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.

1. 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 helps the buyer to reduce the cost of entering into a hedging strategy and the liquidity risk. Nowadays, the installment options are the most actively traded warrant throughout the financial world, such as the installment warrants on Australian stock and a 10-year warrant with 9 annual payments offered by Deutsche bank, and so on. There is very little literature on pricing the installment option, in particular, for pricing the American CI options. Ciurlia and Roko [12], and Ben-Ameur et al. [13] provide numerical procedures for valuing American CI options on stock under the geometric Brownian motion framework. However, in practice the option on bond is more useful than option on stock, and pricing the former is more complicated, because it is dependent on interest rates variable which is modelled by many economical models.

The aim of this paper is to present an approximation method for pricing American CI put option written on default-free, zero-coupon bond under Vasicek interest rate model. This method is based on Kim integral equations using quadrature formula approximations, such as the trapezoidal rule and the Simpson rule. The layout of this paper is as follows. Section 2 introduces the model and provides some preliminary results. In Section 3 we formulate the valuation problem for the American CI put option on bond describe as a free boundary problem and describe the Kim integral equations. Numerical method and results are presented in Section 4. Section 5 concludes.

2. The Model and Preliminary Results

In the one-factor Vasicek model [14], the short-term interest rate 𝑟𝑡 is modeled as a mean-reverting Gaussian stochastic process on a probability space (Ω,ℱ,𝑃) equipped with a filtration (ℱ𝑡)𝑡≥0. Under the the risk-neutral probability measure 𝑄, it satisfies the linear stochastic differential equation (SDE) 𝑑𝑟𝑡𝑟=ğœ…âˆžâˆ’ğ‘Ÿğ‘¡î€¸ğ‘‘ğ‘¡+ğœŽğ‘‘ğ‘Šğ‘¡,(2.1) where (𝑊𝑡)𝑡≥0 is a standard 𝑄-Brownian motion, 𝜅>0 is the speed of mean reversion, ğ‘Ÿâˆž>0 is the long-term value of interest rate, and ğœŽ is a constant volatility.

Consider a frictionless and no-arbitrage financial market which consists of a bank account 𝐴𝑡 with its price process given by 𝑑𝐴𝑡=𝑟𝑡𝐴𝑡𝑑𝑡 and a 𝑇1-maturity default-free, zero-coupon bond 𝐵(𝑡,𝑟,𝑇1)=𝐵𝑡 with its no-arbitrage price at time 𝑡 given by 𝐵𝑡,𝑟,𝑇1=𝐸𝑄𝑒−∫𝑇1𝑡0𝑥0200𝑑𝑟𝑢𝑑𝑢ℱ𝑡=𝐸𝑡𝑄𝑒−∫𝑇1𝑡0𝑥0200𝑑𝑟𝑠𝑑𝑠,(2.2) where 𝐸𝑄 is the expectation under the risk-neutral probability measure 𝑄. Vasicek [14] provides the explicit form of the zero-bond as follows: 𝐵𝑡,𝑟,𝑇1𝑇=ğ‘Ž1𝑒−𝑡−𝑏(𝑇1−𝑡)𝑟𝑡(2.3) with î‚»âˆ’î‚¸ğ‘…ğ‘Ž(𝑢)=expâˆžğ‘¢âˆ’ğ‘…âˆžğœŽğ‘(𝑢)+2𝑏4𝜅2,(𝑢)𝑏(𝑢)=1−𝑒−𝜅𝑢𝜅,ğ‘…âˆž=ğ‘Ÿâˆžâˆ’ğœŽ22𝜅2.(2.4)

From (2.3), we are easy to obtain the following partial differential equation (P.D.E.): 𝜕𝐵𝑡𝑟𝜕𝑡+ğœ…âˆžî€¸âˆ’ğ‘Ÿğœ•ğµğ‘¡+1𝜕𝑟2ğœŽ2𝜕2𝐵𝑡𝜕𝑟2−𝑟𝐵𝑡=0(2.5) with terminal condition 𝐵(𝑇1,𝑟,𝑇1)=1.

The payoff of a European-style put option without paying any dividends written on the zero-coupon bond 𝐵(𝑡,𝑟,𝑇1) with maturity 𝑇(𝑇<𝑇1), and strike price 𝐾 is ℎ(𝑇,𝑟)=max{𝐾−𝐵(𝑇,𝑟,𝑇1),0}. The no-arbitrage price at time 𝑡(0≤𝑡≤𝑇) of this option is denoted by 𝑝𝑒(𝑡,𝑟,𝐾;𝑇). Following Jamshidian [15], the price of this option can generally be expressed as follows: 𝑝𝑒(𝑡,𝑟,𝐾;𝑇)=𝐸𝑡𝑄𝑒−∫𝑇𝑡0𝑥0200ğ‘‘ğ‘Ÿğ‘ ğ‘‘ğ‘ î‚„î€·â„Ž(𝑇,𝑟)=𝐾𝐵(𝑡,𝑟,𝑇)𝑁−𝑑2−𝐵𝑡,𝑟,𝑇1𝑁−𝑑1,(2.6) where 𝑁(⋅) is the 1-dimensional standard cumulative normal distribution, and 𝑑1,2=1ğœŽ0𝐵ln𝑡,𝑟,𝑇1±1𝐾𝐵(𝑡,𝑟,𝑇)2ğœŽ0,ğœŽ0𝑇=ğœŽğ‘1−𝑇1−𝑒−2𝜅(𝑇−𝑡).2𝜅(2.7)

Now we consider a CI option written on the zero-coupon bond 𝐵(𝑡,𝑟,𝑇1). Denote the initial premium of this option to be 𝑉𝑡=𝑉(𝑡,𝑟;ğ‘ž), which depends on the interest rate, time 𝑡, and the continuous-installment rate ğ‘ž. Applying Ito’s Lemma to 𝑉𝑡, the dynamics for the initial value of this option is obtained as follows: 𝑑𝑉𝑡=𝜕𝑉𝑡+1𝜕𝑡2ğœŽ2𝜕2𝑉𝑡𝜕𝑟2𝑟+ğœ…âˆžâˆ’ğ‘Ÿğ‘¡î€¸ğœ•ğ‘‰ğ‘¡îƒ­ğœ•ğ‘Ÿâˆ’ğ‘žğ‘‘ğ‘¡+ğœŽğœ•ğ‘‰ğ‘¡ğœ•ğ‘Ÿğ‘‘ğ‘Šğ‘¡.(2.8)

Theorem 2.1. In the Vasicek interest rates term structure model (2.1). The contingent claim 𝑉(𝑡,𝑟;ğ‘ž) satisfies the inhomogeneous partial differential equation 𝜕𝑉𝑡+1𝜕𝑡2ğœŽ2𝜕2𝑉𝑡𝜕𝑟2𝑟+ğœ…âˆžâˆ’ğ‘Ÿğ‘¡î€¸ğœ•ğ‘‰ğ‘¡ğœ•ğ‘Ÿâˆ’ğ‘Ÿğ‘‰ğ‘¡=ğ‘ž.(2.9)

Proof. We now consider a self-financing trading strategy 𝜓=(𝜓1,𝜓2), where 𝜓1 and 𝜓2 represent positions in bank account and 𝑇1-maturity zero-coupon bonds, respectively. It is apparent that the wealth process 𝜋𝑡 satisfies 𝜋𝑡=𝜓1𝐴𝑡+𝜓2𝐵𝑡=𝑉𝑡,(2.10) 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 𝜋𝑡 also satisfies 𝑑𝜋𝑡=𝜓1𝑑𝐴𝑡+𝜓2𝑑𝐵𝑡,(2.11) so that 𝑑𝜋𝑡=𝜓1𝑟𝑡𝐴𝑡𝑑𝑡+𝜓2𝜕𝐵𝑡𝑟𝜕𝑡+ğœ…âˆžâˆ’ğ‘Ÿğ‘¡î€¸ğœ•ğµğ‘¡+1𝜕𝑟2ğœŽ2𝜕2𝐵𝑡𝜕𝑟2𝑑𝑡+ğœŽğœ“2𝜕𝐵𝑡𝜕𝑟𝑑𝑊𝑡.(2.12) From (2.8) and (2.10), we get 𝜕𝑉𝑡+1𝜕𝑡2ğœŽ2𝜕2𝑉𝑡𝜕𝑟2𝑟+ğœ…âˆžâˆ’ğ‘Ÿğ‘¡î€¸ğœ•ğ‘‰ğ‘¡ğœ•ğ‘Ÿâˆ’ğ‘žâˆ’ğ‘Ÿğ‘‰ğ‘¡îƒ­ğ‘‘ğ‘¡+ğœŽğœ•ğ‘‰ğ‘¡ğœ•ğ‘Ÿğ‘‘ğ‘Šğ‘¡=𝜓2𝜕𝐵𝑡𝑟𝜕𝑡+ğœ…âˆžâˆ’ğ‘Ÿğ‘¡î€¸ğœ•ğµğ‘¡+1𝜕𝑟2ğœŽ2𝜕2𝐵𝑡𝜕𝑟2−𝑟𝐵𝑡𝑑𝑡+𝜓2ğœŽğœ•ğµğ‘¡ğœ•ğ‘Ÿğ‘‘ğ‘Šğ‘¡.(2.13) Setting 𝜓2=(𝜕𝐵𝑡/𝜕𝑟)/(𝜕𝑉𝑡/𝜕𝑟) the coefficient of 𝑑𝑊𝑡 vanishes. It follows from (2.5) that, 𝑉𝑡 satisfies (2.9).

3. Kim Equations for the Price of American CI Put Option

Consider an American CI put option written on the zero-coupon bond 𝐵𝑡 with the same strike price 𝐾 and maturity time 𝑇(𝑇<𝑇1). 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 𝑟𝑢𝑡 above which it is optimal to stop the installment payments by exercising the option early, as well as a lower critical interest rate 𝑟𝑙𝑡 below which it is advantageous to terminate payments by stopping the option contract. We may call 𝑟𝑢𝑡 to be exercising boundary and 𝑟𝑙𝑡 to be stopping boundary. Denote the initial premium of this put option at time 𝑡 by 𝑃(𝑡,𝑟;ğ‘ž)=𝑃𝑡, defined on the domain 𝒟={(𝑟𝑡,𝑡)∈[0,+∞)×[0,𝑇]}. It is known that 𝑃(𝑡,𝑟;ğ‘ž), 𝑟𝑢𝑡 and 𝑟𝑙𝑡 are the solution of the following free boundary problem [4]: 𝜕𝑃𝑡+1𝜕𝑡2ğœŽ2𝜕2𝑃𝑡𝜕𝑟2𝑟+ğœ…âˆžâˆ’ğ‘Ÿğ‘¡î€¸ğœ•ğ‘ƒğ‘¡ğœ•ğ‘Ÿâˆ’ğ‘Ÿğ‘ƒğ‘¡î€·=ğ‘ž,∀(𝑟,𝑡)âˆˆğ’ž,𝑃(𝑡,𝑟;ğ‘ž)=0,(𝑟,𝑡)∈𝒮,𝑃(𝑡,𝑟;ğ‘ž)=𝐾−𝐵𝑡,𝑟,𝑇1𝑃,(𝑟,𝑡)∈ℰ,𝑃(𝑇,𝑟;ğ‘ž)=ℎ(𝑇,𝑟),𝑟≥0,𝑡,𝑟𝑢𝑡;ğ‘ž=𝐾−𝐵𝑡,𝑟𝑢𝑡,𝑇1,𝑃𝑡,𝑟𝑙𝑡[],;ğ‘ž=0,𝑡∈0,𝑇(3.1) where ğ’ž={(𝑟𝑡,𝑡)∈(𝑟𝑙𝑡,𝑟𝑢𝑡)×[0,𝑇)} is a continuation region, 𝒮={((𝑟𝑡,𝑡)∈[0,𝑟𝑙𝑡]×[0,𝑇]} is a stopping region, and ℰ={((𝑟𝑡,𝑡)∈[𝑟𝑢𝑡,+∞)×[0,𝑇]} is a exercise region.

Remark 3.1. Due to the decreasing property of the price 𝐵(𝑡,𝑟,𝑇1) on the state variable 𝑟, the strike price 𝐾 should be strictly less than 𝐵(𝑇,0,𝑇1). 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 𝑟𝑡 satisfy model (2.1). Then the initial premium of the American CI put option , 𝑃(𝑡,𝑟;ğ‘ž), can be written as 𝑃(𝑡,𝑟;ğ‘ž)=𝑝𝑒(𝑡,𝑟,𝐾;𝑇)+ğ‘žğ‘‡ğ‘¡î€·ğ‘’î€·0𝑥0200𝑑𝐵(𝑡,𝑟,𝑠)𝑁𝑟,𝑟𝑙𝑠+𝑑𝑠𝑇𝑡𝑒0𝑥0200𝑑𝐵(𝑡,𝑟,𝑠)âˆ’ğ‘žğ‘ğ‘Ÿ,𝑟𝑢𝑠𝑟+ğ¾ğ‘¢ğ‘ âˆ’ğœŽ1𝑒𝑟,𝑟𝑢𝑠𝑁−𝑒𝑟,𝑟𝑢𝑠+î€¸î€¸ğ¾ğœŽ1√−𝑒2𝜋exp2𝑟,𝑟𝑢𝑠2𝑑𝑠.(3.2) Moreover, the optimal stopping and exercise boundaries, 𝑟𝑢 and 𝑟𝑙, are solutions to the following system of recursive integral equations: 𝐾−𝐵𝑡,𝑟𝑢,𝑇1=𝑝𝑒(𝑡,𝑟𝑢,𝐾;𝑇)+ğ‘žğ‘‡ğ‘¡0𝑥0200𝑑𝐵(𝑡,𝑟𝑢𝑐𝑟,𝑠)𝑁𝑢,𝑟𝑙𝑠+𝑑𝑠𝑇𝑡0𝑥0200𝑑𝐵(𝑡,𝑟𝑢𝑒𝑟,𝑠)âˆ’ğ‘žğ‘ğ‘¢,𝑟𝑢𝑠𝑟+ğ¾ğ‘¢ğ‘ âˆ’ğœŽ1𝑒𝑟𝑢,𝑟𝑢𝑠𝑁𝑟−𝑒𝑢,𝑟𝑢𝑠+î€¸î€¸ğ¾ğœŽ1√−𝑒2𝜋exp2𝑟𝑢,𝑟𝑢𝑠2𝑑𝑠,0=𝑝𝑒𝑡,𝑟𝑙,𝐾;𝑇+ğ‘žğ‘‡ğ‘¡î€·0𝑥0200𝑑𝐵𝑡,𝑟𝑙𝑁𝑐𝑟,𝑠𝑙,𝑟𝑙𝑠+𝑑𝑠𝑇𝑡0𝑥0200𝑑𝐵𝑡,𝑟𝑙𝑒𝑟,ğ‘ âˆ’ğ‘žğ‘ğ‘™,𝑟𝑢𝑠𝑟+ğ¾ğ‘¢ğ‘ âˆ’ğœŽ1𝑒𝑟𝑙,𝑟𝑢𝑠𝑁𝑟−𝑒𝑙,𝑟𝑢𝑠+ğ¾ğœŽ1√−𝑒2𝜋exp2𝑟𝑙,𝑟𝑢𝑠2𝑑𝑠,(3.3) subject to the boundary conditions 𝐵𝑇,𝑟𝑢,𝑇1=𝐾,𝐵𝑇,𝑟𝑙,𝑇1=𝐾,(3.4) where 𝑒(𝑟,𝑟∗𝑠)=((𝑟∗𝑠−𝑟𝑡)−𝜅(ğ‘Ÿğ‘¡âˆ’ğ‘Ÿâˆž)𝑏(𝑠−𝑡)+(1/2)ğœŽ2𝑏2(𝑠−𝑡))/ğœŽ1) and ğœŽ21=(ğœŽ2/2𝜅)(1−𝑒−2𝜅(𝑠−𝑡)).

Proof. Let 𝑍(𝑠,𝑟)=𝑒−∫𝑠00𝑥0200𝑑𝑟𝑢𝑑𝑢𝑃(𝑠,𝑟;ğ‘ž) 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)∈𝐶1,2(𝒟). We can apply Ito Lemma to 𝑍(𝑠,𝑟) and write 𝑍(𝑇,𝑟)=𝑍(𝑡,𝑟)+𝑇𝑡0𝑥0200𝑑𝜕𝑍(𝑠,𝑟)𝜕𝑠𝑑𝑠+𝜕𝑍(𝑠,𝑟)1𝜕𝑟𝑑𝑟+2ğœŽ2𝜕2𝑍(𝑠,𝑟)𝜕𝑟2𝑑𝑠.(3.5) In terms of 𝑃(𝑡,𝑟;ğ‘ž) this means 𝑒−∫𝑇𝑡0𝑥0200𝑑𝑟𝑠𝑑𝑠𝑃(𝑇,𝑟;ğ‘ž)=𝑃(𝑡,𝑟;ğ‘ž)+𝑇𝑡0𝑥0200𝑑𝑒−∫𝑠𝑡0𝑥0200𝑑𝑟𝑢𝑑𝑢𝜕𝑃𝑠+1𝜕𝑠2ğœŽ2𝜕2𝑃𝑠𝜕𝑟2𝑟+ğœ…âˆžî€¸âˆ’ğ‘Ÿğœ•ğ‘ƒğ‘ ğœ•ğ‘Ÿâˆ’ğ‘Ÿğ‘ƒğ‘ îƒ­+𝑑𝑠𝑇𝑡0𝑥0200𝑑𝑒−∫𝑠𝑡0𝑥0200ğ‘‘ğ‘Ÿğ‘¢ğ‘‘ğ‘¢ğœŽğœ•ğ‘ƒğ‘ ğœ•ğ‘Ÿğ‘‘ğ‘Šğ‘ .(3.6) From (3.1) we know that 𝑃(𝑇,𝑟;ğ‘ž)=ℎ(𝑇,𝑟) and 𝑃(𝑠,𝑟;ğ‘ž)=𝑃(𝑠,𝑟;ğ‘ž)𝟏(𝑟,𝑠)âˆˆğ’ž+𝑃(𝑠,𝑟;ğ‘ž)𝟏(𝑟,𝑠)∈𝒮+𝑃(𝑠,𝑟;ğ‘ž)⋅𝟏(𝑟,𝑠)∈ℰ=𝑃(𝑠,𝑟;ğ‘ž)𝟏(𝑟,𝑠)âˆˆğ’ž+[𝐾−𝐵(𝑠,𝑟,𝑇1)]𝟏(𝑟,𝑠)∈ℰ. Substituting and taking expectation under 𝑄 on both sides of (3.6) give 𝑝𝑒(𝑡,𝑟,𝐾;𝑇)=𝐸𝑡𝑄𝑒−∫𝑇𝑡0𝑥0200𝑑𝑟𝑠𝑑𝑠𝑔(𝑇,𝑟)=𝑃(𝑡,𝑟;ğ‘ž)+𝑇𝑡0𝑥0200𝑑𝐸𝑡𝑄𝑒−∫𝑠𝑡0𝑥0200𝑑𝑟𝑢𝑑𝑢𝜕𝑃𝑠+1𝜕𝑠2ğœŽ2𝜕2𝑃𝑠𝜕𝑟2𝑟+ğœ…âˆžî€¸âˆ’ğ‘Ÿğœ•ğ‘ƒğ‘ ğœ•ğ‘Ÿâˆ’ğ‘Ÿğ‘ƒğ‘ î€œîƒ­îƒ°ğ‘‘ğ‘ =𝑃(𝑡,𝑟;ğ‘ž)+ğ‘žğ‘‡ğ‘¡0𝑥0200𝑑𝐸𝑡𝑄𝑒−∫𝑠𝑡0𝑥0200𝑑𝑟𝑢𝑑𝑢1𝑟𝑙𝑠<𝑟𝑠<𝑟𝑢𝑠𝑑𝑠−𝐾𝑇𝑡0𝑥0200𝑑𝐸𝑡𝑄𝑒−∫𝑠𝑡0𝑥0200𝑑𝑟𝑢𝑑𝑢𝑟𝑠1𝑟𝑠≥𝑟𝑢𝑠𝑑𝑠.(3.7) From (2.1), it is easy to obtain that the state variable 𝑟𝑠 follows 𝑟𝑠=𝑟𝑡𝑒−𝜅(𝑠−𝑡)+ğ‘Ÿâˆžî€·1−𝑒−𝜅(𝑠−𝑡)+ğœŽğ‘ ğ‘¡0𝑥0200𝑑𝑒−𝜅(𝑠−𝑢)𝑑𝑊𝑢(3.8) for every 𝑠>𝑡. Then the state variable 𝑟𝑠 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,𝑟𝑢𝑡;ğ‘ž)=𝐾−𝐵(𝑡,𝑟𝑢𝑡,𝑇1) and 𝑃(𝑡,𝑟𝑙𝑡;ğ‘ž)=0. The boundary conditions (3.4) hold since the limitation for (3.3) as 𝑡↑𝑇.

Remark 3.3. From (3.2), when 𝑟𝑙𝑡 and 𝑟𝑢𝑡 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, 𝑟𝑙𝑡 and 𝑟𝑢𝑡, cannot be proved to be monotone function of time 𝑡. So we use trapezoidal rule method to deal with them.

4. 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, 𝑟𝑙𝑡 and 𝑟𝑢𝑡 from the equations approximated above with the Newton-Raphson (NR) iteration approach. When the values of 𝑟𝑙𝑡 and 𝑟𝑢𝑡 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: 𝑡𝑖=𝑖Δ𝑡,𝑖=0,1,2,…,𝑁,Δ𝑡=𝑇/𝑁. Denote 𝑟𝑙𝑡𝑖=𝑟𝑙𝑖 and 𝑟𝑢𝑡𝑖=𝑟𝑢𝑖 for 𝑖=0,1,2,…,𝑁. Since 𝑇𝑁=𝑇, we get by (2.3) and(3.4)

𝑟𝑙𝑁=𝑟𝑢𝑁=1𝑏𝑇1î€¸ğ‘Žî€·ğ‘‡âˆ’ğ‘‡ln1−𝑇𝐾.(4.1) We define the integrand of (3.3) as the following functions: 𝑓𝑡,𝑟,𝑠,𝑟∗𝑠𝑒=𝐵(𝑡,𝑟,𝑠)âˆ’ğ‘žğ‘ğ‘Ÿ,𝑟∗𝑠𝑟+ğ¾âˆ—ğ‘ âˆ’ğœŽ1𝑒𝑟,𝑟∗𝑠𝑁−𝑒𝑟,𝑟∗𝑠+î€¸î€¸ğ¾ğœŽ1√−𝑒2𝜋exp2𝑟,𝑟∗𝑠2,𝑔𝑡,𝑟,𝑠,𝑟∗𝑠𝑐=ğ‘žğµ(𝑡,𝑟,𝑠)𝑁𝑟,𝑟∗𝑠.(4.2) We use the trapezoidal rule to represent the system of recursive integral equations (3.3) as follows: 𝑝𝑡𝑖,𝑟𝑢𝑖1,𝐾;𝑇+Δ𝑡2𝑔𝑡𝑖,𝑟𝑢𝑖,𝑡𝑖,𝑟𝑙𝑖+𝑁−1𝑗=𝑖+1𝑡0𝑥0200𝑑𝑔𝑖,𝑟𝑢𝑖,𝑡𝑗,𝑟𝑙𝑗+12𝑔𝑡𝑖,𝑟𝑢𝑖,𝑡𝑁,𝑟𝑙𝑁1+Δ𝑡2𝑓𝑡𝑖,𝑟𝑢𝑖,𝑡𝑖,𝑟𝑢𝑖+𝑁−1𝑗=𝑖+1𝑡0𝑥0200𝑑𝑓𝑖,𝑟𝑢𝑖,𝑡𝑗,𝑟𝑢𝑗+12𝑓𝑡𝑖,𝑟𝑢𝑖,𝑡𝑁,𝑟𝑢𝑁𝑡+𝐵𝑖,𝑟𝑢𝑖,𝑇1𝑝𝑡−𝐾=0,𝑖,𝑟𝑙𝑖1,𝐾;𝑇+Δ𝑡2𝑔𝑡𝑖,𝑟𝑙𝑖,𝑡𝑖,𝑟𝑙𝑖+𝑁−1𝑗=𝑖+1𝑡0𝑥0200𝑑𝑔𝑖,𝑟𝑙𝑖,𝑡𝑗,𝑟𝑙𝑗+12𝑔𝑡𝑖,𝑟𝑙𝑖,𝑡𝑁,𝑟𝑙𝑁1+Δ𝑡2𝑓𝑡𝑖,𝑟𝑙𝑖,𝑡𝑖,𝑟𝑢𝑖+𝑁−1𝑗=𝑖+1𝑡0𝑥0200𝑑𝑓𝑖,𝑟𝑙𝑖,𝑡𝑗,𝑟𝑢𝑗+12𝑓𝑡𝑖,𝑟𝑙𝑖,𝑡𝑁,𝑟𝑢𝑁=0,𝑖=0,…,𝑁−1.(4.3) Since there are nonlinear system equations, one can solve it using the NR iteration. In a similar way, numerical values of both 𝑟𝑙𝑖 and 𝑟𝑢𝑖, 𝑖=𝑁−1,𝑁−2,…,0 can be obtained recursively from (4.3). We denote the representation of left side in (4.3) by 𝐹1(𝑟𝑙𝑖,𝑟𝑢𝑖) and 𝐹2(𝑟𝑙𝑖,𝑟𝑢𝑖), respectively. Then, by the NR iteration the values (𝑟𝑙𝑖,𝑟𝑢𝑖) have approximations (𝑟𝑙𝑖(𝑘),𝑟𝑢𝑖(𝑘)) of order 𝑘, where 𝑘=0,1,2,…𝑟𝑙𝑖(𝑟𝑘+1)𝑢𝑖=𝑟(𝑘+1)𝑙𝑖(𝑟𝑘)ğ‘¢ğ‘–î‚¶âˆ’âŽ›âŽœâŽœâŽœâŽ(𝑘)𝜕𝐹1𝜕𝑥𝜕𝐹1𝜕𝑦𝜕𝐹2𝜕𝑥𝜕𝐹2âŽžâŽŸâŽŸâŽŸâŽ ğœ•ğ‘¦âˆ’1⋅𝐹1𝐹(𝑥,𝑦)2|(𝑥,𝑦)(𝑥,𝑦)=(𝑟𝑙𝑖(𝑘),𝑟𝑢𝑖(𝑘)),(4.4) where 𝜕𝐹𝑗/𝜕𝑥 and 𝜕𝐹𝑗/𝜕𝑦,𝑗=1,2 are, respectively, partial derivatives of functions 𝐹𝑗(𝑥,𝑦) with respect to 𝑥 and 𝑦. When the values of all (𝑟𝑙𝑖,𝑟𝑢𝑖) for 𝑖=𝑁,…,0 are obtained, using Simpson’s rule for (3.2) we get the approximation, 𝑃0(𝑟,ğ‘ž), 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 𝑃0(𝑟,ğ‘ž)=𝑝(0,𝑟,𝐾;𝑇)+Δ𝑡3𝑔0,𝑟,0,𝑟𝑙0+4𝑔0,𝑟,𝑡1,𝑟𝑙1+2𝑔0,𝑟,𝑡2,𝑟𝑙2+4𝑔0,𝑟,𝑡3,𝑟𝑙3+⋯+2𝑔0,𝑟,𝑡𝑁−2,𝑟𝑙𝑁−2+4𝑔0,𝑟,𝑡𝑁−1,𝑟𝑙𝑁−1+𝑔0,𝑟,𝑇,𝑟𝑙𝑇+Δ𝑡3𝑓0,𝑟,0,𝑟𝑢0+4𝑓0,𝑟,𝑡1,𝑟𝑢1+2𝑓0,𝑟,𝑡2,𝑟𝑢2+4𝑓0,𝑟,𝑡3,𝑟𝑢3+⋯+2𝑓0,𝑟,𝑡𝑁−2,𝑟𝑢𝑁−2+4𝑓0,𝑟,𝑡𝑁−1,𝑟𝑢𝑁−1+𝑓0,𝑟,𝑇,𝑟𝑢𝑇.(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 ğ‘ž=1,10, and 30 with different initial interest rate 𝑟0=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 ğ‘ž. 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.

5. 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 ğ‘ž 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.


The work has been partially supported by the NNSF of China with no. 40675023 and the Guangxi Natural Science Foundation with no. 0991091.