A note on closed-form spread option valuation under log-normal models

A spread option is a contract that gives the owner the right, but not the obligation, to receive the difference, or spread, between the prices of two assets. Spread options can be written on various types of financial products including equities, bonds, currencies, and commodities. Their use is widespread in fixed income, currency, commodity futures, and energy markets. As such pricing and hedging techniques of spread options are very important for the financial industry.

One of the difficulties in pricing spread options is that the exercise boundary is not in a linear form when the strike price is not zero. When the strike price of the spread option is zero, a spread option becomes an exchange option for which a closed-form formula exists. This formula is called Margrabe formula in the literature (see [16]). When the strike price of the spread option is not zero, the exercise boundary is given by a curve and to obtain spread option price one needs to evaluate the risk-neutral probability that the joint log prices lie in one side of that curve, see our Proposition 4.1 below. Due to this non-linearity of the exercise boundary, it is challenging to compute spread option prices efficiently and accurately.

Numerous papers were devoted in pricing spread options in the past either by deriving approximate analytical closed-form formulas or by introducing various numerical methods. The most popular numerical method in pricing spread options is probably the application of the fast Fourier Transformation technique (FFT) which was introduced in [8] initially for pricing European options written on single asset. The idea of this FFT technique was later extended to the case of pricing problems of spread options; see [10] and [13] for example. Both of these papers apply FFT in bi-variate Fourier transformation setting. The recent papers [3, 5, 4] studied pricing problems of spread options under general price dynamics by using Fourier Transformation techniques also. In their papers, they applied the FFT technique to a lower bound for the spread option price that were obtained by using the similar idea in the paper [1]. By doing so they were able to give an approximate semi-closed form formula for the spread option prices. Their approach gives spread prices by a uni-variante Fourier inversion formula which is easier and faster to compute than the formulas that involve bi-variate Fourier Transformation in [10] and [13].

Analytical methods offer closed-form approximate formulas to spread option prices. The first such approximation is the Bachelier approximation (see [19] and [17]). In the Bachelier approximation, the price difference of two assets is approximated by a normal random variable. This approximation gives a closed-form formula for the spread option prices but it does not give very accurate spread prices. Another closed-form formula for spread options was proposed by [15]. This formula has been popular among practitioners as it is simple and gives relatively accurate spread prices. The Kirk’s closed-form formula is obtained by approximating the sum of the second asset with the fixed strike by a log-normal random variable. After this approximation is implemented, Kirk’s formula was obtained by using a similar approach in calculating [16] formula. The papers [7, 6] introduced a new approach in approximating spread option prices in closed-form under log-normal models also. They gave lower and upper bounds to the spread prices. The [7] spread option formula gives very accurate spread option prices, but it involves solving a nonlinear system of equations, which is not trivial.

The recent paper by [1] also developed a closed-form formula for spread option prices under log-normal models. They first gave a lower bound to the spread option price and obtained their closed form formula by computing this lower bound. Numerical tests show that the formula in [1] performs better than the Kirk’s formula, see Section 7 of [1] for this.

The purpose of this paper is to give an alternative closed-form analytical formula for spread options under the log-normal models also. Our formula can be seen as a generalization of the formula in [1], as their formula can be obtained by plugging in some special parameter values into our closed-form formula. Numerical tests show that our formula performs better than the closed formula in [1] for a large range of model parameters.

The paper is organized as follows. In Section 2 we describe our model. In Section 3, we review the [1] and [7] formulas. In Section 4, we present two propositions, Proposition 4.1 and Proposition 4.2, that are useful for our discussions in the remaining sections of the paper. Especially, our Proposition 4.2 gives a numerical procedure which performs faster compared to the Monte Carlo methods. Section 5 presents the main results of this paper. In this section, we obtain a closed-form formula for spread option price (see Proposition 5.4) and show that the closed-form formula for spread options presented in [1] is just a special case of our formula. Section 6 presents numerical results, and the Appendix section presents proofs of our results.

We consider a financial market with one risk-free asset with a constant interest rate $r$ and two stocks in a finite time horizon $[0,T]$. The price dynamics of the two stocks under the risk neutral measure are given by

where $q_{1}$ and $q_{2}$ are instantaneous dividend yields of the two stocks respectively, $\sigma_{1}$ and $\sigma_{2}$ are positive constants that represent the volatilities of these two stocks, and $W_{1}(t)$ and $W_{2}(t)$ are two Brownian motions with correlation $\rho$ under the risk-neutral measure $Q$. For the details of risk-neutral measure and risk-neutral pricing see [9, 12, 14, 2] and the references therein.

We denote by $S_{1}(0)$ and $S_{2}(0)$ the stock price at time $t=0$ respectively and we let $F_{1}=S_{1}(0)e^{(r-q_{1})T}$ and $F_{2}=S_{2}(0)e^{(r-q_{2})T}$ denote the forward prices. Then the stock prices at time $T$ can be written as

The price of a spread option with strike price $K$ is given by

where the expectation is taken under the risk-neutral measure $Q$. In (3) we used a similar notation as in [7] for the spread price. By using the parity relation discussed in [7], we have

This shows that a closed-form formula for (3) when $K>0$ can be used to obtain a closed-form formula for (3) when $K<0$ by using (4). Therefore in our discussions below we assume $K>0$. We also drop the measure $Q$ when we take expectations. All the expectations below are understood under the risk-neutral measure $Q$.

Denote by $(Y,X)$ a two dimensional Gaussian random vector with the same distribution as $\frac{1}{\sqrt{T}}(W_{1}(T),W_{2}(T))$. Then both $X$ and $Y$ are standard normal and $Cov(X,Y)=\rho$. With these new notations, we can express the spread option price as

It can be easily seen from (5) above that when $F_{2}=0$ the spread price $\Pi_{T}$ is given by Black-Scholes formula [2]. When $K=0$, the spread option becomes an exchange option and in this case we have closed form formula, see [16] for this. For general model parameters, there is no exact closed-form solution exits for $\Pi_{T}$. As mentioned in the introduction, a number of papers have presented closed form approximation formulas for the spread prices $\Pi_{T}$ in the past.

Our goal in this paper is to obtain an alternative approximate closed-form formula for the above equation (5). To achieve this goal, we first express the spread price by the probabilities of three events and then approximate these three probabilities separately to obtain an approximate closed form formula for $\Pi_{T}$.

Throughout the paper we use the following notations. For any $x=(x_{1},x_{2},x_{3},x_{4},x_{5},\rho)$ we write

We denote

With these notations the put-call parity relation (4) is written as

We use $\varphi$ to denote the probability density function of the standard normal random variable and by $\Phi$ we denote the cumulative distribution function of the standard normal random variable.

The main purpose of this section is to give a review for the approaches in the two noticeable papers [7] and [1] in pricing spread options under log-normal models. In our Proposition 3.1 below, we explain that the Carmona-Durrleman formula always gives higher spread option prices than the Bjerksund-Stensland formula. This, both of them being lower bounds to the true spread prices, shows that the Carmona-Durrleman formula is more accurate than the Bjerksund-Stensland formula.

First recall that we are interested in computing $EG^{+}$, where

and $Y,X$ are standard normal random variables with $Cov(Y,X)=\rho$, see (5) and the arguments preceding to it for the details of this. The paper [7] gives lower bounds for $EG^{+}$ by using the arguments in their Proposition 2. In their paper they write down $Y$ , adapting to our own notation, as $Y=Z\sin\phi+X\cos\phi$, where $Z$ is a standard normal random variable that is independent from $X$ and $\phi$ is such that $\cos\phi=\rho$. They introduce a family of random variables of the form $Y_{\theta}=Z\sin\theta+X\cos\theta$ and calculate

Then the spread option price $\Pi_{T}$ in (5) is approximated by $e^{-rT}\hat{\Pi}_{T}^{CD}$ with

A simple application of Girsanov theorem (see the proof in Appendix B of [1]) gives

where $\Phi(\cdot)$ denotes the cumulative distribution function of a standard normal random variable. So the problem of finding (11) reduces to maximizing the function on the right hand side of (12) for $\theta$ and $d$. The first order conditions for maximization gives the following two equations

where $\varphi$ denotes the probability density function of a standard normal random variable. These two equations can be solved for $\varphi(d+\sigma_{1}\sqrt{T}\cos(\theta+\phi))$ and $\varphi(d+\sigma_{2}\sqrt{T}\cos\theta)$ as follows

Denoting the solution of (14) by $\theta^{\star}$ and $d^{\star}$, the spread option price presented in [7] equals to

Note here that the equation (14) above is equivalent to the equation (12) in the paper [7].

The above approach provides a method in pricing spread options by constructing proper lower bounds to it and then optimizing these lower bounds. The Carmona-Durrleman formula gives very precise spread price. However, to obtain spread price one needs to solve the equations (14) for $d^{\star}$ and $\theta^{\star}$ numerically. The paper [7] in fact reduced this problem further and one only needs to solve $\theta^{\star}$ numerically from (13) in their paper and use the relation for $d^{\star}$ in Proposition 6 in their paper to obtain $d^{\star}$. While this method reduces the problem a bit, one still needs numerical procedure to obtain the prices of spread options.

Another interesting procedure in pricing spread options were proposed in the recent paper [1]. The paper also introduced lower bounds for the spread call option prices as follows

where $a$ and $b$ are two parameters that needs to be determined by optimizing the lower bounds $\Pi_{T}^{BS}(a,b)$. Their proposed approximate spread option price is given by

In their paper the authors set

By evaluating the equation (16), they obtained the following formula for the spread call option price

where

and

Clearly, the Bjerksund-Stensland formula is simpler than the Carmona-Durrleman formula as one does not have to apply numerical procedure to obtain spread option price in the Bjerksund-Stensland formula. However, as mentioned earlier, the Carmona-Durrleman formula gives more precise spread option prices than the Bjerksund-Stensland formula.

To see this, we write the relation $S_{1}(T)\geq\frac{a(S_{2}(T))^{b}}{E[(S_{2}(T))^{b}]}$ in (16) by using $Y_{\theta}$ defined as in the paper [7]. We obtain that in fact the lower bounds $\Pi_{T}^{BS}(a,b)$ are a subset of the lower bounds $\Pi_{T}^{CD}(\theta,d)$ proposed by [7]. We state this fact as a proposition and write down the proof in the appendix.

In this section, we first express the spread price (3) as a linear combination of the probabilities of three events, see Proposition 4.1 below. Such representation will help us to derive a closed form spread option valuation formula in the next section, see Proposition 5.4. We also write down a numerical approximation formula (see Proposition 4.2 below) of the spread price by using Simpson’s rule for Riemann integrals. At the end of this section, we compare our formula (26) with a related formula discussed in [18].

The following proposition expresses the price of the spread option by probabilities of three events under the risk-neutral measure. The proof of this result uses the Girsanov’s change of measure technique for log-normal models, see the proof in Appendix C.

We remark that $C_{D}^{1}$ above represents the probability (under the risk neutral measure $Q$) that a call option with strike price $K$ and maturity $T$ on the spread between $g_{1}$ stocks with price dynamics $S_{1}(t)$ and $\alpha$ stocks with price dynamics $S_{2}(t)$ is exercised (ends up in the money). $C_{D}^{2}$ can be interpreted as the exercise probability of a call on the spread between $\alpha$ stocks with price dynamics $S_{1}(t)$ and $g_{2}$ stocks with price dynamics $S_{2}(t)$. $C_{D}^{3}$ can be interpreted as the exercise probability of the call option with strike price $K$ of the spread $S_{1}(T)-S_{2}(T)$.

Next, in Proposition 4.2 below, we write down a numerical procedure that approximates the spread price. Before we state our next result, we first fix some notations. Being a jointly Gaussian random variable with $Cov(Y,X)=\rho$, $(Y,X)$ has the property that $X\sim N(0,1)$ and $Y|X=a\sim N(a\rho,1-\rho^{2})$. We use this fact to write down an approximate formula for the spread call option price below. Let $b>0$ be a number that the probability of the event $X\in[-b,b]^{c}$ is very small. Here the notation $c$ represents the complement of an event. We divide the interval $[-b,b]$ into small equally spaced $N$ intervals for a large integer number $N>0$. We denote $a_{i}=b(\frac{2i}{N}-1)$ for $i=1,2,\cdots,N$. With these notations fixed, we state our next result

Another relevant approach to our Proposition 4.2 above is given in the paper [18]. Here we review this approach. Denoting by $E_{Z}$ the expectation with respect to a random variable $Z$, the paper [18] writes the spread price $\Pi_{T}$ in (5) as

The inner expectation $E_{Y}[\big{(}F_{1}e^{-\frac{1}{2}\sigma^{2}_{1}T+\sigma_{1}\sqrt{T}Y}-F_{2}e^{-\frac{1}{2}\sigma^{2}_{2}T+\sigma_{2}\sqrt{T}X}-K)^{+}/X\big{]}$ is evaluated first and then the outer expectation $E_{X}$ is found. This approach leads to an exact formula for the spread price $\Pi_{T}$ as in [18]. The expression (35) can be simplified to (36) (adapted to our notation) in the footnote below. The derivation will be given in Appendix C.

The formula (36) can be evaluated by using the above mentioned trapezoidal rule for Riemann-Stieltjes integrals or by using other Gaussian quadrature methods.

As mentioned earlier, the [1] formula presented in Section 3 above performs highly accurately. In this section, we study the Bjerksund and Stensland formula further and give an alternative derivation for it. Our new approach in deriving the Bjerksund and Stensland formula in this section leads us to a more general formula than the Bjerksund and Stensland formula as our Corollary 5.5 below shows.

As pointed out in Section 4 above, the [7] and [1] spread option prices were obtained by maximizing lower bounds for the spread prices. In this section, instead of constructing lower bounds for spread call option prices, we directly approximate the probabilities $C_{D}^{1},C_{D}^{2},C_{D}^{3}$ in the Proposition 4.1 above.

Before presenting our main result of this paper, we first define the following three curves in the $x$-$y$ coordinate system:

We denote by $\mathcal{C}_{1}^{+},\mathcal{C}_{2}^{+},\mathcal{C}_{3}^{+}$ the domains that lie above the three curves $\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}$ in the $x$-$y$ coordinate system respectively. Namely, the domain $\mathcal{C}_{1}^{+}$, for example, is given by all $(x,y)$ such that

and the other two domains $\mathcal{C}_{2}^{+}$ and $\mathcal{C}_{3}^{+}$ are defined similarly.

Our approach in this section is to approximate the following probabilities

in closed form for each $i=1,2,3$.

The difficulty in obtaining closed form approximations for (39) above is that the boundaries $\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}$ of the domains $\mathcal{C}_{1}^{+},\mathcal{C}_{2}^{+},\mathcal{C}_{3}^{+}$ are not linear functions. Therefore, we wish to find linear functions $y=\kappa_{i}x+\delta_{i}$ with appropriate slope $\kappa_{i}$ and intersection $\delta_{i}$ for each $i=1,2,3,$ so that the following approximations hold

If we can achieve (40), then the latter probabilities in (40) can be written in closed-form by using the following lemma

The application of Lemma 5.1 above leads us to

and this will give us to a closed-form formula for the spread option price due to (26).

To construct lines $y=\kappa_{i}x+\delta_{i}$ that makes the above approximation as precise as possible, we fix some point $(x_{0}^{i},y_{0}^{i})$ on the curve $\mathcal{C}_{i}$ and some slope $\kappa_{i}$ for each $i=1,2,3,$ and use the following lines

In the next lemma, we state some properties of the curves $\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}$ above. The property (a) of this lemma is useful for our discussions in the remaining of this paper.

Part (a) of the above Lemma 5.2 shows that the slopes of the asymptotic lines of all the three curves $\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}$ when $x\rightarrow+\infty$ is the same and is given by $\frac{\sigma_{2}}{\sigma_{1}}$. Part (b) gives lower bounds for each curves when $x\rightarrow-\infty$.