Pricing American interest rate option on zero-coupon bond numerically

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Abstract

In this paper, an American put option on zero-coupon bond is priced numerically by finite volume method (FVM) under a single factor model of the short-term rate. In term of the price of zero-coupon bond, an integral representation of the early exercise rate is derived, which can both locate the exercise rate and be viewed as an error indicator. In our numerical results, the prices of zero-coupon bond and American put option are given and the optimal early interest rate is also provided.

Introduction

It is known that American options can only be evaluated numerically with few exceptions. Our objection in this paper is to develop technique to find the solution for value of American option on pure discount bonds. An option gives the owner the right to buy or sell an underlying asset at a specific time, called the expiration date T, and for a specific price, denominated the exercise price K. In the case of an American option, the option can be exercised at any time prior to the expiration date. Since there is uncertainty in the time at which the option should be exercised, the holder of the option needs to know when he or she should exercise the option. This information is provided by a curve called the optimal exercise curve. Mathematically, this curve is characterized as a free boundary to the model equation.

The pricing of interest rate contingent claim has been the subject of a significant amount of recent research in finance. For European option on pure discount bonds, it is fairly easy to show that prices of these claims can be represented as expected terminal payoffs discounted using the path of the instantaneous risk-free rate r under a risk-adjusted or equivalent martingale measure. In literatures [1], [2], [3], [4], the prices of pure discount bond and European contingent claim on the zero-coupon bonds are discussed. For American options on the zero-coupon bonds, numerical approach needs to be developed. In literature [6], an integral representation and computation for the solution of American option on stock are developed.

In this paper, we discuss an American put option on pure discount bonds where spot interest rate follows a single-factor model of the short-term rate. To our knowledge, the prices of both zero-coupon bond and European option on the bond are not derived when the spot rate follows the single-factor model. We make use of a numerical method called finite volume method (FVM) to approximate the solution of American put option on the pure discount bond and the optimal exercise curve γ(t), which is called early exercise interest rate, and give an integral representation of γ(t) in term of the price of pure discount bond. Our numerical results also show that the FVM method is stable and convergent in our cases. The FVM method considered in our paper is different from the existent literatures in three aspects:

  • (1)

    the more general discretization schemes in state are considered;

  • (2)

    discrete linear complementary problems are written in symmetric forms and can be solved rapidly by using the Brennan–Schwartz algorithm (see [10]);

  • (3)

    to price the American put option, we make use of the FVM method to price the pure discount bond numerically.

The balance of the paper is organized as follows. Section 2 describes the model. We introduce the finite volume method (FVM) to derive the solution of the American put option on pure discount bond and the early exercise rate in Section 3. In Section 4, we get the prices of zero-coupon bonds numerically by FVM method. Section 5 provides an integral representation of the early exercise rate in term of the price of zero-coupon bond. Numerical results is in Section 6. Section 7 concludes this paper.

Section snippets

The model

We discuss a single factor model of the short-term rate. Most existing single factor structure can be nested within the following stochastic differential equation describing moments in the spot interest rate r:drt=(a+br)dt+σrβdWt,where (W(t))t⩾0 is a standard Brownian motion, a, b and σ are positive constants, and β is a specified number as being either 0, 1/2 or 1.

It is well known that the price of any interest rate contingent claim on the pure discount bonds, denoted by V(r, t), must satisfy a

Numerical solution of American put option on zero-coupon bond

In this section, we discuss the numerical solution of the American put option on the zero-coupon bond. We adopt a finite volume method (FVM). FVM method is also called the box method or generalized difference method by other authors. Since γ(t) is bounded, we can restrict r in an interval [0, R] for a sufficiently large number R. Letv(r,t)=P(r,T-t),G(r,t)=g(r,T-t).(2.1), (2.2), (2.3), (2.4), (2.5) become the following parabolic complementarity problem:vt-12σ2r22vr2-k(θ-r)vr+rv=0,vG,0<r<R,0

Numerical price of zero-coupon bond

In fact, we cannot know the price B(r, t, T) of zero-coupon bond when β = 1, especially B(r, T, T). To get price of American put option on the zero-coupon bond, first we must solve B(r, t, T) numerically. We also use FVM method because the method is stable unconditionally from Theorem 2.

The price of zero-coupon bond, B(r, t, T), satisfies the terminal value problem of following partial differential equation:Bt+12σ2r22Br2+k(θ-r)Br-rB=0,r0,0t<T,B(r,T,T)=1.Letf(r,t)=B(r,T-t,T)then we havef

An integral representation of free boundary

In this section, we discuss how to derive the free boundary of the American style bonds option. Letu(r,t)=v(r,t)+f(r,t).

Because v(r,t) and f(r,t) satisfy (3.1), it follows that u(r,t) satisfies the following free boundary problemut-12σ2r22ur2-k(θ-r)ur+ru=0,u(r,t)>K,0<tT,u(γ(T-t),t)=K,ur(γ(T-t),t)=0,0<tT,u(r,t)=K,r>γ(T-t),0tT,u(0,t)=max(K,B(0,T-t)),0tT,v(r,0)=max(K,B(r,T)),r0.LetD={(r,s)0rγ(T-t),0st}for t  [0,T]. Integrating (5.1) over D and using integration by parts, we getD

Numerical results

In this section, we present numerical results for our algorithm to give the prices of American put option on the zero-coupon bond and the optimal early exercise interest rate. We consider a one-year American put option on a five-year zero-coupon bond, which pays one unit of currency to the holder of the bond at maturity time. The strike price of the American option is K = 0.6 units. R is taken to be 0.5. For simplicity, we use the equal distance partition in r, that is, [0, R] will be divided into

Conclusions

In this paper, we use FVM method to discuss the price of an American put option on a pure discount bond when the spot rate follows a single factor model of the short-term rate. To our knowledge, so far the price of the pure discount bond has not been given as a closed forms when the interest rate follows the single -factor model where β = 1, it is difficult to get the price of American options on the pure discount bond due to no closed form of zero-coupon bond. In this paper, at first the price

Acknowledgements

The authors gratefully acknowledge the support provided by JiangSu Altitude College Natural Science Foundation. The authors also gratefully thank the editors and referees for theirs useful help to improve this paper.

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