Double Barrier Options Research Articles

Multiperiod conditional valuation of barrier options with incomplete information

In this paper, we provide analytical valuation results for barrier options with three different types of information: continuous observations of the underlying asset value, delayed continuous observations and multiple discrete observations. Market, counterparty credit risk management and economic risk capital implications of the valuations with incomplete information are also discussed. We obtain precise analytical solutions using all the information generated by the previous discrete observations of the underlying asset, generated in a consistent fashion by path-dependent simulation. Our results have far-reaching implications for economic capital charges on path-dependent derivatives. We demonstrate that using non-conditional valuation mis-estimates the credit charges on the barrier options with respect to the precise credit charges computed with valuations conditional on the actual available information. The rest of this paper is organized as follows. Section 2 describes the classical case with full, continuous information about the underlying asset. Section 3 deals with the case of delayed continuous information. Section 4 investigates the case of discrete observations at multiple discrete times. Section 5 offers explicit closed-form solutions for the main types of single-barrier options with discrete information and parity results. Section 6 provides valuation results for double-barrier options. Section 7 presents risk management applications for foreign exchange (FX) barrier options, while section 8 concludes our paper.

A Complete Analytical Solution of the Asian Option Pricing within the Heston Model for Stochastic Volatility: A Probability Density Function Approach

The first ever explicit formulation of the concept of the option’s probability density functions has been introduced in our publications “Breakthrough in Understanding Derivatives and Option Based Hedging - Marginal and Joint Probability Density Functions of Vanilla Options - True Value-at-Risk and Option Based Hedging Strategies” and “Complete Analytical Solution of the Asian Option Pricing and Asian Option Value-at-Risk Problems. A Probability Density Function Approach.” See links: http://ssrn.com/abstract=2489601 and http://ssrn.com/abstract=2546430.The first ever explicit formulation of the concept of the options’ probability density functions within the framework of stochastic volatility (Heston model) has been introduced in our publications “Complete Analytical Solution of the Heston Model for Option Pricing and Value-at-Risk Problems: A Probability Density Function Approach”, “Complete Analytical Solution of the American Style Option Pricing with Constant and Stochastic Volatilities: A Probability Density Function Approach” and “A Complete Analytical Resolution of the Double Barrier Option’s Pricing Within the Heston Model. A Probability Density Approach.” See links:http://ssrn.com/abstract=2549033 and http://ssrn.com/abstract=2554038 and http://ssrn.com/abstract=2605948.In this paper we report complete analytical closed-form results for the European style Asian Options considered within the Heston model for Stochastic Volatility (SV). Our discovery of the probability density function of the European style Asian Options with SV enables exact closed-form representation of its expected value (price) for the first time ever. Our formulation of the probability density function for the European style Asian Options with SV is expressive enough to enable derivation for the first time ever of corollary analytical closed-form results for such Value-At-Risk characteristics as the probabilities that an Asian Option with SV will be below or above any threshold at any future time before or at termination. Such assessments are absolutely out of reach of the current published methods for treating Asian Options even in the framework of constant volatility.All numerical evaluations based on our analytical results are practically instantaneous and absolutely accurate.

A Complete Analytical Resolution of the Double Barrier Optionss Pricing within the Heston Model. A Probability Density Function Approach

• The first ever explicit formulation of the concept of an option’s probability density functions has been introduced in our publications “Breakthrough in Understanding Derivatives and Option Based Hedging - Marginal and Joint Probability Density Functions of Vanilla Options - True Value-at-Risk and Option Based Hedging Strategies” and “Complete Analytical Solution of the Asian Option Pricing and Asian Option Value-at-Risk Problems. A Probability Density Function Approach” (see links http://ssrn.com/abstract=2489601 and http://ssrn.com/abstract=2546430).• The first ever explicit formulation of the concept of an options’ probability density functions within the framework of stochastic volatility (Heston model) has been introduced in our publications “Complete Analytical Solution of the Heston Model for Option Pricing and Value-at-Risk Problems: A Probability Density Function Approach” and “Complete Analytical Solution of the American Style Option Pricing with Constant and Stochastic Volatilities: A Probability Density Function Approach” (see links http://ssrn.com/abstract=2549033 and http://ssrn.com/abstract=2554038).The probability density approach has allowed complete analytical resolution of not only all the pricing problems but for the first time complete analytical resolution of all the associated Value-At-Risk (VAR) problems by specifying probabilities of options, as stochastic quantities, to be below (above) of any threshold. • In this paper we report analogous results for pricing Double Barrier options in the presence of stochastic volatility (Heston model), even enabling complete analytical resolution of all problems associated with these options. • Our discovery of the analytical closed-form for the probability density function for the Double Barrier options with stochastic volatility enables exact results for pricing of these options for the first time without depending on approximate numerical methods. • Our formulation of the density function for the Double Barrier options with stochastic volatility within the Heston model is expressive enough to enable derivation for the first time ever of corollary closed-form analytical results for such Value-At-Risk characteristics as the probabilities that options with stochastic volatility will be below or above any set of thresholds at termination. Such assessments are absolutely out of reach of the current published methods for treating options within or outside the Heston model.• All numerical evaluations based on our analytical results are practically instantaneous and absolutely accurate.

Analytically pricing double barrier options based on a time-fractional Black–Scholes equation

This paper investigates the pricing of double barrier options when the price change of the underlying is considered as a fractal transmission system. In this scenario, the option price is governed by a modified Black–Scholes equation with a time-fractional derivative. In comparison with standard derivatives of integer order, the fractional-order derivatives are characterized by their “globalness”, i.e., the rate of change of a function near a point is affected by the property of the function defined in the entire domain of definition rather than just near the point itself. The existence of the time-fractional derivative, in conjunction with the presence of two barriers of the double-barrier options, has added an additional degree of difficulty not only when a purely numerical solution is sought but also when an analytical method is attempted. Albeit difficult, we have managed to find an explicit closed-form analytical solution for double-barrier options, which has been taken to price the single barrier options and European path-independent options under the same framework as a special case of the current solution. In addition, not only have we provided a theoretical proof for the convergence of the newly-found analytic solution in series form, which is a vital step to show the closeness of our solution, we have also proposed an efficient numerical evaluation technique to facilitate the implementation of our formula so that it can be easily used in trading practice.

Pricing discrete double barrier options with a numerical method

Most of traded double barrier options are monitored in discrete time, their pricing being more challenging than in continuous time. A few solutions are analytical with a correction for continuity; the remaining solutions are numerical with lattices, grids or Monte Carlo (MC) simulation. We apply an original variance reduction technique to the pricing of European double barrier options monitored in discrete time. This technique speeds up significantly the MC simulation. The computational algorithm repeats each experiment with an increasing number of trials at a logarithmic rate and calculates a weighted average of the options values. We test the accuracy of the Login (Logarithmic Increment) model with an analytical solution adjusted for discretization and a naive numerical model using Monte Carlo simulation. We show that the Login model accurately prices double barrier options. Market participants in need of selecting a reliable numerical method for pricing double barrier options monitored in discrete time will find our article appealing. Moreover, the idea behind the method is simple and can be applied to the pricing of plain-vanilla or more complex derivatives, easing and speeding the valuation step significantly.

Implementation of the modified Monte Carlo simulation for evaluate the barrier option prices

In this paper, we apply an improved version of Monte Carlo methods to pricing barrier options. This kind of options may match with risk hedging needs more closely than standard options. Barrier options behave like a plain vanilla option with one exception. A zero payoff may occur before expiry, if the option ceases to exist; accordingly, barrier options are cheaper than similar standard vanilla options. We apply a new Monte Carlo method to compute the prices of single and double barrier options written on stocks. The basic idea of the new method is to use uniformly distributed random numbers and an exit probability in order to perform a robust estimation of the first time the stock price hits the barrier. Using uniformly distributed random numbers decreases the estimation of first hitting time error in comparison with standard Monte Carlo or similar methods. It is numerically shown that the answer of our method is closer to the exact value and the first hitting time error is reduced.

Pricing external barrier options in a regime-switching model

External barrier options are two-asset options where the payoff is defined on one asset and the barrier is defined on another asset. In this paper, we derive the Laplace transforms of the prices and deltas for the external single and double barrier options where the underlying asset prices follow a regime-switching model with finite regimes. The derivation is made possible because we can obtain the joint Laplace transform of the first passage time of one asset value and the value of the other asset. Numerical inversion of the Laplace transforms is used to calculate the prices of external barrier options.

Optimal limit methods for computing sensitivities of discontinuous integrals including triggerable derivative securities

We introduce an approach to computing sensitivities of discontinuous integrals. The methodology is generic in that it only requires knowledge of the simulation scheme and the location of the integrand’s singularities. The methodology is proven to be optimal in terms of minimizing the variance of the measure changes. For piecewise constant payoffs this minimizes the variance of Monte Carlo sensitivities. An efficient adjoint implementation is discussed, and the method is shown to be effective for a number of natural financial examples including double barrier options and triggerable interest rate derivative securities.

A Hybrid Finite Difference Method for Pricing Two-Asset Double Barrier Options

The pricing of the two-asset double barrier option is modeled as an initial-boundary value problem of the two-dimensional Black-Scholes partial differential equation. We use the hybrid finite different method to solve the problem. The hybrid method is a combination of the Laplace transform and a finite difference method. It is more efficient than a traditional finite difference method to obtain a solution without a step-by-step process. The method is implemented on a computer. Two numerical examples are calculated to verify the performance of the hybrid method. In our numerical examples, the convergence rate of the method is approximately two. We conclude that the method is efficient for pricing two-asset barrier options.

Pricing and static hedging of European-style double barrier options under the jump to default extended CEV model

This paper develops two novel methodologies for pricing and hedging European-style barrier option contracts under the jump to default extended constant elasticity of variance (JDCEV) model, namely: a stopping time approach based on the first passage time densities of the underlying asset price process through the barrier levels; and a static hedging portfolio approach in which the barrier option is replicated by a portfolio of plain-vanilla and binary options. In doing so, both valuation methodologies are extended to a more general set-up accommodating endogenous bankruptcy, time-dependent barriers and the commonly observed stylized facts of a positive link between default and equity volatility and of a negative link between volatility and stock price. The two proposed numerical methods are shown to be accurate, easy to implement and efficient under both the JDCEV model and the nested constant elasticity of variance model.

A numerical method to price discrete double Barrier options under a constant elasticity of variance model with jump diffusion

We develop a numerical method to price discrete barrier options on an underlying described by the constant elasticity of variance model with jump-diffusion (CEVJD). In particular, the partial integro differential equation associated to this model is discretized in time using an operator splitting scheme whose accuracy is enhanced by repeated Richardson extrapolation. Such an approach allows us to approximate the differential terms and the jump integral by means of two different numerical techniques. Precisely, the spatial derivatives, which exist only in the weak sense, are discretized using a finite element method based on piecewise quadratic polynomials, whereas the jump integral is directly collocated at the mesh points, so that it can be easily evaluated by Simpson numerical quadrature. As shown by extensive numerical simulation, the proposed approach is very efficient from the computational standpoint, and performs significantly better than the finite difference scheme developed in Wade et al. [On smoothing of the Crank–Nicolson scheme and higher order schemes for pricing barrier options, J. Comput. Appl. Math. 204 (2007), pp. 144–158].

Credit spreads with dynamic debt

This paper extends the baseline Merton (1974) structural default model, which is intended for static debt spreads, to a setting with dynamic debt, where leverage can be ratcheted up as well as written down through pre-specified exogenous policies. We provide a different and novel solution approach to dynamic debt than in the extant literature. For many dynamic debt covenants, ex-ante credit spread term structures can be derived in closed-form using modified barrier option mathematics, whereby debt spreads can be expressed using combinations of single barrier options (both knock-in and knock-out), double barrier options, double-touch barrier options, in-out barrier options, and one-touch double barrier binary options. We observe that debt principal swap down covenants decrease the magnitude of credit spreads but increase the slope of the credit curve, transforming downward sloping curves into upward sloping ones. On the other hand, ratchet covenants increase the magnitude of ex-ante spreads without dramatically altering the slope of the credit curve. These covenants may be optimized by appropriately setting restructuring boundaries, which entails a trade-off between the reduction in spreads against restructuring costs. Overall, explicitly modeling this latent option to alter debt leads to term structures of credit spreads that are more consistent with observed empirics.

THE BINOMIAL INTERPOLATED LATTICE METHOD FOR STEP DOUBLE BARRIER OPTIONS

We consider the problem of pricing step double barrier options with binomial lattice methods. We introduce an algorithm, based on interpolation techniques, that is robust and efficient, that treats the "near barrier" problem for double barrier options and permits the valuation of step double barrier options with American features. We provide a complete convergence analysis of the proposed lattice algorithm in the European case.

Probability of multiple crossings and pricing of double barrier options

This paper derives pricing formulas of standard double barrier option, generalized window double barrier option and chained option. Our method is based on probabilitic approach. We derive the probability of multiple crossings of curved barriers for Brownian motion with drift, by repeatedly applying the Girsanov theorem and the reflection principle. The price of a standard double barrier option is presented as an infinite sum that converges very rapidly. Although the price formula of standard double barrier option is the same with Kunitomo and Ikeda (1992), our method gives an intuitive interpretation for each term in the infinite series. From the intuitive interpretation we present the way how to approximate the infinite sum in the pricing formula and an error bound for the given approximation. Guillaume (2003) and Jun and Ku (2013) assumed that barriers are constant to price barrier options. We extend constant barriers of window double barrier option and chained option to curved barriers. By employing multiple crossing probabilities and previous skills we derive closed formula for prices of 16 types of the generalized chained option. Based on our analytic formulas we compute Greeks of chained options directly.

An integro-differential parabolic variational inequality arising from the valuation of double barrier American option

This paper studies the nonlinear variational inequality with integro-differential term arising from valuation of American style double barrier option. First, the authors use the penalty method to transform the variational inequality into a nonlinear parabolic initial boundary problem (i.e., penalty problem). Second, the existence and uniqueness of solution to the penalty problem are proved by using the Scheafer fixed point theory. Third, the authors prove the existence of variational inequality’ solution by showing the fact that the penalized PDE converges to the variational inequality. The uniqueness of solution to the variational inequality is also proved by contradiction.

Closed-Form Pricing of Two-Asset Barrier Options with Stochastic Covariance

Single and double barrier options on more than one underlying with stochastic volatility are usually priced via Monte Carlo simulation due to the non-existence of closed-form solutions for their value. In this paper, for a special dependence structure, the prices of some two-asset barrier derivatives, like double-digital options and correlation options can be derived analytically using generalized Fourier transforms and some conditions on the characteristic functions. We study the influence of the various parameters on these prices and show that these formulas can be easily and quickly computed. We also extend our approach to further allow for a random correlation structure.

DIGITAL DOUBLE BARRIER OPTIONS: SEVERAL BARRIER PERIODS AND STRUCTURE FLOORS

We determine the price of digital double barrier options with an arbitrary number of barrier periods in the Black–Scholes model. This means that the barriers are active during some time intervals, but are switched off in between. As an application, we calculate the value of a structure floor for structured notes whose individual coupons are digital double barrier options. This value can also be approximated by the price of a corridor put.

Pricing discrete barrier options and credit default swaps under Lévy processes

We consider discretely monitored barrier options under Lévy models, including single and double barrier options and first-touch digitals, as well as CDS and defaultable bonds. At each step of backward induction, we use piece-wise polynomial interpolation and an efficient version of the Fourier transform technique, which allows for efficient error control. We derive accurate recommendations for the choice of parameters of the numerical scheme, and produce numerical examples showing that oversimplified prescriptions in other methods can result in large errors.

Credit Spreads with Dynamic Debt

This paper extends the baseline Merton (1974) structural default model, which is intended for static debt spreads, to a setting with dynamic debt, where leverage can be ratcheted up as well as written down through pre-specified exogenous policies. We provide a different and novel solution approach to dynamic debt than in the extant literature. For many dynamic debt covenants, ex-ante credit spread term structures can be derived in closed-form using modified barrier option mathematics, whereby debt spreads can be expressed using combinations of single barrier options (both knock-in and knock-out), double barrier options, double-touch barrier options, in-out barrier options, and one-touch double barrier binary options. We observe that debt principal swap down covenants decrease the magnitude of credit spreads but increase the slope of the credit curve, transforming downward sloping curves into upward sloping ones. On the other hand, ratchet covenants increase the magnitude of ex-ante spreads without dramatically altering the slope of the credit curve. These covenants may be optimized by appropriately setting restructuring boundaries, which entails a trade-off between the reduction in spreads against restructuring costs. Overall, explicitly modeling this latent option to alter debt leads to term structures of credit spreads that are more consistent with observed empirics.

Double-barrier first-passage times of jump-diffusion processes

Abstract. Required in a wide range of applications in, e.g., finance, engineering, and physics, first-passage time problems have attracted considerable interest over the past decades. Since analytical solutions often do not exist, one strand of research focuses on fast and accurate numerical techniques. In this paper, we present an efficient and unbiased Monte-Carlo simulation to obtain double-barrier first-passage time probabilities of a jump-diffusion process with arbitrary jump size distribution; extending single-barrier results by [Journal of Derivatives 10 (2002), 43–54]. In mathematical finance, the double-barrier first-passage time is required to price exotic derivatives, for example corridor bonus certificates, (step) double barrier options, or digital first-touch options, that depend on whether or not the underlying asset price exceeds certain threshold levels. Furthermore, it is relevant in structural credit risk models if one considers two exit events, e.g., default and early repayment.