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

<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>

An efficient and provable sequential quadratic programming method for American and swing option pricing

High-order methods for the option pricing under multivariate rough volatility models

We first recall the well-known expression of the price of barrier options, and compute double barrier options by the mean of the iterated mirror principle. The formula for double barriers provides an intraday volatility estimator from the information of high-low-close prices. Then we give explicit formulas for the probability distribution function and the expectation of the exit time of single and double barrier options. These formulas allow to price time independent and time dependent rebates. They are also helpful to hedge barrier and double barrier options, when taking into account variations of the term structure of interest rates and of volatility. We also compute the price of rebates of double knock-out options that depend on which barrier is hit first, and of the BOOST, an option which pays the time spent in a corridor. All these formulas are either in closed form or double infinite series which converge like e-α n2.

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.

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

In this paper we summarize the main idea and results of Yuen and Yang (2009, 2010a, 2010b) and provide some results on pricing of Parisian options under the Markov regime-switching model (MRSM). The MRSM allows the parameters of the market model depending on a Markovian process, and the model can reflect the information of the market environment which cannot be modeled solely by linear Gaussian process. However, when the parameters of the stock price model are not constant but governed by a Markovian process, the pricing of the options becomes complex. We present a fast and simple trinomial tree model to price options in MRSM. In recent years, the pricing of modern insurance products, such as Equity-Indexed annuity (EIA) and variable annuities (VAs), has become a popular topic. We show here that our trinomial tree model can been used to price EIA with strong path dependent exotic options in the regime switching model.

This article derives the bivariate joint probability distribution functions of a geometric Brownian motion and the extreme values of another geometric Brownian. Based on the probability distribution functions, the authors develop the analytical pricing formulas of three exotic double barrier options (DBOs) with continuously monitored barriers, including rainbow DBOs, protected DBOs, and protected rainbow DBOs. By using the continuity correction of barriers proposed in research by Doobae Jun, the aforementioned pricing formulas can be further extended to price the three exotic DBOs with discretely monitored barriers. Some numerical examples are also provided for end-users to examine the pricing accuracy and efficiency and the properties of the exotic DBOs. TOPICS:Derivatives, options Key Findings ▪ This article derives the bivariate joint probability distribution functions of a geometric Brownian motion and the extreme values of another geometric Brownian. ▪ This article develops the analytical pricing formulas of three exotic double barrier options (DBOs) with continuously monitored barriers, including rainbow DBOs, protected DBOs, and protected rainbow DBOs. ▪ By using the continuity correction of barriers proposed in research by Doobae Jun, the three exotic double barrier options with discretely monitored barriers are also developed.

We present a very fast and accurate algorithm for calculating prices of finite lived double barrier options with arbitrary terminal payoff functions under regime-switching hyper-exponential jump-diffusion (HEJD) models, which generalize the double-exponential jump-diffusion model pioneered by Kou and Lipton. Numerical tests demonstrate an excellent agreement of our results with those obtained using other methods, as well as a significant increase in computation speed (sometimes by a factor of 5). The first step of our approach is Carr's randomization, whose convergence we prove for barrier and double barrier options under strong Markov processes of a wide class. The resulting sequence of perpetual option pricing problems is solved using an efficient iteration algorithm and the Wiener-Hopf factorization.

Price dispersion and vanilla options in a financial market game

In recent years, option pricing has garnered significant attention as a critical topic in financial engineering. Classical models, such as Black-Scholes, exhibit limitations in accurately capturing the dynamic changes in complex market environments. Consequently, machine learning techniques and neural network methods have emerged as pivotal research directions in option pricing. This paper provides a systematic review of the advances in option pricing based on optimization methods, machine learning techniques, and neural network models. Optimization methods, incorporating techniques like genetic algorithms and particle swarm optimization, significantly enhance the adaptability and accuracy of classical models. Machine learning methods, such as support vector machines and decision trees, demonstrate their advantages in nonlinear data modeling. Neural networks and deep learning models, with their robust nonlinear fitting capabilities, have become central tools in pricing complex financial products. Additionally, this review summarizes the application of hybrid models and evaluates the strengths and weaknesses of various approaches in practical applications. Finally, the paper discusses future research directions, including the integration of reinforcement learning techniques, improvement in model interpretability, and deep fusion of heterogeneous multi-source data. This work aims to provide theoretical references and technical support for further research in the field of option pricing.

One of the possible improvements of the classical Black–Scholes option pricing model is to incorporate the stochastic nature of the short rate dynamics in option valuation. In this paper, we present the numerical scheme, based on the discontinuous Galerkin method, for European option pricing when the short rate follows the Merton model. The pricing function satisfies a partial differential equation with two underlying variables — stock price and short rate value. With a localization to a bounded spatial domain, including setting the proper boundary conditions, the governing equation is discretized by the discontinuous Galerkin method over a finite element grid and Crank-Nicolson time integration is applied, consequently. As a result the numerical scheme is represented by a sequence of linear algebraic systems with sparse matrices. Moreover, the numerical simulations reflect the capability of the scheme presented.

SummaryDouble barrier options are important path-dependent derivatives in the financial market. How to price them efficiently and accurately is thus important. Until now, no simple closed-form pricing formula for double barrier options is reported. Double barrier options can be priced on a lattice that divides a certain time interval (from option initial date to maturity date) into n equal-length time steps. The pricing results obtained by the lattice algorithm converge to the true option value as n → ∞, and the results oscillate significantly especially when n is not large enough. To obtain an accurate pricing result without suffering from price oscillation, n is required to be a large number. Unfortunately, the lattice pricing algorithm runs in O(n 2) time. This paper proposes a linear-time combinatorial algorithm that can generate the same pricing results as the lattice algorithm. Thus our algorithm can handle very large n’s efficiently. This algorithm uses a novel technique based on the reflection principle and the inclusion-exclusion principle. Numerical experiments are given to verify the excellent performance of our algorithm.

From Continuous to Discrete: Studies on Continuity Corrections and Monte Carlo Simulation with Applications to Barrier Options and American Options Menghui Cao This dissertation 1) shows continuity corrections for first passage probabilities of Brownian bridge and barrier joint probabilities, which are applied to the pricing of two-dimensional barrier and partial barrier options, and 2) introduces new variance reduction techniques and computational improvements to Monte Carlo methods for pricing American options. The joint distribution of Brownian motion and its first passage time has found applications in many areas, including sequential analysis, pricing of barrier options, and credit risk modeling. There are, however, no simple closed-form solutions for these joint probabilities in a discrete-time setting. Chapter 2 shows that, discrete two-dimensional barrier and partial barrier joint probabilities can be approximated by their continuous-time probabilities with remarkable accuracy after shifting the barrier away from the underlying by a factor of βσ √ ∆t. We achieve this through a uniform continuity correction theorem on the first passage probabilities for Brownian bridge, extending relevant results in Siegmund (1985a). The continuity corrections are applied to the pricing of two-dimensional barrier and partial barrier options, extending the results in Broadie, Glasserman & Kou (1997) on one-dimensional barrier options. One interesting aspect is that for type B partial barrier options, the barrier correction cannot be applied throughout one pricing formula, but only to some barrier values and leaving the other unchanged, the direction of correction may also vary within one

A Note on the Option Price and 'Mass at Zero in the Uncorrelated SABR Model and Implied Volatility Asymptotics'