British put option on stocks under stochastic interest rate

Abstract

The British option was first introduced by G. Peskir and F. Samee (2011). In a British option, the holder can enjoy the early exercise feature of American option whereupon his payoff is the ’best prediction’ of the European payoff given all the information up to the exercise date under the hypothesis that the true drift of the stock equals a specified contract drift. Consistent with the plain vanilla option, the authors considered the constant interest rate. In this paper, we will consider the pricing of the British put option in a stochastic interest rate environment, particularly the Vasicek model. We will derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary.

Keywords

British put option american put option european put option arbitrage-free price vasicek model rational exercise boundary geometric Brownian motion optimal stopping time free boundary problem2010 Mathematics Subject Classiffcation: 91G20 60G40 35R35

1. Introduction

Plain vanilla options such as European options and American options are widely used in the market and their pricing mechanisms are well studied. An option gives the holder the right, but not the obligation, to buy or sell an underlying asset for a specified price, called strike price, on or before a specified future date, called maturity date or expiration date. The option is European if the holder can exercise it only at expiration date; it is American if the option can be exercised anytime even prior to the expiration date.

One of the pricing mechanisms for European option is provided by the well-known Black-Scholes-Merton formula. This mathematical model assumes, among other things, the absence of arbitrage opportunities and that lending and borrowing are possible at the same risk-free rate. Such method falls within the so-called risk-neutral pricing framework.

Peskir and Samee (2011) introduced a new type of option, called British option, which is American in nature because it can be exercised prior to maturity but with European payoff. The motivation for this new financial product stems from the disparity between the expected value of the option buyer’s investment, in the form of premium paid, and the expected value of his payoff when the actual drift rate of the underlying stock price deviates from the risk-free rate. An added feature is built into this instrument which aims at both providing protection against unfavourable price movements as well as securing higher returns when these movements are favourable (Peskir and Samee, 2011).

The derivation of the British option price by Peskir and Samee (2011) assumes the usual model as in the Black-Scholes-Merton formula: a geometric Brownian motion for the dynamics of the underlying stock and a constant risk-free interest rate. This paper aims to extend the results of Peskir and Samee (2011) by assuming non-constant interest rate. In particular, it will be assumed that the short rate follows the Vasicek model. The paper is organized as follows. In Section 2 we present the definition of the British put option as given in Peskir and Samee (2011) and the financial setting, particularly the assumptions on the dynamics of the underlying stock price and the short-term interest rate. In Section 3 we discuss the payoff function, the premium and the price process for the British put option. In section 4 we derive the equations for the free-boundary problem satisfied by the option price. We present the main result in Section 5, then conclude.

2. Setting of the problem

Consider the financial market consisting of a risky stock with price process $X=(X_{t}:t\in[0,T])$ and a zero-coupon bond with price process $P=(P_{t}:t\in[0,T])$ , where the prices evolve as

$\displaystyle dX_{t}=\mu X_{t}dt+\sigma_{1}X_{t}dW_{t}$ (1) $\displaystyle dP_{t}=r_{t}P_{t}dt-\lambda(t,T)\sigma_{2}P_{t}d\overline{W}_{t}$ (2)

where $\mu\in\mathbb{R}$ is the appreciation rate, $\sigma_{1}$ and $\sigma_{2}$ are volatility coefficients, $W=(W_{t})_{t\geqslant 0}$ and $\overline{W}=(\overline{W}_{t})_{t\geqslant 0}$ are standard Wiener processes defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ with

$\displaystyle cov(dW_{t},d\overline{W}_{t})=\rho dt,\quad(|\rho|<1),$

$r_{t}$ is the short rate that follows the Vasicek model

$\displaystyle dr_{t}=a(\theta-r_{t})dt+\sigma_{2}d\overline{W}_{t},$ (3)

where $a,\theta\in\mathbb{R}$ , $a>0$ and $\displaystyle\lambda(t,T)=\frac{1-e^{-a(T-t)}}{a}$ . It is well-known that the price $P_{t}=P(t,r_{t};T)$ of a zero-coupon bond has $P(t,r;T)$ satisfying the partial differential equation (see Fang (2012) for example)

$\displaystyle\frac{\partial P}{\partial t}+a(\theta-r)\frac{\partial P}{% \partial r}+\frac{1}{2}\sigma_{2}^{2}\frac{\partial^{2}P}{\partial r^{2}}-rP=0$ (4)

subject to the terminal condition that $P(T,r;T)=1$ for all $r\in\mathbb{R}$ . As in Mamon (2004), we use the fact that the process of the interest rate $r_{u}$ is Markovian. That is, $r_{u}$ is dependent on $r_{t}$ for $u>t$ . Thus,

$\displaystyle P(t,r_{t};T)=E\left[e^{-\int_{t}^{T}r_{u}du}\Big{|}\mathcal{F}_{% t}\right]=E\left[e^{-\int_{t}^{T}r_{u}du}\Big{|}r_{t}\right]=E\left[e^{-\int_{% t}^{T}r_{u}(r_{t})du}\right],$ (5)

where $E$ is taken with respect to the probability measure $\mathbb{P}$ . From Eq. (13) in Mamon (2004), we have

$\displaystyle P(t,r_{t};T)=e^{-\lambda(t,T)r_{t}+B(t,T)},$ (6)

where

$\displaystyle B(t,T)=\left(\theta-\frac{\sigma_{2}^{2}}{2a^{2}}\right)\left[% \lambda(t,T)-(T-t)\right]-\frac{\sigma_{2}^{2}\lambda^{2}(t,T)}{4a}.$ (7)

We will consider the British put option on the stock in the aforementioned financial market. British put option with strike price $K$ and time to maturity $T$ in years is defined in Peskir and Samee (2011) as follows.

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The British put option is a financial contract between a seller/hedger and a buyer/holder entitling the latter to exercise at any (stopping) time $\tau$ prior to maturity $T$ whereupon his payoff (deliverable immediately) is the ’best prediction’ of the European payoff $(K-X_{T})^{+}$ given all the information up to time $\tau$ under the hypothesis that the true drift of the stock price equals the contract drift $\mu_{c}$ .

In Peskir and Samee (2011), the price of the British put option is derived under the hypothesis that the risk-free rate is constant, that is, $r_{t}=r$ for all $t\in[0,T]$ . Hence, this paper presents an extension of the results in Peskir and Samee (2011).

Let $\displaystyle\beta(t):=\frac{\mu_{c}-\mu}{\sigma_{1}}$ , with $\mu_{c}\neq\mu$ and $Z_{t}:=exp\left[\int_{0}^{t}\beta(u)dW_{u}-\frac{1}{2}\int_{0}^{t}\beta^{2}(u)% du\right]$ . Define an equivalent measure $\mathbb{P}^{\mu_{c}}$ via the following:

$\displaystyle\frac{d\mathbb{P}^{\mu_{c}}}{d\mathbb{P}}=Z_{T}.$ (8)

By It $\hat{\text{o}}$ ’s formula, we have

$\displaystyle\frac{dZ_{t}}{Z_{t}}=\beta(t)dW_{t}$

showing that $Z_{t}$ is a local martingale with $E[Z_{t}]=1$ for $0\leqslant t\leqslant T$ . From Lemma 1 in Yao et al. (2006), we have

$\displaystyle W^{\mu_{c}}_{t}=W_{t}-\int_{0}^{t}\beta(u)du,$ (9)

which is a $\mathbb{P}^{\mu_{c}}$ -Brownian motion. Under the probability $\mathbb{P}^{\mu_{c}}$ , the stock price process Eq. (1) becomes

$\displaystyle dX_{t}=\mu_{c}X_{t}dt+\sigma_{1}X_{t}dW^{\mu_{c}}_{t}$ (10)

where $0\leqslant t\leqslant T$ with $X_{0}=x\in(0,\infty)$ . Thus, making use of Eq. (8), we have $E^{\mu_{c}}(X)=E(Z_{T}X)=E(Z_{T})E(X)=E(X)$ for any random variable $X$ .

Let $(F_{t})_{0\leqslant t\leqslant T}$ be the filtration generated by the process $X$ . Then the payoff of the British put option at a given stopping time $t=\tau$ is given by

$\displaystyle E^{\mu_{c}}\left[(K-X_{T})^{+}\Big{|}F_{\tau}\right]$ (11)

where the conditional expectation is taken with respect to a new (equivalent) probability measure $\mathbb{P}^{\mu_{c}}$ under which the stock price $X$ evolves as in Eq. (10). Thus, the effect of exercising the British put option is to substitute the contract drift $\mu_{c}$ to the true (unknown) drift $\mu$ of the stock price for the remaining time of the contract. Note that the value of the contract drift $\mu_{c}$ must be equivalent to the buyer’s tolerance level for the deviation of the true drift $\mu$ from his original belief (See Peskir and Samee (2011)).

3. Payoff, premium and the price process of the british put option

Using the properties of the Wiener process $W^{\mu_{c}}$ , then

$\displaystyle E^{\mu_{c}}\left[(K-X_{T})^{+}\Big{|}\mathcal{F}_{t}\right]=G^{% \mu_{c}}(t,X_{t})$ (12)

where $G^{\mu_{c}}$ is the payoff function defined by

$\displaystyle G^{\mu_{c}}(t,x)=E\left(K-xZ_{T-t}^{\mu_{c}}\right)^{+}$ (13)

for $t\in[0,T]$ and $x>0$ , $Z_{T-t}^{\mu_{c}}$ is given by

$\displaystyle Z_{T-t}^{\mu_{c}}=e^{(\mu_{c}-\frac{\sigma_{1}^{2}}{2})(T-t)+% \sigma_{1}W_{T-t}}$ (14)

and the expectation $E$ is taken under the equivalent actual probability measure $\mathbb{P}$ . It can be verified that

$\displaystyle G^{\mu_{c}}(t,x)=K\Phi(-d_{2})-xe^{\mu_{c}(T-t)}\Phi(-d_{1}),$

where $d_{1}=\frac{\ln\frac{x}{K}+(\mu_{c}+\frac{1}{2}\sigma_{1}^{2})(T-t)}{\sigma_{1% }\sqrt{T-t}}$ , $d_{2}=d_{1}-\sigma_{1}\sqrt{T-t}$ for $t\in[0,T)$ , $x>0$ and $\Phi(\cdot)$ is the standard normal cumulative distribution function.

Applying the usual hedging scheme, then the arbitrage-free price of the British put option at deal date (time 0) is given by

$\displaystyle V=\sup_{0\leqslant\tau\leqslant T}\tilde{E}\left[e^{-\int_{0}^{% \tau}r_{u}du}E^{\mu_{c}}\left((K-X_{T})^{+}\Big{|}\mathcal{F}_{\tau}\right)\right]$ (15)

where the supremum is taken over all stopping time $\tau\in[0,T]$ of $X$ and $\tilde{E}$ is taken with respect to the (unique) equivalent martingale measure $\tilde{\mathbb{P}}$ . Note that in the expression above, $E\left[e^{-\int_{0}^{\tau}r_{u}du}\right]$ takes the role of the discounting factor which brings the payoff $E^{\mu_{c}}\left[(K-X_{T})^{+}\Big{|}\mathcal{F}_{\tau}\right]$ at exercise date $\tau$ to time $0$ , where $E$ is taken as in Eq. (5).

Now, fix $t\in[0,T]$ . We want a general expression for the price, denoted by $V(t,r_{t},x)$ , of the British put option at any time $t$ at which the stock price $X_{t}=x$ and the short rate is $r_{t}$ . If the exercise date is at time $t+\tau$ , where $\tau\in[0,T-t]$ , then based on Eq. (12) and extending the argument in Eq. (15), we have

$\displaystyle V(t,r_{t},X_{t})=\sup_{0\leqslant\tau\leqslant T-t}\tilde{E}_{t,% x}\left[e^{-\int_{t}^{t+\tau}r_{u}du}G^{\mu_{c}}(t+\tau,X_{t+\tau})\ \Big{|}\ % \mathcal{F}_{t}\right]$ (16)

where the supremum is taken over all stopping times $\tau\in[0,T-t]$ of $X$ and $\tilde{E}_{t,x}$ is taken with respect to the (unique) equivalent martingale measure $\tilde{\mathbb{P}}_{t,x}$ under which $X_{t}=x$ and $r_{t}\in\mathbb{R}$ . Note that $V(0,r_{0},x)=V$ . Since the supremum in Eq. (16) is attained at the first entry time of $X$ to the closed set where $V=G^{\mu_{c}}$ , and Law $(X(\mu)|\mathbb{P})$ is the same as Law $(X(r_{t})|\tilde{\mathbb{P}})$ , then

$\displaystyle V(t,r,x)=\sup_{0\leqslant\tau\leqslant T-t}E\left[e^{-\int_{t}^{% t+\tau}r_{u}du}G^{\mu_{c}}(t+\tau,xX_{\tau})\ \Big{|}\ X_{t}=x,r_{t}=r\right],$ (17)

where the process $X=(X_{t}(t,r_{t}))_{t\in[0,T]}$ under $\mathbb{P}$ evolves as

$\displaystyle dX_{t}=r_{t}X_{t}dt+\sigma_{1}X_{t}dW_{t},\quad(X_{0}=1).$ (18)

Moreover, the British put option price at maturity time $T$ is

$\displaystyle V(T,r_{T},X_{T})=G^{\mu_{c}}(T,X_{T})=E(K-X_{T})^{+}.$ (19)

This means that the price of the British put option at maturity coincides with the payoff from a classical European put whose stock price dynamics follows the stochastic differential Eq. (18) above. Let $G^{r_{t}}(t,x)=E(K-X_{T})^{+}$ , where $X$ follows Eq. (18) with $X_{t}=x$ . Thus, $V(T,r_{T},X_{T})=G^{r_{t}}(t,x)$ .

It may be noted that if the interest rate is constant ( $r_{t}=r$ for all $t\in[0,T])$ , i.e., the coefficients $a$ and $\sigma_{2}$ in Eq. (3) are all equal to zero, then the expression for $G^{r_{t}}(t,x)$ multiplied by $e^{-r(T-t)}$ coincides with the Black-Scholes formula for the arbitrage-free price of the European put option at time $t$ with maturity time $T$ .

Define the set

$\displaystyle D:=\left\{(t,r_{t},x)\in[0,T]\times\mathbb{R}\times(0,\infty)\ % \Big{|}\ V(t,r_{t},x)=G^{\mu_{c}}(t,x)\right\}.$ (20)

By Eq. (19), we say that $\{T\}\times\{r_{T}\}\times(0,\infty)\subset D$ , which is consistent with the fact that the supremum in Eq. (16) is taken over $(\mathcal{F}_{t})_{t\in[0,T]}$ -stopping times $\tau\in[t,T]$ . Furthermore, by Corollary 2.9 (page 46) in Peskir and Shiryaev (2006), the $(\mathcal{F}_{t})_{t\in[0,T]}$ -stopping time defined by

$\displaystyle\tau_{D}(t,r_{t},x):=\inf\{s\in[0,T-t]\ :\ (t+s,r_{t+s},X_{t+s})% \in D\}$ (21)

with $X_{t+s}=x\in(0,\infty)$ and $r_{t+s}\in\mathbb{R}$ , is an optimal stopping time for the option price in Eq. (16) since both $x\mapsto V(t,r_{t},x)$ and $x\mapsto G^{\mu_{c}}(t,x)$ are continuous on $(0,\infty)$ and $G^{\mu_{c}}(t,x)\leqslant K$ for all $t\in[0,T]$ and $r_{t}\in\mathbb{R}$ .

We next derive the following continuity results to show that the set $D$ is closed.

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The mapping $(t,x)\mapsto G^{\mu_{c}}(t,x)$ is jointly continuous on $[0,T]\times(0,\infty)$ .