithms.

随机波动率模型下离散几何平均亚式障碍期权定价

随机波动率模型下离散几何平均亚式障碍期权定价

Numerical algorithm for discrete barrier option pricing in a Black-Scholes model with stationary process

In this article, we propose a numerical algorithm for computing price of discrete single and double barrier option under the emph{Black-Scholes} model. In virtue of some general transformations, the partial differential equations of option pricing in different monitoring dates are converted into simple diffusion equations. The present method is fast compared to alternative numerical methods presented in previous papers.

A Spectral Approach to Pricing of Arbitrage-Free SABR Discrete Barrier Options

Market volatility smile risk in derivative pricing can be modelled by the Stochastic Alpha Beta Rho (SABR) model. Once calibrated to market data, prices of European and continuously monitored barrier options can be obtained using equivalent Black’s implied volatility approximations. However these prices are only accurate for options with short maturities. On the other-hand, discretely monitored barrier options cannot be priced using this approach and a numerical technique is required. A novel computational method based on a spectral discretisation of the pricing equation is proposed for the solution of these problems. The high accuracy of the method is first established for special cases of the SABR model where analytical solutions are available and the method is then applied to the pricing of discrete barriers under the arbitrage-free SABR model.

An importance sampling-based smoothing approach for quasi-Monte Carlo simulation of discrete barrier options

Handling discontinuities in financial engineering is a challenging task when using quasi-Monte Carlo (QMC) method. This paper develops a so-called sequential importance sampling (SIS) method to remove multiple discontinuity structures sequentially for pricing discrete barrier options. The SIS method is a smoothing approach based on importance sampling, which yields an unbiased estimate with reduced variance. However, removing discontinuities still may not recover the superiority of QMC when the dimensionality of the problem is high. In order to handle the impact of high dimensionality on QMC, one promising strategy is to reduce the effective dimension of the problem. To this end, we develop a good path generation method with the smoothed estimator under the Black–Scholes model and models based on subordinated Brownian motion (e.g., Variance Gamma process). We find that the order of path generation influences the variance of the SIS estimator, and show how to choose optimally the first generation step. As confirmed by numerical experiments, the SIS method combined with a carefully chosen path generation method can significantly reduce the variance with improved rate of convergence. In addition, we show that the effective dimension is greatly reduced by the combined method, explaining the superiority of the proposed procedure from another perspective. The SIS method is also applicable for general models (with the Euler discretization). The smoothing effect of the SIS method facilitates the use of general dimension reduction techniques in reclaiming the efficiency of QMC.

Pricing discrete barrier options under jump-diffusion model with liquidity risk

Classical option pricing theories are usually built on the paradigm of competitive and frictionless markets, while ignoring the impact of market liquidity on underlying asset prices. In this paper, the importance of liquidity risk on discrete barrier option pricing is analyzed. First, we propose a new model for describing the asset price dynamics in the presence of jumps and liquidity risks, which is able to capture empirically observed patterns. Based on the COS method, we then derive the analytical approximation formulas for the prices of the discrete barrier options. Numerical experiments demonstrate the accuracy of our proposed pricing model by comparing the analytical approximation solutions with Monte Carlo simulation. Finally, empirically studies are carried out to show the superiority of our model based on SSE 50 ETF options. The numerical and empirical results support our idea of introducing liquidity risk and jumps into the underlying asset price dynamics.

Hedging and Pricing European-Type, Early-Exercise and Discrete Barrier Options Using an Algorithm for the Convolution of Legendre Series

This paper applies an algorithm for the convolution of compactly supported Legendre series (the CONLeg method) (cf. Hale and Townsend 2014a), to pricing/hedging European-type, early-exercise and discrete- monitored barrier options under a Levy process. The paper employs Chebfun (cf. Trefethen et al. 2014) in computational finance and provides a quadrature-free approach by applying the Chebyshev series in financial modelling. A significant advantage of using the CONLeg method is to formulate option pricing and option Greek curves rather than individual prices/values. Moreover, the CONLeg method can yield high accuracy in option pricing and hedging when the risk-free smooth probability density function (PDF) is smooth/non-smooth. Finally, we show that our method can accurately price/hedge options deep in/out of the money and with very long/short maturities. Compared with existing techniques, the CONLeg method performs either favourably or comparably in numerical experiments.

Fast quadrature methods for options with discrete dividends

Discontinuities in the stock price at ex-dividend dates make it hard to derive mathematically elegant solutions for European-style options with discrete dividends under the piecewise lognormal model. Numerical schemes such as non-recombining trees are either computationally intensive, lack accuracy when computations are fast, or are unable of handling discrete barriers. Quadrature techniques circumvent the latter two problems but due to the occurrence of time-consuming multifold integrals for multi-dividend options, their implementations have been limited to three-dividend European options only. This research proposes a faster computational method where the integrand is approximated by a Chebyshev polynomial and the fast Fejér quadrature is used to evaluate integral representations of European, American, Bermudan, continuous and discrete barrier options. The method has the capability of accurately handling an arbitrary number of discrete dividend payments in an efficient manner. The computational superiority and higher accuracy of the proposed approach is demonstrated via an extensive set of numerical results. • A fast method for pricing options with discrete dividend payments is developed. • The method can price options with different exercise and barrier features. • The method can efficiently handle more discrete dividends than existing methods. • A wide set of numerical results is given to illustrate the merits of the method.

A numerical method for pricing discrete double barrier option by Legendre multiwavelet

In this Article, a fast numerical numerical algorithm for pricing discrete double barrier option is presented. According to Black-Scholes model, the price of option in each monitoring date can be evaluated by a recursive formula upon the heat equation solution. These recursive solutions are approximated by using Legendre multiwavelets as orthonormal basis functions and expressed in operational matrix form. The most important feature of this method is that its CPU time is nearly invariant when monitoring dates increase. Besides, the rate of convergence of presented algorithm was obtained. The numerical results verify the validity and efficiency of the numerical method.

Efficient and fast numerical method for pricing discrete double barrier option by projection method

In this paper, we introduce a new and considerably fast numerical method based on projection method in pricing discrete double barrier option. According to the Black–Scholes framework, the price of option in each monitoring dates is the solution of well-known partial differential equation that can be expressed recursively upon the heat equation solution. These recursive solutions are approximated by projection method and expressed in operational matrix form. The most important advantage of this method is that its computational time is nearly fixed against monitoring dates increase. Afterward, in implementing projection method we use Legendre polynomials as an orthogonal basis. Finally, the numerical results show the validity and efficiency of presented method in comparison with some others.

Pricing early-exercise and discrete barrier options by Shannon wavelet expansions

We present a pricing method based on Shannon wavelet expansions for early-exercise and discretely-monitored barrier options under exponential Levy asset dynamics. Shannon wavelets are smooth, and thus approximate the densities that occur in finance well, resulting in exponential convergence. Application of the Fast Fourier Transform yields an efficient implementation and since wavelets give local approximations, the domain boundary errors can be naturally resolved, which is the main improvement over existing methods.

Numerical Methods for Discrete Double Barrier Option Pricing Based on Merton Jump Diffusion Model

As a kind of weak-path dependent options, barrier options are an important kind of exotic options. Because the pricing formula for pricing barrier options with discrete observations cannot avoid computing a high dimensional integral, numerical calculation is time-consuming. In the current studies, some scholars just obtained theoretical derivation, or gave some simulation calculations. Others impose underlying assets on some strong assumptions, for example, a lot of calculations are based on the Black-Scholes model. This thesis considers Merton jump diffusion model as the basic model to derive the pricing formula of discrete double barrier option; numerical calculation method is used to approximate the continuous convolution by calculating discrete convolution. Then we compare the results of theoretical calculation with simulation results by Monte Carlo method, to verify their efficiency and accuracy. By comparing the results of degeneration constant parameter model with the results of previous models we verified the calculation method is correct indirectly. Compared with the Monte Carlo simulation method, the numerical results are stable. Even if we assume the simulation results are accurate, the time consumed by the numerical method to achieve the same accuracy is much less than the Monte Carlo simulation method.