Pricing Barrier Options Research Articles

Pricing barrier options with discrete dividends

The presence of discrete dividends complicates the derivation and form of pricing formulas even for vanilla options. Existing analytic, numerical, and theoretical approximations provide results of varying quality and performance. Here, we compare the analytic approach, developed and effective for European puts and calls, of Buryak and Guo with the formulas, designed in the context of barrier option pricing, of Dai and Chiu.

Collocation Boundary Element Method for the pricing of Geometric Asian Options

The Semi-Analytical method for pricing of Barrier Options (SABO) already applied in the context of European options is here extended to the evaluation of geometric Asian options with barriers. The validity of this approximation method, based on the use of collocation Boundary Element Method, is illustrated by numerical examples, where accuracy and stability of the presented approach are analyzed.

Pricing and hedging barrier options under a Markov-modulated double exponential jump diffusion-CIR model

A semi-closed-form valuation model is presented for barrier options whose underlying asset follows a mean-reverting and regime-switching double exponential jump diffusion process, and the interest rate is modulated by a mean-reverting square root model. The proposed model captures the impact of regime-switching uncertainty on barrier option prices and their hedge parameters in long and short business cycles. The model provides richer economic insight and is more appropriate for valuing barrier options in commodity markets as well as in equity and foreign-exchange markets, when an economy faces regime-switching uncertainty.

Barrier option pricing under the 2-hypergeometric stochastic volatility model

We investigate the pricing of financial options under the 2-hypergeometric stochastic volatility model. This is an analytically tractable model that reproduces the volatility smile and skew effects observed in empirical market data. Using a regular perturbation method from asymptotic analysis of partial differential equations, we derive an explicit and easily computable approximate formula for the pricing of barrier options under the 2-hypergeometric stochastic volatility model. The asymptotic convergence of the method is proved under appropriate regularity conditions, and a multi-stage method for improving the quality of the approximation is discussed. Numerical examples are also provided. • We investigate the pricing of barrier options under stochastic volatility. • Using an asymptotic expansion method, an explicit pricing formula is derived. • The convergence of the asymptotic solution is proved. • The evaluation of the asymptotic prices is fast and suitable for practical uses. • Our numerical examples demonstrate that the approach leads to a small error.

ULTRA-FAST PRICING BARRIER OPTIONS AND CDSs

We construct a new approximate method for pricing barrier options and CDSs. In many cases, prices of barrier options and CDS of maturities [Formula: see text] years, at the log-distance 0.1 from the barrier and farther, for eight spots, can be calculated adding up 4–16 fairly simple terms, with relative errors of order [Formula: see text] and smaller, in 4–12[Formula: see text]msc.

The Heston Stochastic Volatility Model with Piecewise Constant Parameters - Efficient Calibration and Pricing of Window Barrier Options

We present a simple and numerically efficient approach to the calibration of the Heston stochastic volatility model with piecewise constant parameters. Extending the original ansatz for the characteristic function, proposed in the seminal paper by Heston, to the case of piecewise constant parameters, we show that the resulting set of ordinary differential equations can still be integrated semi-analytically. Our numerical scheme is based on the calculation of the characteristic function using Gauss-Kronrod quadrature, additionally supplying a Black-Scholes control variate to stabilize the numerical integrals. We apply our method to the problem of calibration of the Heston model with piecewise constant parameters to the foreign exchange (FX) options market. Finally, we demonstrate cases in which window barrier option prices calculated using the Heston model with piecewise constant parameters are consistent with the market, while those calculated with a plain Heston model are not.

Robust barrier option pricing by frame projection under exponential Lévy dynamics

ABSTRACTWe present an efficient method for robustly pricing discretely monitored barrier and occupation time derivatives under exponential Lévy models. This includes ordinary barrier options, as well as (resetting) Parisian options, delayed barrier options (also known as cumulative Parisian or Parasian options), fader options and step options (soft-barriers), all with single and double barriers, which have yet to be priced with more general Lévy processes, including KoBoL (CGMY), Merton’s jump diffusion and NIG. The method’s efficiency is derived in part from the use of frame-projected transition densities, which transform the problem into the Fourier domain and accelerate the convergence of intermediate expectations. Moreover, these expectations are approximated by Toeplitz matrix-vector multiplications, resulting in a fast implementation. We devise an augmentation approach that contributes to the method’s robustness, adding protection against mis-specifying a proper truncation support of the transition density. Theoretical convergence is verified by a series of numerical experiments which demonstrate the method’s efficiency and accuracy.

An analytical approximation for single barrier options under stochastic volatility models

The aim of this paper is to derive an approximation formula for a single barrier option under local volatility models, stochastic volatility models, and their hybrids, which are widely used in practice. The basic idea of our approximation is to mimic a target underlying asset process by a polynomial of the Wiener process. We then translate the problem of solving first hit probability of the asset process into that of a Wiener process whose distribution of passage time is known. Finally, utilizing the Girsanov’s theorem and the reflection principle, we show that single barrier option prices can be approximated in a closed-form. Furthermore, ample numerical examples will show the accuracy of our approximation is high enough for practical applications.

Pricing Continuously Monitored Barrier Options under the SABR Model: A Closed-Form Approximation

The stochastic alpha beta rho (SABR) model introduced by Hagan et al. (2002) is widely used in both fixed income and the foreign exchange (FX) markets. Continuously monitored barrier option contracts are among the most popular derivative contracts in the FX markets. In this paper, we develop closed-form formulas to approximate various types of barrier option prices (down-and-out/in, up-and-out/in) under the SABR model. We first derive an approximate formula for the survival density. The barrier option price is the one-dimensional integral of its payoff function and the survival density, which can be easily implemented and quickly evaluated. The approximation error of the survival density is also analyzed. To the best of our knowledge, it is the first time that analytical (approximate) formulas for the survival density and the barrier option prices for the SABR model are derived. Numerical experiments demonstrate the validity and efficiency of these formulas.

Efficient Simulation for Pricing Barrier Options with Two-Factor Stochastic Volatility and Stochastic Interest Rate

This paper presents an extension of the double Heston stochastic volatility model by combining Hull-White stochastic interest rates. By the change of numeraire and quadratic exponential scheme, this paper develops a new simulation scheme for the extended model. By combining control variates and antithetic variates, this paper provides an efficient Monte Carlo simulation algorithm for pricing barrier options. Based on the differential evolution algorithm the extended model is calibrated to S&P 500 index options to obtain the model parameter values. Numerical results show that the proposed simulation scheme outperforms the Euler scheme, the proposed simulation algorithm is efficient for pricing barrier options, and the extended model is flexible to fit the implied volatility surface.

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.

A Wiener-Hopf factorization approach for pricing barrier options in the Heston model

A Wiener-Hopf factorization approach for pricing barrier options in the Heston model

이항분포모형에 일반화된 카탈란 수를 적용한 배리어 옵션의 가격 산정

이항분포모형에 일반화된 카탈란 수를 적용한 배리어 옵션의 가격 산정

A GENERAL ORNSTEIN–UHLENBECK STOCHASTIC VOLATILITY MODEL WITH LÉVY JUMPS

We present a general class of stochastic volatility models with jumps where the stochastic variance process follows a Lévy-driven Ornstein–Uhlenbeck (OU) process and the jumps in the log-price process follow a Lévy process. This financial market model is a true extension of the Barndorff-Nielsen–Shephard (BNS) model class and can establish a weak link between log-price jumps and volatility jumps. Furthermore, we investigate the weak-link [Formula: see text]-OU-BNS model as a special case, where we calculate the characteristic function of the logarithmic price in closed form. The classical [Formula: see text]-OU-BNS model can be obtained as a limit of weak-link [Formula: see text]-OU-BNS models in the Skorokhod topology. We highlight that the weak-link property may be a useful model extension in the case of pricing barrier options.

Valuation of barrier options using sequential Monte Carlo

Sequential Monte Carlo (SMC) methods have successfully been used in many applications in engineering, statistics and physics. However, these are seldom used in financial option pricing literature and practice. This paper presents SMC method for pricing barrier options with continuous and discrete monitoring of the barrier condition. Under the SMC method, simulated asset values rejected due to barrier condition are re-sampled from asset samples that do not breach the barrier condition improving the efficiency of the option price estimator; while under the standard Monte Carlo many simulated asset paths can be rejected by the barrier condition making it harder to estimate option price accurately. We compare SMC with the standard Monte Carlo method and demonstrate that the extra effort to implement SMC when compared with the standard Monte Carlo is very little while improvement in price estimate can be significant. Both methods result in unbiased estimators for the price converging to the true value as $1/\sqrt{M}$, where $M$ is the number of simulations (asset paths). However, the variance of SMC estimator is smaller and does not grow with the number of time steps when compared to the standard Monte Carlo. In this paper we demonstrate that SMC can successfully be used for pricing barrier options. SMC can also be used for pricing other exotic options and also for cases with many underlying assets and additional stochastic factors such as stochastic volatility; we provide general formulas and references.

PRICING EXTERNAL-CHAINED BARRIER OPTIONS WITH EXPONENTIAL BARRIERS

External barrier options are two-asset options with stochastic variables where the payoff depends on one underlying asset and the barrier depends on another state variable. The barrier state variable determines whether the option is knocked in or out when the value of the variable is above or below some prescribed barrier level. This paper derives the explicit analytic solution of the chained option with an external single or double barrier by utilizing the probabilistic methods - the reflection principle and the change of measure. Before we do this, we examine the closed-form solution of the external barrier option with a single or double-curved barrier using the methods of image and double Mellin transforms. The exact solution of the external barrier option price enables us to obtain the pricing formula of the chained option with the external barrier more easily.

Efficiently pricing continuously monitored barrier options under stochastic volatility model with jumps

ABSTRACTStochastic volatility model with jumps fits almost perfectly the empirical implied volatility surface. Under this model, this paper considers continuously monitored barrier options pricing by Monte Carlo simulation. Based on quadratic exponential scheme, this paper develops an algorithm for pricing barrier options and provides convergence of the algorithm by moment-matching techniques. Variance reduction technique based on control variates further improves the efficiency of the algorithm. The algorithm is also extended to stochastic volatility model with contemporaneous jumps in variance and stock price. Simulations show that the proposed algorithm is efficient and easy to implement. Compared to contemporaneous jumps in variance and stock price, only jumps in stock price produce more profound impact on barrier options prices.

Practical Applications of Forced Liquidations, Fire Sales, and the Cost of Illiquidity

Many investors do not understand the true risk—or cost—of illiquidity until a forced liquidation or fire sale occurs. Then it’s too late. Most investors fail to account for forced liquidations and fire sales when estimating their risk-adjusted returns. In Forced Liquidations, Fire Sales, and the Cost of Illiquidity, Richard Lindsey and Andrew Weisman of Janus Capital describe a new barrier option pricing method to adjust returns for the probability that such events will occur. In this interview with Institutional Investor Journals, Lindsey and Weisman discuss how this option pricing method gives investors a practical way to calculate an illiquid investment’s reported returns, looking beyond serial correlations of returns to enable investors to estimate the probable cost of illiquidity in advance. TOPICS:Exchanges/markets/clearinghouses, VAR and use of alternative risk measures of trading risk, real assets/alternative investments/private equity

Digital barrier options pricing: an improved Monte Carlo algorithm

A new Monte Carlo method is presented to compute the prices of digital barrier options on stocks. The main idea of the new approach is to use an exceedance probability and uniformly distributed random numbers in order to efficiently estimate the first hitting time of barriers. It is numerically shown that the answer of this method is closer to the exact value and the first hitting time error of the modified Monte Carlo method decreases much faster than of the standard Monte Carlo methods.

A very efficient approach for pricing barrier options on an underlying described by the mixed fractional Brownian motion

We deal with the problem of pricing barrier options on an underlying described by the mixed fractional Brownian model. To this aim, we consider the initial-boundary value partial differential problem that yields the option price and we derive an integral representation of it in which the integrand functions must be obtained solving Volterra equations of the first kind. In addition, we develop an ad-hoc numerical procedure to solve the integral equations obtained. Numerical simulations reveal that the proposed method is extremely accurate and fast, and performs significantly better than the finite difference method.