_{1}

^{*}

In this study we propose an approach to solve a partial differential equation (PDE), the boundary integral method, for the valuation of both discrete and continuous window barrier options, as well as multi-window barrier options within a deterministic term structure of volatility and interest rates. Numerical results reveal that the proposed method yields rapid and highly accurate closed-form approximate solutions. In addition, the term structure will have a significant impact on the valuation.

Barrier options are widely used and traded in financial markets to manage the risk related to foreign exchange, interest rates, and equity in the global market. A barrier option is a path-dependent option that is activated (i.e. knocked in), or extinguished (i.e. knocked out) when the price of the underlying asset breaches the pre-specified barrier level at any time before maturity. There are two main reasons for the prevalence of barrier options. First, barrier options are useful for limiting the risk exposure of clients, specifically in the foreign exchange market. Second, barrier options are cheaper than vanilla options with similar attributes. Therefore, they are more affordable financial instruments that can match investors’ views regarding the degree of volatility in the underlying asset price change. Additionally, they offer an appropriate level of downside protection for hedging purposes. For example, if long hedgers believe that only a mild volatility exists in the market over a given period, the hedgers will prefer to buy a knock-out option with fewer premiums than to pay the full premium for a plain vanilla option. Alternatively, when speculators believe that the price change will be volatile for a period of time, they will prefer to purchase a knock-in option rather than to obtain the plain option with a higher price. Since an ordinary option can be decomposed into two otherwise identical knock-in and knock-out options, this feature makes the barrier option a highly suitable instrument for tailor-made structured deals.

Partial barrier options are the extension of barrier options, however there is a major difference between the two. Partial barrier options assume that the barrier prevails only for some fraction of the option’s lifetime, while barrier options prevail through the entire life of the option. Heynan and Kat [

A window barrier option incorporates a barrier which commences at an arbitrary date after the starting date and terminates at an arbitrary date before the expiration. Window barrier options are more flexible than standard barrier options because the adaptable monitoring period provides traders with the full flexibility to lock volatility risks during a specific time period. Window barrier options not only offer investors a hedge instrument maneuvered by investors’ views on the range of volatility, but these options also provide building blocks to create various types of partial exotic options embedded in structured products. In recent years they have become more popular with investors, particularly in foreign exchange markets. Meanwhile, academics and practitioners have turned their attention to the more complicated structures of barrier options.

Since Merton [

The binomial and trinomial lattice models developed respectively by Cox, Ross and Rubinstein [

Broadie, Glasserman, and Kou [

Heynen and Kat [

Most of the partial barrier models mentioned previously, with the exception of that of Heynen and Kat [

This paper proposes the boundary integral method (BIM) [

This paper is organized as follows: in Section 2 we present the valuation algorithm in terms of the initial value problem and boundary value problem. We also explain how to deal with discontinuities caused by the window barrier feature, and demonstrate how to recursively obtain the option price. Section 3 discusses the valuation of discrete window barrier options, and defines the pricing problem as a sequence of the initial value problem. Section 4 contains numerical results, and Section 5 concludes with a summary and some suggestions for future research.

Since a knock-in window barrier option plus a knock-out window barrier option will be equal to the value of an otherwise equivalent vanilla option, we will concentrate only on the knock-out window barrier option. Let 0, t_{1}, t_{2}, and T denote the option’s starting date, the time to the start of the barrier, the time to the end of the barrier, and the option’s maturity date, respectively, and . The valuation of the early-ending window barrier option can be calculated by letting t_{1} approach 0, and the valuation of the forward-start window barrier option can be recovered by letting t_{2} approach T.

Our objective is to apply the PDE (partial differential equation) approach to the valuation of the window barrier option in Black-Scholes economy assumptions. When the underlying is assumed to follow a lognormal random walk, under no arbitrage argument, the PDE governing the window barrier option will be as follows:

where C is the option value, S is the underlying asset price, is stock’s volatility, is the risk-free interest rate, is the dividend yield, and t is the time. To allow for the case of greatest generality, the stock’s volatility and the risk-free interest rate may change deterministically across the barrier monitoring period. We allow for three different governing PDEs, or we may say, the Black-Scholes equation with different coefficients for time intervals [0, t_{1}), [t_{1}, t_{2}], and (t_{2}, T]. As in Armstrong [

,

,

.

When the underlying asset price does not touch or breach the barrier level through the monitoring period, the payoff of the window barrier option at the maturity date is given as the following equation:

where K denotes the strike price.

Oppositely, if the underlying asset price triggers the barrier level throughout the monitoring period, the option will be knocked out and clients will receive an immediate rebate R_{b}. The payoff is as follows:

where B denotes the barrier price and R_{b} denotes the immediate rebate.

The equation (1) is the governing equation and conditions (2) and (3) are the initial condition and boundary condition for the call price C(S, t) respectively.

Let

Making the following variable transformations:

equation (1) can be simplified into the heat equation with a constant diffusion coefficient as follows:

The payoff of the window barrier option at the maturity date will be transformed into:

Since the asset price is assumed to change continuously with time, if the asset price S(τ) breaches the barrier level between time interval, it will first touch the barrier level. Therefore the payoff of the option can be transformed, as in equation (7).

where and.

On the time interval, there is not any boundary condition; hence the solution of the PDE is uniquely determined by the initial condition (6). The problem of finding the unique solution with PDE (5) and initial condition (6) is termed the initial value problem. In our notation, the integral representation of the solution will be:

where the function G is called Green’s function with the infinite domain or the fundamental solution of the heat equation. It can be expressed as follows:

where is the Heaviside step function, and is defined by:

Equation (8) has some interesting interpretations with respect to the risk-neutral approach of Cox and Rubinstein [

As in Black-Scholes [

where N (.) is the cumulative normal distribution function. and

.

However, if the transformed underlying asset price x_{2 }_{} is within the range atthe instant after it passes the monitoring date, i.e., the transformed window barrier price will still change continuously across the barrier monitoring date. If we denote as the instant after, the continuity assumption will guarantee that will be equal to. Therefore, the initial condition for equation (5) between will be as follows:

,(12)

where.

If the underlying asset price breaches the barrier level during the period, the window barrier option will be knocked out, and clients will receive an immediate rebate R_{b}. The continuity property will guarantee that the asset price first touches the barrier level before it breaches the barrier level. Thus, the boundary condition for Equation (5) between will be specified as in Equation (13):

There exists only one solution that satisfies the PDE (5) and is subject to the initial condition (12) and the boundary condition (13). The integral representation of a solution for the heat equation at is as follows:

If, and, then

where and

.

G is the green function with the boundary.

where b_{1} is the transformed barrier price at, , and

The functions G(.) and G_{x}(.) in equation (14) can be interpreted as the transition probability density function and the hitting probability density function of the transformed underlying asset price, respectively. The first term in equation (14) is the expected payoff when the barrier is never breached throughout the monitoring interval [,], and the second term is the expected payoff when the underlying asset price breaches the barrier in the time interval [,]. Once again the solution can be interpreted as the expected payoff in a risk-neutral environment.

Finally, we will discuss how to cope with parameter discontinuities and obtain the ultimate solution . Let denote the instant after; the parameter discontinuity happening between time to maturity and can be overcome with the same logic as in equation (12). That is, when approaches, will be equal to. The solution problem once again is an initial value problem analogous to the first period valuation from to. Therefore the initial condition for differential equation (5) between time interval has to be, and the integral representation of the closed-form approximate solution for is as follows:

where Green’s function is denoted by equation (17),

Finally, the theoretical value for the window barrier option, , can easily be obtained by the following inverse transformation:

where.

A key benefit of adopting the PDE approach is that it reduces the pricing problem for the window barrier option to two initial value problems and one boundary value problem, for all of which standard mathematical engineering numerical algorithms are well developed. These algorithms allow a straightforward numerical integration of highly accurate numerical values for window barrier options pricing. Since the Black-Scholes equation can be simplified into a homogenous linear equation such as the heat equation, the boundary integral method proposed in this paper will be a highly efficient algorithm to calculate window barrier options’ numerical solution.

In brief, the valuation of the standard partial barrier call can be divided into a three-phase process, as demonstrated in

, has to be continuous at. Thus the parameter discontinuity at can be handled accordingly, and the initial condition and the boundary condition imposing on the heat equation can be denoted as in equations (12) and (13). Since the Green function under such boundary constraints exists and is available, the boundary integral method can be applied to acquire the unique solution for differential equation (5) at time to maturity. In the third phase, the discontinuity problem between and can be handled in the same manner as in phase 2. Thus

will be the initial condition imposing on the differential equation between. Hence, the problem of finding will be an initial value problem again, and the integral representation of the solution can be obtained easily, as in phase 1. The transformation of by Equation (18) will ultimately obtain the numerical solution of the window barrier option. In Equations (8), (14) and (16), u(.) and G(.) are very smooth functions. Thus, a simple integration scheme such as the Simpson integral will obtain exceptionally precise closed-form approximate values for both and. Finally, a highly accurate estimation of can be obtained by recursively integrating backward through time.

The boundary integral method proposed in this paper can be easily extended to a multi-window barrier option or window ladder option. It also accommodates the pricing of early-end and late-start barrier options. The extra flexibility makes the proposed method an applicable way to calculate options with more complex features.

In practice, barrier options are frequently monitored only at specific discrete dates. The discrete feature will cause a knock-out window barrier option to be more expensive and a knock-in window barrier option to be less expensive than their respective continuous monitoring counterparts. In this section, we still assume the Black-Scholes economy, but it is not necessary to assume a flat term-structure and constant volatility. As in the pricing of a continuous window barrier option, the term-structure interest rates and volatility can be multi-partite step functions that accommodate the location of the discrete barrier. The PDE approach also allows that the discrete barrier level may change deterministically during the monitoring period, and monitoring need not necessarily take place at equal-spaced points in time. Since the multipartite term structure of interest rates and volatility can be easily handled in the standard setting, for the purpose of simplifying the notations and focusing on the main idea of the recursive integral method, we will assume constant volatility and a flat term structure in this section.

Assume the option monitoring period [t_{1}, t_{2}] is partitioned into m − 1 discrete time intervals, and the option is subject to m times of discrete monitoring. If denotes the set of discrete monitoring date, and is for the corresponding set of time to maturity. In addition, we assume that the relations between the discrete monitoring dates are, .

denotes the instant after the corresponding discrete monitoring dates.

Under these assumptions, equation (6) is still the initial condition for differential equation (5) between, and is still the unique solution that satisfies the differential equation (5) at subject to the initial condition (6). Since no monitoring is required during time interval, the mathematical problem of finding the solution for difference equation (5) at will be simplified to become an initial value problem. The solution that satisfies the partial differential equation (5) subject to initial condition (6) will be as follows:

The discrete monitoring feature will introduce m discontinuities into solutions at discrete monitoring dates M^{*}. Following the same logic as discussed in the section on pricing a continuous window barrier option, the instant after the discrete monitoring date if

, the value of the window barrier option will change continuously across the discrete monitoring date, thus will be equal to. Otherwise, it will be equal to according to the pre-specified contract rebate specification. Thus the solution can be defined as follows:

Equation (20) will be the initial condition for Equation (5) between, and the unique solution for equation (5) at is given by equation (21).

By applying the same argument, the integral representations of solutions for differential equation (5) at,_{ }, , are given by the recursive integral method as follows:

for.

for.

Finally, will be equal to as in the continuous window barrier option case in Section 2, and can be obtained by using as the initial condition, and the unique solution for equation (5) subject to initial condition will be:

and can be obtained by dividing with as specified in equation (4).

The valuation of the discrete window barrier option can be defined as a sequence of initial value problems. The proposed recursive integral method is quite analogous to the lattice models and finite difference algorithm. All approaches involve the initial value and work backward to find a solution one step back in time, but there are no intermediate time steps between two discrete monitoring dates for the integral approach. The key advantage of the PDE approach is that most of the standard techniques for solving the PDE in engineering mathematics can be applied to calculate the option pricing, and it can provide practitioners with a more flexible and applicable way to accommodate the complexities of OTC exotic options. Furthermore, when the composite Simpson’s rule is applied to estimate the numerical solution of, Wang and Hsiao [

This section provides some numerical examples, and studies the performance of the BIM algorithm for different choices of parameter settings. To assess the validity of our approach, we compare our continuous window barrier option results with numerical results presented in Armstrong [