Double Barrier Options Research Articles

A simplified Wiener–Hopf factorization method for pricing double barrier options under Lévy processes

A simplified Wiener–Hopf factorization method for pricing double barrier options under Lévy processes

The Boyle–Romberg Trinomial Tree, a Highly Efficient Method for Double Barrier Option Pricing

Oscillations in option price convergence have long been a problematic aspect of tree methods, inhibiting the use of repeated Richardson extrapolation that could otherwise greatly accelerate convergence, a feature integral to some of the most efficient modern methods. These oscillations are typically caused by the fluctuating positions of nodes around the discontinuities in the payoff function or its derivatives. Our paper addresses this crucial gap that typically prohibits the use of lattice methods when high efficiency is needed. Focusing on double barrier options, we develop a trinomial tree in which the positions of the nodes are precisely adjusted to align with these discontinuities throughout the option’s lifespan and across various time steps. This alignment enables the use of repeated extrapolation to achieve high order convergence, including near barriers, a well-known challenge in many tree methods. Maintaining the inherent simplicity and adaptability of tree methods, our approach is easily applicable to other models and option types.

EFFICIENT EVALUATION OF DOUBLE-BARRIER OPTIONS

In the paper, we develop a very fast and accurate method for pricing double barrier options with continuous monitoring in wide classes of Lévy models; the calculations are in the dual space, and the Wiener–Hopf factorization is used. For wide regions in the parameter space, the precision of the order of [Formula: see text] is achievable in seconds, and of the order of [Formula: see text]–[Formula: see text] — in fractions of a second. The Wiener–Hopf factors and repeated integrals in the pricing formulas are calculated using sinh-deformations of the lines of integration, the corresponding changes of variables and the simplified trapezoid rule. If the Bromwich integral is calculated using the Gaver–Wynn Rho acceleration instead of the sinh-acceleration, the CPU time is typically smaller but the precision is of the order of [Formula: see text]–[Formula: see text], at best. Explicit pricing algorithms and numerical examples are for no-touch options, digitals (equivalently, for the joint distribution function of a Lévy process and its supremum and infimum processes), and call options. Several graphs are produced to explain fundamental difficulties for accurate pricing of barrier options using time discretization and interpolation-based calculations in the state space.

Alternative Models for FX: Pricing Double Barrier Options in Regime-switching Lévy Models With Memory

Alternative Models for FX: Pricing Double Barrier Options in Regime-switching Lévy Models With Memory

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.

An Efficient Numerical Method for Pricing Double-Barrier Options on an Underlying Stock Governed by a Fractal Stochastic Process

After the discovery of the fractal structures of financial markets, enormous effort has been dedicated to finding accurate and stable numerical schemes to solve fractional Black-Scholes partial differential equations. This work, therefore, proposes a numerical scheme for pricing double-barrier options, written on an underlying stock whose dynamics are governed by a non-standard fractal stochastic process. The resultant model is time-fractional and is herein referred to as a time-fractional Black-Scholes model. The presence of the time-fractional derivative helps to capture the time-decaying effects of the underlying stock while capturing the globalized change in underlying prices and barriers. In this paper, we present the construction of the proposed scheme, analyse it in terms of its stability and convergence, and present two numerical examples of pricing double knock-in barrier-option problems. The results suggest that the proposed scheme is unconditionally stable and convergent with order O(h2+k2).

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.

Numerical approximation and fast implementation to a generalized distributed-order time-fractional option pricing model

We present a fully-discrete finite element scheme to a generalized distributed-order time-fractional option pricing model, which adequately describes, e.g., the valuation of the European double barrier option. Due to the dependence of the density function on the stock price, the temporal discretization coefficients from the generalized distributed-order time-fractional derivative will be coupled with the inner product of the finite element method, which significantly complicates the analysis and traditional numerical analysis techniques do not apply. Novel techniques are developed to prove error estimates of this fully-discrete numerical scheme, which not only resolves the above difficulty, but indeed simplifies existing methods by avoiding the mathematical induction procedure. Based on the structure of the all-at-once coefficient matrix of the proposed numerical scheme, a fast divide and conquer algorithm is developed to reduce the computational cost of solving the numerical scheme from O(LNt2Nx) to O(LNtlogNtNx), where L, Nt and Nx refer to numbers of the degree of freedom of discretizations for the distributed-order integral, the spatial domain and the time period, respectively. Numerical experiments are performed to demonstrate the accuracy of the proposed numerical scheme and its applications in the valuation of the option price.

Pricing multi-step double barrier options by the efficient non-crossing probability

This paper considers pricing multi-step double barrier options. The non-crossing probability for a multi-step double boundary is vital in valuing the options. We extend an explicit formula for the non-crossing probability using the solutions to the relevant Fokker–Planck equations. The derived formula not only provides a new look at the non-crossing probability but also outperforms the existing probability formula relying on multivariate normal distribution functions in terms of efficiency by avoiding multi-dimensional numerical integrals. We demonstrate the merits of the new expression of the non-crossing probability via numerical experiments and apply it to valuing multi-step double barrier options.

The Research on the Pricing of Double Barrier Option Based on the Black-Scholes Model

The two barrier values in the double barrier option are the focus in the evaluation simulation and they affect the knock-in and knock-out of the option. This paper, simulates a double barrier option and calculates its option value based on the Black-Scholes Model, using Coca-Cola stock data from January 2022 to the end of July, and verifies the feasibility of the Black-Scholes model. This paper also analyzes the sensitivity of the influence of the two barrier values on the option value, outlining the option value changes under different barrier values. The main finding of this paper is that the Black-Scholes formula is used to verify the value of the Coca-Cola option product. This study enriches the practical examples of the validation of double barrier option products in the Black-Scholes formula.

Alternative models for FX, arbitrage opportunities and efficient pricing of double barrier options in L\'evy models

Alternative models for FX, arbitrage opportunities and efficient pricing of double barrier options in L\'evy models

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.

On pricing double-barrier options with Markov regime switching

We propose an efficient method for valuation of double-barrier options in a Markov regime switching diffusion model. This model incorporates three factors: Structural changes in economic conditions, business and investment environments into the diffusion process, and capturing some important properties of asset returns such as asymmetry and heavy tails. Under a such framework, we provide the closed-form upper and lower bounds of double-barrier options using Fourier series expansion. Furthermore, the bounds can also be systemically improved to get higher accuracy. In addition, we carry out extensive numerical experiments, and the results show that the solution we derive is tight.

The valuation of barrier options under a threshold rough Heston model

In this paper, we propose a novel model for pricing double barrier options, where the asset price is modeled as a threshold geometric Brownian motion time changed by an integrated activity rate process, which is driven by the convolution of a fractional kernel with the CIR process. The new model both captures the leverage effect and produces rough paths for the volatility process. The model also nests the threshold diffusion, Heston and rough Heston models. We can derive analytical formulas for the double barrier option prices based on the eigenfunction expansion method. We also implement the model and numerically investigate the sensitivities of option prices with respect to the parameters of the model.

Pricing European Double Barrier Option with Moving Barriers Under a Fractional Black–Scholes Model

Pricing European Double Barrier Option with Moving Barriers Under a Fractional Black–Scholes Model

Pricing double-barrier Parisian options

Abstract Parisian options are a useful real-option tool in risk management, particularly for corporate bankruptcy protection. However, in the past, only one barrier with the Parisian feature was studied in the literature, possibly due to the additional complication that arises with the co-existence of two barriers. In this paper, we present an analytic solution for the valuation of European-style double-barrier Parisian options by casting the pricing problem into three inter-coupled partial differential equations. These are then solved using a dimension reduction procedure and the ‘moving window’ technique. Our solution is in an explicit and analytical form that is written in terms of multiple integrals. This represents an important advantage over the purely numerical approaches previously published in the literature.

A simple and efficient numerical method for pricing discretely monitored early-exercise options

We present a simple, fast, and accurate method for pricing a variety of discretely monitored options in the Black-Scholes framework, including autocallable structured products, single and double barrier options, and Bermudan options. The method is based on a quadrature technique, and it employs only elementary calculations and a fixed one-dimensional uniform grid. The convergence rate is O(1/N4) and the complexity is O(MNlogN), where N is the number of grid points and M is the number of observation dates. Besides Black-Scholes, our method is also applicable to more general frameworks such as Merton’s jump diffusion model.

Double-barrier option pricing equations under extended geometric Brownian motion with bankruptcy risk

We consider an extended geometric Brownian motion with bankruptcy risk and solve its double barrier option pricing problem. We establish its partial differential equation and provide its numerical solution. Then we discuss the influence of bankruptcy omission by simulation.

A Valuation Formula for Chained Options with n‐Barriers

This study examines chained options that are connected in the sense that another barrier option becomes active continuously after the underlying asset price crosses a primary barrier. These barrier options have several advantages. First, they preserve the merit of regular barrier options, but demand far lower option premiums, which appeal to option traders. Second, they reduce the higher risk of loss of double barrier options, making option strategies more profitable in certain cases. Third, they have closed‐form pricing formulas, unlike double‐barrier options, and, thus, avoid the complexity of option pricing. Therefore, they help to enlarge the range of trader’s choice according to a variety of demand of buyers. The values of chained options are compared to those of similar single‐ and double‐barrier options. This study extends the chained option with two barriers to a generalized chained option with n‐barriers. In addition, this paper proves the closed formulas of generalized chained options with n‐barriers using mathematical induction.