Valuation for an American Continuous-Installment Put Option on Bond under Vasicek Interest Rate Model

The valuation for an American continuous-installment put option on zero-coupon bond is considered by Kim’s equations under a single factor model of the short-term interest rate, which follows the famous Vasicek model. In term of the price of this option, integral representations of both the optimal stopping and exercise boundaries are derived. A numerical method is used to approximate the optimal stopping and exercise boundaries by quadrature formulas. Numerical results and discussions are provided.


Introduction
Although there has been a large literature dealing with numerical methods for American options on stocks 1 and references cited therein, 2 , there are not many papers for American options on default-free bonds, see, for example, 3-7 , and so on.Numerical methods such as finite differences, binomial tree methods and Least-Square Monte Carlo simulations are still widely used.However, these methods have several shortcomings including time consuming, unbounded domain and discontinuous derivative with respect to the variate of payoff function.The most recent papers, like 8-11 provide different types of methods.
In this paper we consider an alternative form of American option in which the buyer pays a smaller up-front premium and then a constant stream of installments at a certain rate per unit time.So the buyer can choose at any time to stop making installment payments by either exercising the option or stopping the option contract.This option is called American continuous-installment CI option.Installment options are a recent financial innovation that

The Model and Preliminary Results
In the one-factor Vasicek model 14 , the short-term interest rate r t is modeled as a meanreverting Gaussian stochastic process on a probability space Ω, F, P equipped with a filtration F t t≥0 .Under the the risk-neutral probability measure Q, it satisfies the linear stochastic differential equation SDE where W t t≥0 is a standard Q-Brownian motion, κ > 0 is the speed of mean reversion, r ∞ > 0 is the long-term value of interest rate, and σ is a constant volatility.Consider a frictionless and no-arbitrage financial market which consists of a bank account A t with its price process given by dA t r t A t dt and a T 1 -maturity default-free, zerocoupon bond B t, r, T 1 B t with its no-arbitrage price at time t given by where E Q is the expectation under the risk-neutral probability measure Q. Vasicek 14 provides the explicit form of the zero-bond as follows:

2.4
From 2.3 , we are easy to obtain the following partial differential equation P.D.E.: with terminal condition B T 1 , r, T 1 1.The payoff of a European-style put option without paying any dividends written on the zero-coupon bond B t, r, T 1 with maturity T T < T 1 , and strike price K is h T, r max{K − B T, r, T 1 , 0}.The no-arbitrage price at time t 0 ≤ t ≤ T of this option is denoted by p e t, r, K; T .Following Jamshidian 15 , the price of this option can generally be expressed as follows: where N • is the 1-dimensional standard cumulative normal distribution, and

2.7
Now we consider a CI option written on the zero-coupon bond B t, r, T 1 .Denote the initial premium of this option to be V t V t, r; q , which depends on the interest rate, time t, and the continuous-installment rate q.Applying Ito's Lemma to V t , the dynamics for the initial value of this option is obtained as follows: 2.9 Proof.We now consider a self-financing trading strategy ψ ψ 1 , ψ 2 , where ψ 1 and ψ 2 represent positions in bank account and T 1 -maturity zero-coupon bonds, respectively.It is apparent that the wealth process π t satisfies where the second equality is a consequence of the assumption that the trading strategy ψ replicate the option.Furthermore, since ψ is self-financing, its wealth process π t also satisfies From 2.8 and 2.10 , we get

2.13
Setting ψ 2 ∂B t /∂r / ∂V t /∂r the coefficient of dW t vanishes.It follows from 2.5 that, V t satisfies 2.9 .

Kim Equations for the Price of American CI Put Option
Consider an American CI put option written on the zero-coupon bond B t with the same strike price K and maturity time T T < T 1 .Although the underlying asset is the bond, the independent variable is the interest rate.Similar to American continuous-installment option on stock 12 , there is an upper critical interest rate r u t above which it is optimal to stop the installment payments by exercising the option early, as well as a lower critical interest rate r l t below which it is advantageous to terminate payments by stopping the option contract.We may call r u t to be exercising boundary and r l t to be stopping boundary.Denote the initial premium of this put option at time t by P t, r; q P t , defined on the domain It is known that P t, r; q , r u t and r l t are the solution of the following free boundary problem 4 : Remark 3.1.Due to the decreasing property of the price B t, r, T 1 on the state variable r, the strike price K should be strictly less than B T, 0, T 1 .Otherwise, exercise would never be optimal.
It should be noted that although the value of the American CI put option has been expressed through the use of PDEs and their boundary conditions, there is still no explicit solution for the P.D.E. in 3.1 .Numerical methods must be applied to value the price of the American CI option on bond.In the following we will solve this problem 3.1 with the integral equation method discussed in 8-12 .This method expresses the price of the American option as the sum of the price of the corresponding European option and the early exercise gains depending on the optimal exercise boundary.Jamshidian 3 uses this method to value the American bond option in Vasicek model.Theorem 3.2.Let the short interest rate r t satisfy model 2.1 .Then the initial premium of the American CI put option , P t, r; q , can be written as P t, r; q p e t, r, K; T q T t B t, r, s N e r, r l

3.2
Moreover, the optimal stopping and exercise boundaries, r u and r l , are solutions to the following system of recursive integral equations: Proof.Let Z s, r e − s 0 r u du P s, r; q be the discounted initial premium function of the American CI put option in the domain D. It is known that the function Z s, r ∈ C

3.6
From 3.1 we know that P T, r; q h T, r and P s, r; q P s, r; q 1 r,s ∈C P s, r; q 1 r,s ∈S P s, r; q •1 r,s ∈E P s, r; q 1 r,s ∈C K−B s, r, T for every s > t.Then the state variable r s follows the normal distribution.Furthermore, using s-forward measures discussed in 16 and the normal distribution produces the representation 3.2 .The recursive equations 3.3 for the optimal stopping and exercise boundaries are obtained by imposing the boundary conditions P t, r u t ; q K−B t, r u t , T 1 and P t, r l t ; q 0. The boundary conditions 3.4 hold since the limitation for 3.3 as t ↑ T .
Remark 3.3.From 3.2 , when r l t and r u t are obtained by 3.3 , the value of American CI put option is also derived.However, 3.3 are Volterra integral equations and can be solved numerically.Notice that the stopping and exercise boundary functions, r l t and r u t , cannot be proved to be monotone function of time t.So we use trapezoidal rule method to deal with them.

Numerical Method and Results
In this section we provide our method for pricing American CI put option by solving the Kim equations and present numerical results.This method consists of the following three steps.The first is to approximate the quadrature representations in 3.3 by using the trapezoidal rule.The second step is needed to find the numerical values of both the stopping and exercise boundaries, r l t and r u t from the equations approximated above with the Newton-Raphson NR iteration approach.When the values of r l t and r u t are obtained, the third step, numerical integration of 3.2 , yields the value of a given American CI put option.This method is widely used to value American option by several authors, for example, 8, 11 .
We now divide the time interval 0, T into N subintervals: t i iΔt, i 0, 1, 2, . . ., N, Δt T/N.Denote r l t i r l i and r u t i r u i for i 0, 1, 2, . . ., N. Since T N T , we get by 2.3 and 3.4

Journal of Applied Mathematics and Decision Sciences
We define the integrand of 3.3 as the following functions:

4.2
We use the trapezoidal rule to represent the system of recursive integral equations 3.3 as follows: Since there are nonlinear system equations, one can solve it using the NR iteration.In a similar way, numerical values of both r l i and r u i , i N−1, N−2, . . ., 0 can be obtained recursively from 4.3 .We denote the representation of left side in 4.3 by F 1 r l i , r u i and F 2 r l i , r u i , respectively.Then, by the NR iteration the values r l i , r u i have approximations r l i k , r u i k of order k, where k 0, 1, 2, . . .
where ∂F j /∂x and ∂F j /∂y, j 1, 2 are, respectively, partial derivatives of functions F j x, y with respect to x and y.When the values of all r l i , r u i for i N, . . ., 0 are obtained, using Simpson's rule for 3.2 we get the approximation, P 0 r, q , of the value at time t 0 for the American CI put bond option in the following way: assuming N is an even number we have

4.5
In Table 1, we describe the parameters in this section.In our example, we take N 6.Table 2 provides the initial premium of this put option on bond for different installment rate q 1, 10, and 30 with different initial interest rate r 0 0.04, 0.10, and 0.15.
Table 2 shows that the larger the initial interest rate is, the higher the price of American CI put option on bond is.However, the larger the installment rate is, the lower the price of this option is.
Figure 1 displays the curves of both the optimal stopping and exercise boundaries versus different installment rates q.We find out that the two boundaries decrease when the installment rate is arising.That shows that the larger the installment rate is, the higher probability the exercising of the option is.

Conclusions
A simple approximated method for pricing the American CI option written on the zero-bond under Vasicek model is proposed.Numerical example is provided to analyze the effects of the installment rate q on the price of this option and the optimal stopping and exercise boundaries.However, the Vasicek model allows for negative values of interest rate.This property is manifestly incompatible with reality.For this reason, work is ongoing to extend them to other models.

Figure 1 :
Figure 1: Optimal stopping and exercise boundaries for different installment rates.
1 1 r,s ∈E .Substituting and taking expectation under Q on both sides of 3.6 give

Table 1 :
Value of parameters.

Table 2 :
Initial premium of option on bond.