_{1}

^{*}

A window reset option is a kind of reset options with continuous reset constraints. The issue is very important for applying to employee stock options in finance or reservation options on truck-only toll lanes in traffic management. Our contribution of this study is that we proposed an accurate and simple method to price window reset options. The option price is formulated as the solution of a boundary value problem of the Black-Scholes PDE. The problem is then transformed into an initial-boundary value problem of the heat equation. Then Green’s function is applied to solve the heat equation problem. Finally, the option price is calculated numerically. A numerical example and some discussions are presented in this paper.

A reset option is a path-dependent derivative whose payoff depends on the historical prices of underlying assets. The strike price can be adjusted according to pre-specified conditions during the specified discrete monitoring dates or a continuous period. The main function of the reset clause is to protect investors from downside risk that affects the underlying prices during bear or uncertain financial markets.

A window reset option is a reset option with a continuous reset period which is called the reset window. The strike price of a window reset option is reset to a prespecified level provided the underlying asset price being below the reset threshold during the pre-specified window period. This study presents a simple and effective method for pricing window reset options under BlackScholes economy assumptions.

Derivatives with reset clauses are extremely popular. For example, the Chicago Board Options Exchange (CBOE) and the New York Stock Exchange (NYSE) traded the S&P 500 index put warrants with a threemonth reset period in 1996. In June 1997, Morgan Stanley issued a warrant containing a reset clause on the Hong Kong stock market [

Since Black and Scholes [

In recent years, it became more and more popular for companies to issue employee stock options with reset constraints on a lock-up period. When the employee stock option is embedded with a continuous reset period, it is called the window reset option. However, there are few studies about pricing continuous reset options since there is no closed-form exact solution for the valuation of window reset options. Liao and Wang [

Our contribution of this study is that we proposed an accurate and simple method to formulate window reset option as the solution to a boundary value problem of the Black-Scholes PDE. The problem was simplified into an initial-boundary value problem of a heat equation. The solution could be represented explicitly via a boundary integral representation [

The rest of this paper is organized as follows. The window reset option is detailed in Section 2. In Section 3, we formulate the mathematical model, while in Section 4, we transform it into a boundary value problem of the heat equation. The boundary integral representation of the solution is then obtained. Next, in Section 5 we present a numerical example. Finally, in the last section there are conclusions and discussions.

A window reset option is a reset option with a partial continuous reset window. The strike price of a window reset option is reset to a new level if the price of underlying asset is below the threshold at a pre-specific period. This period is called the reset window. The path of stock value will determine the strike price of the option. In this article, the period, the strike price and the reset level are denoted by, E and respectively. The reset price is assumed to be the same as the reset level. Therefore, at expiration T, the option price C(s,T) is

where

In this section, we establish the mathematical model for a window reset option. Under the Black-Scholes [

where is the stock’s volatility, r_{f} is the risk-free interest rate, and t is the time.

At the end of the reset window, the strike price is determined. Thus, by the Black-Scholes formula, the price has to be

where N(.) is the standard normal cumulative distribution function,

,.

Let C_{1}(s,t) denote the option price with strike E and C_{2}(s,t) with strike. Then the formulae for C_{1}(s,t) and C_{2}(s,t) are as follows:

where,

.

where,

.

During the reset window, if the stock price has touched the reset level, the strike price is reset to and never changes until expiration. In this situation, the option price can be expressed by the Black-Scholes formula, i.e. C(s,t) = C_{2}(s,t). Thus, we only need to evaluate the option price C(s,t) before the stock price touches the reset level during the window. The option price C(s,t) fulfills the Black-Sholes Equation (2). When the stock price touches the reset level, the price C(s,t) = C_{2}(s,t) . If the stock prices do not touch the reset level through the window, at the end of the window, the price has to be C_{1}(s,t). Therefore

and

In the domain of and. Equation (2) is the PDE, Formula (6) is the boundary condition and Formula (7) is the initial condition.

In this section, we employ a variables transformation to simplify the boundary value problem established in the last section. After the variables transformation as in Equation (8), the Black -Scholes PDE will be simplified into the homogeneous heat equation as Equation (9), and the time variables, t_{0}, t_{1}, and T will be transformed into time to maturity as, , and 0.

Let

The Black-Scholes PDE becomes

The boundary condition becomes:

where,.

The initial condition becomes

, for

.

The transformed boundary value problem is an initialboundary value problem of the heat equation. The solution to the problem can be represented by Shen and Wang [

Let,

and

.

where,.

The function G(.) is a fundamental solution of the heat equation. The formula (10) is called the integral representation of the heat equation. There exists a fundamental solution G such that G is banished on the boundary, i.e.. The fundamental solution G is called Green’s function with boundary.

where b_{1} is the transformed reset threshold at,

,.

is the Heaviside step function and is defined in Equation (12).

The partial derivative of the Green function about the variable x is as follows.

When we use the characteristics of the Green function at the boundary condition, we can further simplify the integral representation (10) to equation (14). Therefore, we substitute Green’s function with boundary into the integral representation (10), and the integral representation is as follows.

where, , and

. The solution u is expressed explicitly in formula (14).

The first term of Equation (14) is the contribution of initial data, and it can be calculated easily via Simpson’s rule. The second term of Equation (14) is the contribution of boundary data, and we use the Gaussian quadrature^{1}, which is a simile accurate method to approach the solution of the integration, to calculate it as follows:

(15)

where, , ,.

Thus, we obtain the transformed valuation of window reset option,. The option price is

where and.

Considering a window reset option, the original strike price E = 100, the reset rate, the risk-free rate r_{f} = 0.02. Option life begins at time t_{0}. The reset period is the interval [t_{0}, t_{1}], t_{1} = t_{0} + 0.25 year. The time to maturity of the contract is T = t_{0} + 1 year. The accuracy, calculation time and numerical orders of option price with respect to the volatilities 0.25, 0.5 and 0.75 are showed in

On the other hand, we compare the valuation of window reset option with different volatility to vanilla option in Figures 1-3. The value of a window reset option increases with the current stock price.

Note: The current stock price S_{0} = 100, the original strike price E = 100; the reset rate α = 90%; the risk-free rate r_{f} = 0.02; option life begins at time t_{0}; the reset window is from t_{0} to t_{1} = t_{0} + 0.25; time to maturity T = t_{0} + 1.

Note: The current stock price S_{0} = 100, the original strike price E = 100; the reset rate α = 90%; the risk-free rate r_{f} = 0.02; option life begins at time t_{0}; the reset window is from t_{0} to t_{1} = t_{0} + 0.25; time to maturity T = t_{0} + 1.

Note: the current stock price S_{0} = 100, the original strike price E = 100; the reset rate α = 90%; the risk-free rate r_{f} = 0.02; option life begins at time t_{0}; the reset window is from t_{0} to t_{1} = t_{0} + 0.25; time to maturity T = t_{0} + 1.

Emerging derivatives are more and more popular in recent years. In this paper, we focus on the valuation of a window reset option which is a kind of reset options with continuous reset constraints. The issue is very important for applying to employee stock options in finance or reservation options on truck-only toll lanes in traffic management. Our contribution of this study is that we proposed an accurate and simple method to price window reset options. We use the efficient integral representation of PDE to present the valuation of window reset option and use the numerical method, Simpson’s rule and Gauss integral method, to calculate it.

Our pricing method is useful to obtain accurate numerical results fast. The difference between a vanilla option and a window reset option is significant when the stock volatility is at a low level and the current stock price is close to the reset threshold. In our numerical example, when the current stock price is about more than twofolds standard deviations, the price of window reset option is close to the value of vanilla option. The numerical result is related to the reset period and the volatility of stock price, since a short reset period or a low level of stock volatility will decrease the effect of the reset constraints. Therefore, the price difference volatilizes when the stock volatility is at a low level as shown in