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

Kai He Department of Financial Mathematics, Xi’an Jiaotong Liverpool
University, Suzhou, China.
Email<EMAIL_ADDRESS>Dongdong Hu  Department of Financial
Mathematics, Xi’an Jiaotong Liverpool University, Suzhou, China.
Email<EMAIL_ADDRESS>Nuerxiati Abudurexiti Department of
Financial Mathematics, Xi’an Jiaotong Liverpool University, Suzhou, China.
Email<EMAIL_ADDRESS>Hasanjan Sayit Department of
Financial Mathematics, Xi’an Jiaotong Liverpool University, Suzhou, China.
Email: Hasanjan<EMAIL_ADDRESS>
###### Abstract

In the papers Carmona and Durrleman [7] and Bjerksund and Stensland [1],
closed form approximations for spread call option prices were studied under
the log normal models. In this paper, we give an alternative closed form
formula for the price of spread call options under the log-normal models also.
Our formula can be seen as a generalization of the closed-form formula
presented in Bjerksund and Stensland [1] as their formula can be obtained by
selecting special parameter values to our formula. Numerical tests show that
our formula performs better for a certain range of model parameters than the
closed-form formula presented in Bjerksund and Stensland [1].

Keywords: Spread options; Log-normal models; Option pricing; Excise boundary

JEL Classifications: C00, C02, G10, G13

##  1 Introduction

One of the difficulties in pricing spread options is that the exercise
boundary is not in a linear form when the spread is not zero. When the spread
is zero, a spread option becomes an exchange option for which a closed form
formula exists. This formula is called Margrabe’s [17] formula in the
literature. When the spread 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 2.1 below. Due to this non-linear exercise boundary for spread
options, 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 (FTT)
which was introduced in Carr and Madan [8] initially for pricing European
options written on single assets. The idea of this FFT technique was later
extended to the case of pricing problems of spread options, see Dempster and
Hong [12] and Hurd and Zhou [14] for example. Both of these papers apply FFT
in bi-variate Fourier transformation setting. The recent papers Caldana and
Fusai [3, 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 Bjerksund and
Stensland [1]. By doing so they were able to give an approximate semi-closed
form formula for the spread option prices. Unlike the bi-variate Fourier
transformation technique used in spread option pricing in Dempster and Hong
[12] and Hurd and Zhou [9], the papers Caldana and Fusai [3, 4] were able to
express spread prices by a uni-variante Fourier inversion formula which is
easier and faster to compute.

Analytical methods offer closed form approximate formulas to spread options.
The first such approximation is the Bachelier approximation, see Wilcox [20]
and Poitras [18]. 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 price. However the closed form
formula does not give very accurate spread prices.

Another closed form formula for spread options were proposed by Kirk [16].
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, the Kirk’s
formula was obtained by using the similar approach in calculating Margrabe’s
formula [17].

The recent papers Carmona and Durrleman [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 Carmona and
Durrleman spread option formula gives very accurate spread option prices and
it performs better than the closed form formula introduced in Bjerksund and
Stensland [1]. This fact can also be seen by our Proposition 3.2 below as the
Carmona and Durrleman lowever bound for spread option price is always larger
than the value of the Bjerksund and Stensland formula. However, the only
drawback of the carmona and Durrleman approach is that one needs to solve a
nonlinear system of equations which is not trivial.

The recent paper Bjerksund and Stensland [1], gave another closed form formula
for spread option prices. They also introduced a lower bound to the spread
option price and computed this lower bound which then resulted in a closed
form formula for the spread option price. Numerical tests show that the
formula in Bjerksund and Stensland [1] gives more accurate spread prices than
the Kirk’s formula.

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 Bjerksund and Stensland [1], as
their formula can be obtained by plugging in some special parameter values to
our closed form formula. Numerical tests show that our formula performs better
than the closed formula in Bjerksund and Stensland [1] for a larger class of
parameter values of the model. Our formula involves three parameters that can
be picked freely depending on the model parameters.

The paper is organized as follows. In section 2 we describe our model and
present two propositions, proposition 2.1 and proposition 2.3, that are useful
for our discussions in the remaining sections of the paper. Especially, our
proposition 2.3 gives a numerical procedure which performs faster and more
accurate in computing spread option prices than the numerical methods
introduced in earlier papers. In section 3, we review existing closed-form
analytical formulas of spread options. Section 4 presents our main result of
this paper. In this section, we obtain a closed form formula for spread
options, see proposition 4.3, and show that the closed-form formula for spread
options presented in Bjerksund and Stensland [1] is just a special case of our
formula. Section 5 discusses Greeks that corresponds to our new formula and
section 6 presents the numerical performance of our formula. Section 7
concludes the paper. In Appendix 8, we present the proofs to our results.

##  2 The Model

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

$\begin{split}dS_{1}(t)=(r-r_{div}^{1})S_{1}(t)dt+\sigma_{1}S_{1}(t)dW_{1}(t)\\\
dS_{2}(t)=(r-r_{div}^{2})S_{2}(t)dt+\sigma_{2}S_{2}(t)dW_{2}(t)\end{split}$
(1)

were $r_{div}^{1}$ and $r_{div}^{2}$ are instantaneous dividend yields of the
two stocks respectively, $\sigma_{1}$ and $\sigma_{2}$ are positive constants
that represent the volatilises 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 [10, 13, 15, 19, 2, 5] and the references there.

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

$S_{1}(T)=F_{1}e^{-\frac{1}{2}\sigma^{2}_{1}T+\sigma_{1}W_{1}(T)},\;S_{2}(T)=F_{2}e^{-\frac{1}{2}\sigma^{2}_{2}T+\sigma_{2}W_{2}(T)}.$
(2)

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

$\Pi_{T}=\Pi_{T}(F_{1},\sigma_{1},F_{2},\sigma_{2},K,\rho)=e^{-rT}E^{Q}[(S_{1}(T)-S_{2}(T)-K)^{+}].$
(3)

Here we used a similar notation as in Carmona and Durrleman [7] for the spread
price. By using the parity relation discussed in Carmona and Durrleman [7], we
have

$\Pi_{T}(F_{1},\sigma_{1},F_{2},\sigma_{2},K,\rho)=\Pi_{T}(F_{2},\sigma_{2},F_{1},\sigma_{1},-K,\rho)+e^{-rT}(F_{1}-F_{2}-K).$
(4)

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 $Q$.

We remark that proposition 1 of Carmona and Durrleman [7] shows the following
interesting symmetry property for $\Pi_{T}$

$\begin{split}\Pi_{T}(F_{1},\sigma_{1},F_{2},\sigma_{2},K,\rho)&=\Pi_{T}(F_{1},\sigma,K,\sigma_{2},F_{2},\frac{\sigma_{2}-\rho\sigma_{1}}{\sigma})\\\
&=\Pi_{T}(-K,\sigma_{1},F_{2},\sigma,-F_{1},\frac{\sigma_{1}-\rho\sigma_{2}}{\sigma}),\\\
\end{split}$ (5)

where $\sigma=\sqrt{\sigma_{1}^{2}-2\rho\sigma_{1}\sigma_{2}+\sigma_{2}^{2}}$.
Therefore, a closed form formula for (3) can be used to derive a closed form
formula for the options on the sum of two assets

$e^{-rT}E^{Q}[(S_{1}(T)+S_{2}(T)-K)^{+}],$ (6)

by using the above symmetry relations.

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

$\Pi_{T}=e^{-rT}E[(F_{1}e^{-\frac{1}{2}\sigma^{2}_{1}T+\sigma_{1}\sqrt{T}Y}-F_{2}e^{-\frac{1}{2}\sigma^{2}_{2}T+\sigma_{2}\sqrt{T}X}-K)^{+}].$
(7)

Our goal in this paper is to obtain an approximate closed form formula for the
above equation (7).

There is no exact closed form solution exists for the spread option price
$\Pi_{T}$ other than for some special cases of model parameters. 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 Margrabe formula [17]. For general model parameters, there are some
closed form approximation formulas, see Kirk [16], Carmona and Durrleman [7,
6], and Bjerksund and Stensland [1].

Our method of derivation of a closed form formula for the spread option price
in this paper is different from the approaches in aforementioned papers. We
first express the spread price with probabilities of three events and then we
approximate these three probabilities to achieve a closed form formula.

Next we state few results that will be used in the later sections. The
following proposition expresses the price of the spread option by the
probabilities of three events under the risk-neutral measure

######  Proposition 2.1.

The price of the spread option is given by

$\Pi_{T}=e^{-rT}F_{1}C_{D}^{1}-e^{-rT}F_{2}C_{D}^{2}-e^{-rT}KC_{D}^{3},$ (8)

where

$\begin{split}C_{D}^{1}&=Q\left(g_{1}\bar{F}_{1}e^{\sigma_{1}\sqrt{T}Y}-\alpha\bar{F}_{2}e^{\sigma_{2}\sqrt{T}X}-K\geq
0\right),\\\
C_{D}^{2}&=Q\left(\alpha\bar{F}_{1}e^{\sigma_{1}\sqrt{T}Y}-g_{2}\bar{F}_{2}e^{\sigma_{2}\sqrt{T}X}-K\geq
0\right),\\\
C_{D}^{3}&=Q\left(\bar{F}_{1}e^{\sigma_{1}\sqrt{T}Y}-\bar{F}_{2}e^{\sigma_{2}\sqrt{T}X}-K\geq
0\right),\\\ \end{split}$

and $\alpha=e^{\rho\sigma_{1}\sigma_{2}T}$, $g_{1}=e^{\sigma_{1}^{2}T}$,
$g_{2}=e^{\sigma_{2}^{2}T}$,
$\bar{F}_{1}=F_{1}e^{-\frac{1}{2}\sigma_{1}^{2}T}$,
$\bar{F}_{2}=F_{2}e^{-\frac{1}{2}\sigma_{2}^{2}T}$.

The proof of this proposition will be given in the Appendix. We remark here
that the $C_{D}^{1}$ above represents the price of a digital call option with
payoff $1_{\\{S_{1}(T)-S_{2}(T)-K>0\\}}$ under the price dynamics in (1) with
initial stock prices given as $\bar{S}_{1}(0)=g_{1}S_{1}(0)$ and
$\bar{S}_{2}(0)=\alpha S_{2}(0)$. Similarly, the $C_{D}^{2}$ represents the
price of a digital call option with the same payoff and same price dynamics as
in (1) but with initial prices $\bar{S}_{1}(0)=\alpha S_{1}(0)$ and
$\bar{S}_{2}(0)=g_{2}S_{2}(0)$. The $C_{D}^{3}$ represents the price of a
digital call option with same payoff and with price dynamics as in (1) and
with initial prices $\bar{S}_{1}(0)=S_{1}(0)$ and $\bar{S}_{2}(0)=S_{2}(0)$.
Therefore, to obtain an approximate closed form formula for the spread call
option price, it is sufficient to find closed form approximate formulas for
digital options and this is the approach that we take in this paper.

The formula (8) has a simple form when $K=0$. In fact, when $K=0$, the formula
(8) reduces to Margrabe’s formula [17].

######  Corollary 2.2.

If we let $K=0$ in (8) we obtain the following Margrabe’s formula

$\Pi_{T}^{M}=e^{-rT}F_{1}\Phi\left(\frac{\ln\frac{g_{1}\bar{F}_{1}}{\alpha\bar{F}_{2}}}{\sqrt{T}\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}-2\sigma_{2}\sigma_{2}\rho}}\right)-e^{-rT}F_{2}\Phi\left(\frac{\ln\frac{\alpha\bar{F}_{1}}{g_{2}\bar{F}_{2}}}{\sqrt{T}\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}-2\sigma_{2}\sigma_{2}\rho}}\right)$

Our next result, gives a formula for spread call option price that performs
well for all the model parameters. We compared the performance of this formula
with the Monte Carlo approach in finding the price of spread call option price
and the results match very closely as can be seen in the corresponding table 1
in the Appendix.

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

######  Proposition 2.3.

For a sufficiently large positive integer $N>0$ and a sufficiently large
positive number $b>0$, the value $\Pi_{T}$ of the spread option approximately
equals to $\Pi_{T}^{d}$ that is given by as follows

$\begin{split}\Pi_{T}^{d}=&\;e^{-rT}F_{1}\underset{i=0}{\overset{N-1}{\Sigma}}\Phi\left(d_{i}^{1}\right)(\Phi(a_{i+1})-\Phi(a_{i}))\\\
-&e^{-rT}F_{2}\underset{i=0}{\overset{N-1}{\Sigma}}\Phi\left(d_{i}^{2}\right)(\Phi(a_{i+1})-\Phi(a_{i}))\\\
-&e^{-rT}K\underset{i=0}{\overset{N-1}{\Sigma}}\Phi\left(d_{i}^{3}\right)(\Phi(a_{i+1})-\Phi(a_{i})),\\\
\end{split}$ (9)

where for each $i=0,1,\cdots,N$, $a_{i}=b(\frac{2i}{N}-1)$, and

$\begin{split}d_{i}^{1}&=\frac{-ln\left(\frac{\alpha\bar{F}_{2}}{g_{1}\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}\theta_{i}}+\frac{K}{g_{1}\bar{F}_{1}}\right)+\sigma_{1}\rho\sqrt{T}\theta_{i}}{\sigma_{1}\sqrt{T}\sqrt{1-\rho^{2}}}\\\
d_{i}^{2}&=\frac{-ln\left(\frac{g_{2}\bar{F}_{2}}{\alpha\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}\theta_{i}}+\frac{K}{\alpha\bar{F}_{1}}\right)+\sigma_{1}\rho\sqrt{T}\theta_{i}}{\sigma_{1}\sqrt{T}\sqrt{1-\rho^{2}}}\\\
d_{i}^{3}&=\frac{-ln\left(\frac{\bar{F}_{2}}{\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}\theta_{i}}+\frac{K}{\bar{F}_{1}}\right)+\sigma_{1}\rho\sqrt{T}\theta_{i}}{\sigma_{1}\sqrt{T}\sqrt{1-\rho^{2}}}\\\
\end{split}$ (10)

with $\theta_{i}=\frac{a_{i}+a_{i+1}}{2}$ and with $\bar{F}_{1}$,
$\bar{F}_{2}$, $\alpha,g_{1},g_{2}$ are given as in proposition 2.1 above.

Here we used the notation $\Pi_{T}^{d}$ with $d$ to reflect the fact that the
above formula is obtained by discretizing the standard normal random
variables. As stated earlier, the above proposition 2.3 gives us very fast and
accurate prices for spread call options for all the model parameters. To
implement this formula one can choose $b=5$ and $N=3000$, and with these
parameters the approximated spread prices are very close to the spread prices
obtained by Monte Carlo methods.

##  3 Existing closed-form formulas

In the past various methods for pricing spread options were studied. Some of
the methods propose numerical approaches in obtaining spread option prices.
The most notable numerical method probably is the application of fast Fourier
transformation as in Dempster and Hong [11]. Some of the methods propose to
approximate the spread option price by some closed form formulas. The earliest
closed form formula for spread option is obtained by Bachelier approximation.
In the Bachelier approximation the discounted price spread
$e^{-rT}(S_{1}(T)-S_{2}(T))$ is approximated by a normal random variable
$N(\mu_{B},\sigma_{B}^{2})$ with

$\mu_{B}=S_{1}(0)-S_{2}(0),\;\;\sigma_{B}=\sqrt{S_{1}^{2}(0)(g_{1}-1)-2S_{1}(0)S_{2}(0)(g_{2}-1)+S_{2}^{2}(0)(\alpha-1)},$
(11)

where $g_{1},g_{2},\alpha,$ are given as in Proposition 8. After implementing
this approximation, the approximate spread option price becomes a European
call option and it is given by

$\Pi^{B}=(\mu_{B}-Ke^{-rK})\Phi(d_{B})+\sigma_{B}\phi(d_{B}),\;\;d_{B}=\frac{\mu_{B}-Ke^{-rT}}{\sigma_{B}}.$
(12)

This formula for spread option is simple and easy to implement. However, the
approximation of the discounted spread price by a normal random variable is
far from precise and therefore the Bachelors formula does not perform very
well in numerical tests.

In Kirk’s approximation the sum $S_{T}(2)+K$ of the second asset price
$S_{T}(2)$ with the constant strike $K$ is approximated by a log normal random
variable. This approach reduces the spread price with strike $K$ into an
exchange option and hence the spread price can be approximated by Margrabe’s
formula [17].

More precisely, if one implements the following approximation

$F_{2}e^{-\frac{1}{2}\sigma_{2}^{2}T+\sigma_{2}W_{2}(T)}+K\approx(F_{2}+K)e^{-\frac{1}{2}\theta^{2}T+\theta
W_{2}(T)},$ (13)

with $\theta=\frac{F_{2}}{F_{2}+K}\sigma_{2}$ to the spread price in (7) and
then applies the Margrabe’s formula one can obtain Kirk’s following formula

$\begin{split}\Pi_{T}^{K}&=E\big{[}F_{1}e^{-\frac{1}{2}\sigma_{1}^{2}T+\sigma_{1}W_{T}^{2}}-(F_{2}+K)e^{-\frac{1}{2}\theta^{2}T+\theta
W_{T}^{2}}\big{]}^{+}\\\
&=e^{-rT}\big{\\{}F_{1}\Phi(d_{K,1})+(F_{2}+K)\Phi(d_{K,2})\big{\\}},\\\
\end{split}$

where

$\begin{split}&d_{K,1}=\frac{\ln\frac{F_{1}}{F_{2}+K}+\frac{1}{2}\sigma_{K}^{2}T}{\sigma_{K}\sqrt{T}},\\\
&d_{K,2}=d_{K,1}-\sigma_{K}\sqrt{T},\\\
&\sigma_{K}=\sqrt{\sigma_{1}^{2}-2\frac{F_{2}}{F_{2}+K}\rho\sigma_{1}\sigma_{2}+\big{(}\frac{F_{2}}{F_{2}+K}\big{)}^{2}\sigma_{2}^{2}}.\end{split}$
(14)

######  Remark 3.1.

We remark that the approximation (13) uses the weighted average
$\frac{F_{2}}{F_{2}+K}\sigma_{2}+\frac{K}{F_{2}+K}0$ of the volatility
$\sigma_{2}$ of $\sigma_{2}W_{T}^{2}$ and the volatility $0$ (which
corresponds to K) for $\theta$ in (13).

A similar approximation to the one in Remark 3.1 above will be applied to
derive our main result in the next section.

The next important formula for spread price that we review in this section is
Carmona-Durrleman formula. The Carmona-Durrleman approach provides lower
bounds for spread option prices and the Carmona-Durrleman formula is obtained
by maximizing these lower bounds. Their formula gives very accurate spread
prices.

Before presenting the formula, we first recall that we are interested in
computing $EG^{+}$, where

$G=F_{1}e^{-\frac{1}{2}\sigma^{2}_{1}T+\sigma_{1}\sqrt{T}Y}-F_{2}e^{-\frac{1}{2}\sigma^{2}_{2}T+\sigma_{2}\sqrt{T}X}-K,$
(15)

and $Y,X$ are standard normal random variables with $Cov(Y,X)=\rho$, see (7)
and the arguments preceding to it for the details of this. The paper Carmona
and Durrleman [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 $Y=Z\sin\phi+Xcos\phi$, 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
$Y_{\theta}=Zsin\theta+Xcos\theta$ and calculate

$\Pi_{T}^{CD}(\theta,d)=E[(S_{1}(T)-S_{2}(T)-K)1_{\\{Y_{\theta}\leq d\\}}].$
(16)

Then the spread option price $\Pi_{T}$ in (7) is approximated by
$\hat{\Pi}_{T}^{CD}$ which is given by

$\Pi_{T}^{CD}=\sup_{\theta\in R}\sup_{d\in R}\Pi_{T}^{CD}(\theta,d).$ (17)

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

$\Pi_{T}^{CD}(\theta,d)=F_{1}\Phi(d+\sigma_{1}cos(\theta+\phi))-F_{2}\Phi(d+\sigma_{2}cos\theta)-K\Phi(d).\\\
$ (18)

So the problem of finding (17) reduces to maximizing the function on the right
hand side of (18) for $\theta$ and $d$. The first order conditions for
maximization gives the following two equations

$\begin{split}&F_{1}\varphi(d+\sigma_{1}cos(\theta+\phi))-F_{2}\varphi(d+\sigma_{2}cos\theta)-K\varphi(d)=0,\\\
&F_{1}\sigma_{1}sin(\theta+\phi)\varphi(d+\sigma_{1}cos(\theta+\phi))-F_{2}\sigma_{2}sin\theta\varphi(d+\sigma_{2}cos\theta)+K\varphi(d)=0.\\\
\end{split}$ (19)

These two equations can be solved for $\varphi(d+\sigma_{1}cos(\theta+\phi))$
and $\varphi(d+\sigma_{2}cos\theta)$ as follows

$\begin{split}\varphi(d+\sigma_{1}cos(\theta+\phi))&=\frac{\sigma_{2}sin\theta+1}{\sigma_{2}sin\theta-\sigma_{1}sin(\theta+\phi)}\frac{K\varphi(d)}{F_{1}}\\\
\varphi(d+\sigma_{2}cos\theta)&=\frac{1+\sigma_{1}sin(\theta+\phi)}{\sigma_{2}\sin\theta-\sigma_{1}sin(\theta+\phi)}\frac{K\varphi(d)}{F_{2}}.\end{split}$
(20)

Denoting the solution of (20) by $\theta^{\star}$ and $d^{\star}$, the spread
option price presented in Carmona and Durrlman [7] equals to

$\Pi_{T}^{CD}=e^{-rT}F_{1}\Phi(d^{\star}+\sigma_{1}cos(\theta^{\star}+\phi))-e^{-rT}F_{2}\Phi(d^{\star}+\sigma_{2}cos\theta^{\star})-e^{-rT}K\Phi(d^{\star}).$
(21)

Note here that the equation (20) above is equivalent to the equation (12) in
the paper Carmona and Durrlman [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 accurate spread prices. However, to obtain spread
prices one needs to solve the equation (20) for $d^{\star}$ and
$\theta^{\star}$ numerically. The paper Carmona and Durrlman [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 Bjerksund and Stensland [1]. The paper also introduced lower
bounds for the spread call option prices as follows

$\Pi_{T}^{BS}(a,b)=e^{-rT}E\big{[}(S_{1}(T)-S_{2}(T)-K)I\big{(}S_{1}(T)\geq\frac{a(S_{2}(T))^{b}}{E[S_{2}(T)]^{b}}\big{)}\big{]},$
(22)

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

$\Pi_{T}^{BS}=\sup_{a,b}\Pi_{T}^{BS}(a,b).$ (23)

In their paper the authors set

$a=F_{2}+K,\;\;\;b=\frac{F_{2}}{F_{2}+K}.$ (24)

By evaluating the equation (22), they obtained the following formula for the
spread call option price

$\Pi_{T}^{BS}=e^{-rT}[F_{1}\Phi(\bar{d}_{1})-F_{2}\Phi(\bar{d}_{2})-K\Phi(\bar{d}_{3})],$
(25)

where

$\begin{split}\bar{d}_{1}&=\frac{\ln\frac{F_{1}}{a}+(\frac{1}{2}\sigma_{1}^{2}+\frac{1}{2}b^{2}\sigma_{2}^{2}-b\rho\sigma_{1}\sigma_{2})T}{\sigma\sqrt{T}},\\\
\bar{d}_{2}&=\frac{\ln\frac{F_{1}}{a}+(-\frac{1}{2}\sigma_{1}^{2}-b\sigma_{2}^{2}+\frac{1}{2}b^{2}\sigma_{2}^{2}+\rho\sigma_{1}\sigma_{2})T}{\sigma\sqrt{T}},\\\
\bar{d}_{3}&=\frac{\ln\frac{F_{1}}{a}+(-\frac{1}{2}\sigma_{1}^{2}+\frac{1}{2}b^{2}\sigma_{2}^{2})T}{\sigma\sqrt{T}},\\\
\end{split}$ (26)

and

$\sigma=\sqrt{\sigma_{1}^{2}+b^{2}\sigma_{2}^{2}-2\rho
b\sigma_{1}\sigma_{2}}.$ (27)

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, we should mention
that the Carmona-Durrleman formula gives more accurate 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 (22) by using
$Y_{\theta}$ defined as in the paper Carmona and Durrlman [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 Carmona and Durrlman [7]. We state this
fact as a proposition and write down the proof in the Appendix.

######  Proposition 3.2.

We have

$\Pi_{T}^{BS}(a,b)=e^{-rT}E\big{[}(S_{1}(T)-S_{2}(T)-K)I_{\\{Y_{\theta_{0}}\leq
d_{0}\\}}],$ (28)

where the parameters $\theta_{0}$ and $d_{0}$ are given by the following
relations

$\sin\theta_{0}=\frac{-\sigma_{1}sin\phi}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}b^{2}-2\sigma_{1}\sigma_{2}bcos\phi}},\;\;\cos\theta_{0}=\frac{\sigma_{2}b-\sigma_{1}cos\phi}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}b^{2}-2\sigma_{1}\sigma_{2}bcos\phi}},$
(29)

where $\rho=\cos\phi$ and

$d_{0}=\frac{\ln\frac{F_{1}}{a}-\frac{1}{2}\sigma_{1}^{2}T+\frac{1}{2}b\sigma_{2}^{2}T}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}b^{2}-2\sigma_{1}\sigma_{2}bcos\phi}\sqrt{T}}.$
(30)

This shows the following relation

$\Pi_{T}^{BS}\leq\Pi_{T}^{CD}.$ (31)

We remark that the relation (31) shows that the Carmonna-Durrleman spread
option price is more accurate than the Bjerksund and Stensland spread option
price. The advantage of the Bjerksund-Stensland formula over the Carmona-
Durrleman formula however is that it is in closed-form and one does not need
numerical procedure to obtain the spread option price.

##  4 Generalization of the Bjerksund and Stensland formula

Our numerical tests show that, the Bjerksund and Stensland formula presented
in section 3 above performs better than Kirk’s formula in pricing spread
options. In this section, we study the Bjerksund and Stensland formula further
and give an alternative derivation for it. In fact, 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
4.5 below shows. In corollary 4.5 the Bjerksund and Stensland formula is
obtained by selecting special parameter values to our closed formula in this
section.

As pointed out in section 4 above, the Carmonna-Durrleman and Bjerksund and
Stensland 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 2.1 above.

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

$\begin{split}\mathcal{C}_{1}:&\;\;g_{1}\bar{F}_{1}e^{\sigma_{1}\sqrt{T}y}-\alpha\bar{F}_{2}e^{\sigma_{2}\sqrt{T}x}-K=0,\\\
\mathcal{C}_{2}:&\;\;\alpha\bar{F}_{1}e^{\sigma_{1}\sqrt{T}y}-g_{2}\bar{F}_{2}e^{\sigma_{2}\sqrt{T}x}-K=0,\\\
\mathcal{C}_{3}:&\;\;\bar{F}_{1}e^{\sigma_{1}\sqrt{T}y}-\bar{F}_{2}e^{\sigma_{2}\sqrt{T}x}-K=0.\\\
\end{split}$ (32)

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

$g_{1}\bar{F}_{1}e^{\sigma_{1}\sqrt{T}y}-\alpha\bar{F}_{2}e^{\sigma_{2}\sqrt{T}x}-K\geq
0,$ (33)

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

$Q((X,Y)\in\mathcal{C}_{i}^{+})$ (34)

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

The difficulty in obtaining closed form approximations for (34) above lies in
the fact 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 slopes $\kappa_{i}$ and
intersections $\delta_{i}$ for each $i=1,2,3,$ so that the following
approximations hold

$Q((x,y)\in\mathcal{C}_{i}^{+})\approx
Q(Y\geq\kappa_{i}X+\delta_{i}),i=1,2,3.$ (35)

If we can achieve (35), then the latter probabilities in (35) can be written
in closed form by using the following Lemma

######  Lemma 4.1.

For any two standard normal random variables $Y$ and $X$ with $Cov(Y,X)=\rho$,
and any three real numbers $m,n,\ell,$ we have

$Q(mY-nX\geq\ell)=\Phi(\frac{-\ell}{\sqrt{m^{2}+n^{2}-2\rho mn}}).$ (36)

The application of Lemma 4.1 above leads us to

$Q((x,y)\in\mathcal{C}_{i}^{+})\approx
Q(Y\geq\kappa_{i}X+\delta_{i})=\Phi\big{(}\frac{-\delta_{i}}{\sqrt{1+\kappa_{i}^{2}-2\rho\kappa_{i}}}\big{)},i=1,2,3,$
(37)

and this will give us closed-form formula for the spread option price due to
(8).

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 employ the following lines

$y=\kappa_{i}(x_{0})x+\delta_{i}(x_{0}),\;\;\delta_{i}(x_{0})=y_{0}-\kappa_{i}(x_{0})x_{0},\;i=1,2,3.$
(38)

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.

######  Lemma 4.2.

The three curves $\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}$ have the
following properties

  1. (a)

For all the three curves $\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}$ we
have

$\lim_{x\rightarrow+\infty}\frac{y}{x}=\frac{\sigma_{2}}{\sigma_{1}}.$

  2. (b)

When $x\rightarrow-\infty$, the limit $\lim_{x\rightarrow-\infty}y$ exists and

$\lim_{x\rightarrow-\infty}y=\left\\{\begin{array}[]{ll}\frac{1}{\sigma_{1}\sqrt{T}}\ln\frac{K}{\bar{F}_{1}}-\sigma_{1}\sqrt{T}&\mbox{for
$\mathcal{C}_{1}$},\\\
\frac{1}{\sigma_{1}\sqrt{T}}\ln\frac{K}{\bar{F}_{1}}-\rho\sigma_{2}\sqrt{T}&\mbox{for
$\mathcal{C}_{2}$},\\\
\frac{1}{\sigma_{1}\sqrt{T}}\ln\frac{K}{\bar{F}_{1}}&\mbox{for
$\mathcal{C}_{3}$}.\\\ \end{array}\right.$

Part (a) of the above lemma (4.2) shows that 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
$y=\frac{\sigma_{2}}{\sigma_{1}}x$. Part (b) gives lower bounds for each
curves when $x\rightarrow-\infty$.

We remark that the part (a) of the above lemma suggests us to use a fraction
(a multiplication by a positive number between $0$ and $2$) of the slope
$\frac{\sigma_{2}}{\sigma_{1}}$ to construct the lines in (38).

###  4.1 Alternative derivation of the Bjerksund and Stensland formula

The purpose of this section is to derive the Bjerksund and Stensland spread
option formula presented in Bjerksund and Stensland [1] by using the arguments
discussed in (35), (37), (38), above. In the next subsection, we apply a
similar approach as in this section and obtain a generalization of the
Bjerksund and Stensland spread option formula.

To start with, we consider two points $(x_{0}^{i},y_{0}^{i})$ and
$(x_{\ell}^{i},y_{\ell}^{i})$ on the curve $\mathcal{C}^{i}$ for each
$i=1,2,3$. The equation of the line that passes through these two points are
given by

$y^{i}=\frac{y_{\ell}^{i}-y_{0}^{i}}{x_{\ell}^{i}-x_{0}^{i}}x^{i}+y_{0}^{i}-\frac{y_{\ell}^{i}-y_{0}^{i}}{x_{\ell}^{i}-x_{0}^{i}}x_{0}^{i},i=1,2,3.$

From part (a) of Lemma 4.2, we see that when $x_{\ell}^{i}$ goes to $+\infty$,
$\frac{y_{\ell}^{i}-y_{0}^{i}}{x_{\ell}^{i}-x_{0}^{i}}$ monotonically
increases to $\frac{\sigma_{2}}{\sigma_{1}}$. So we introduce a parameter
$0<b\leq 1$ and define the following lines

$\ell^{i}:y^{i}=b\frac{\sigma_{2}}{\sigma_{1}}x^{i}+y_{0}^{i}-b\frac{\sigma_{2}}{\sigma_{1}}x_{0}^{i},i=1,2,3.$
(39)

Our goal is to use these lines $\ell_{i}$ in (37) to obtain a closed form
formula for the spread option. For this we need to discuss how to choose
appropriate points $(x_{0}^{i},y_{0}^{i})$. Since $(x_{0}^{i},y_{0}^{i})$ lies
on the curve $\mathcal{C}_{i}$ for each $i=1,2,3$, we plug in these points to
the equations of the three curves and obtain the expressions for
$y_{0}^{i},i=1,2,3,$ as follows

$\begin{split}y_{0}^{1}&=\frac{1}{\sigma_{1}\sqrt{T}}\ln\frac{F_{2}e^{\rho\sigma_{1}\sigma_{2}T+(\sigma_{2}\sqrt{T}x_{0}^{1}-\frac{1}{2}\sigma_{2}^{2}T)}+K}{F_{1}}-\frac{\sigma_{1}\sqrt{T}}{2}\\\
y_{0}^{2}&=\frac{1}{\sigma_{1}\sqrt{T}}\ln\frac{F_{2}e^{\sigma_{2}^{2}T+(\sigma_{2}\sqrt{T}x_{0}^{2}-\frac{1}{2}\sigma_{2}^{2}T)}+K}{F_{1}}+(\frac{1}{2}\sigma_{1}-\rho\sigma_{2})\sqrt{T}\\\
y_{0}^{3}&=\frac{1}{\sigma_{1}\sqrt{T}}\ln\frac{F_{2}e^{\sigma_{2}\sqrt{T}x_{0}^{1}-\frac{1}{2}\sigma_{2}^{2}T}+K}{F_{1}}+\frac{\sigma_{1}\sqrt{T}}{2}\\\
\end{split}$ (40)

To simply the expressions for $y_{0}^{i},i=1,2,3,$ above we apply the
following approximations

$\begin{split}F_{2}e^{\rho\sigma_{1}\sigma_{2}T+(\sigma_{2}\sqrt{T}x_{0}^{1}-\frac{1}{2}\sigma_{2}^{2}T)}+K&\approx
a_{1}e^{\rho\sigma_{1}q_{1}T+q_{1}\sqrt{T}x_{0}^{1}-\frac{1}{2}q^{2}_{1}T}\\\
F_{2}e^{\sigma_{2}^{2}T+(\sigma_{2}\sqrt{T}x_{0}^{2}-\frac{1}{2}\sigma_{2}^{2}T)}+K&\approx
a_{2}e^{q_{2}\sigma_{2}T+q_{2}\sqrt{T}x_{0}^{2}-\frac{1}{2}q^{2}_{2}T}\\\
F_{2}e^{\sigma_{2}\sqrt{T}x_{0}^{1}-\frac{1}{2}\sigma_{2}^{2}T}+K&\approx
a_{3}e^{q_{3}\sqrt{T}x_{0}^{1}-\frac{1}{2}q^{2}_{3}T}\\\ \end{split}$ (41)

for some appropriate $a_{1},a_{2},a_{3}$ and $q_{1},q_{2},q_{3}$. We remark
that the approximations (41) are similar to Kirk’s approximation that was
discussed in remark 3.1 above. For example, one can let

$a_{1}=a_{2}=a_{3}=F_{2}+K,q_{1}=q_{2}=q_{3}=\frac{F_{2}}{F_{2}+K}$ (42)

as in Kirk approximation.

We would like to implement the approximations (41) to (40) and obtain
approximated values for these points $y_{0}^{1},y_{0}^{2},y_{0}^{3}$. To this
end, we define the following functions

$\begin{split}z_{0}^{1}(q,a,h)&=\frac{1}{\sigma_{1}\sqrt{T}}\ln\frac{ae^{\rho\sigma_{1}qT+q\sqrt{T}h-\frac{1}{2}q^{2}T}}{F_{1}}-\frac{\sigma_{1}\sqrt{T}}{2},\\\
z_{0}^{2}(q,a,h)&=\frac{1}{\sigma_{1}\sqrt{T}}\ln\frac{ae^{q\sigma_{2}T+q\sqrt{T}h-\frac{1}{2}q^{2}T}}{F_{1}}+(\frac{1}{2}\sigma_{1}-\rho\sigma_{2})\sqrt{T},\\\
z_{0}^{3}(q,a,h)&=\frac{1}{\sigma_{1}\sqrt{T}}\ln\frac{ae^{q\sqrt{T}h-\frac{1}{2}q^{2}T}}{F_{1}}+\frac{\sigma_{1}\sqrt{T}}{2}.\\\
\end{split}$ (43)

Then with (41) and (42), we have

$\begin{split}y_{0}^{1}&\approx
z_{0}^{1}(\frac{F_{2}}{F_{2}+K},F_{2}+K,x_{0}^{1}),\;\;\;y_{0}^{2}\approx
z_{0}^{1}(\frac{F_{2}}{F_{2}+K},F_{2}+K,x_{0}^{2}),\\\ y_{0}^{3}&\approx
z_{0}^{1}(\frac{F_{2}}{F_{2}+K},F_{2}+K,x_{0}^{3}).\end{split}$ (44)

Now we are ready to state our next result. The following proposition shows
that the spread option price $\Pi^{BS}$ presented in Bjerksund and Stensland
[1] can also be obtained by using the ideas in relations (35) and (37) above.

######  Proposition 4.3.

We have

$\begin{split}\Pi^{BS}&=e^{-rT}F_{1}\Phi(\frac{-\delta_{1}}{\sqrt{1+\kappa^{2}-2\rho\kappa}})-e^{-rT}F_{2}\Phi(\frac{-\delta_{2}}{\sqrt{1+\kappa^{2}-2\rho\kappa}})\\\
&-e^{-rT}K\Phi(\frac{-\delta_{3}}{\sqrt{1+\kappa^{2}-2\rho\kappa}}),\\\
\end{split}$ (45)

with $\kappa=b\frac{\sigma_{2}}{\sigma_{1}}$ and
$\delta_{i}=z_{0}^{i}(b\sigma_{2},a,x_{0}^{i})-b\frac{\sigma_{2}}{\sigma_{1}}x_{0}^{i},i=1,2,3$,
where $z_{0}^{i}$ are given as in (43) and $x_{0}^{i}$ are any points.

We remark that the above relation (45) is obtained by using (37) and the line
equations given as in (38) with the points $(x_{0}^{i},y_{0}^{i})$ given by
$(x_{0}^{i},z_{0}^{i}(b\frac{\sigma_{2}}{\sigma_{1}},a))$ and these latter
points do not have to lie on the curves $\mathcal{C}_{i}$ for each $i=1,2,3$.
The proof of this proposition will be given in the appendix. It is also worthy
to note that in the above proposition the choices of $x_{0}^{i}$ are
irrelevant. It will be shown in the proof that, this is due to the choice of
the approximations (43).

###  4.2 Extension of the Bjerksund and Stensland formula

From the relations (35), (37), and (38) above we see that appropriate choices
of the slopes $\kappa_{i}$ and the points $(x_{0}^{i},y_{0}^{i})$ on the
corresponding curves $\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}$ are
important for the spread option pricing formula. From the above subsection, we
see that slopes of the form $\kappa_{i}=b\frac{\sigma_{2}}{\sigma_{1}}$ and
the points of the form $(x_{0}^{i},z_{0}^{i}(q,a))$ give us simple formula for
the spread prices. In this section, we elaborate on the choice of the
parameter $b$ in our Proposition 4.3 above further. Our analysis will enable
us to give a closed form formula for the spread option price which is more
general than the Bjerksund and Stensland spread option formula.

To this end, consider points $(x_{\ell}^{i},y_{\ell}^{i})$ on the curves
$\mathcal{C}_{i},i=1,2,3,$ respectively. By solving for $y_{\ell}^{i}$ for
each $i=1,2,3$ we obtain

$\begin{split}y_{\ell}^{1}&=\frac{1}{\sigma_{1}\sqrt{T}}\ln\frac{\alpha\bar{F}_{2}e^{\sigma_{2}\sqrt{T}x_{\ell}^{1}}+K}{g_{1}\bar{F}_{1}},\\\
y_{\ell}^{2}&=\frac{1}{\sigma_{1}\sqrt{T}}\ln\frac{g_{2}\bar{F}_{2}e^{\sigma_{2}\sqrt{T}x_{\ell}^{2}}+K}{\alpha\bar{F}_{1}},\\\
y_{\ell}^{3}&=\frac{1}{\sigma_{1}\sqrt{T}}\ln\frac{\bar{F}_{2}e^{\sigma_{2}\sqrt{T}x_{\ell}^{3}}+K}{\bar{F}_{1}}.\end{split}$
(46)

From Lemma (4.2) and Hopital’s rule we obtain

$\begin{split}\lim_{x_{\ell}^{1}\rightarrow\infty}\frac{y_{\ell}^{1}}{x_{\ell}^{1}}&=\lim_{x_{\ell}^{1}\rightarrow\infty}(y_{\ell}^{1})^{\prime}=\frac{\sigma_{2}}{\sigma_{1}}\lim_{x_{\ell}^{1}\rightarrow\infty}\frac{\alpha\bar{F}_{2}e^{\sigma_{2}\sqrt{T}x_{\ell}^{1}}}{\alpha\bar{F}_{2}e^{\sigma_{2}\sqrt{T}x_{\ell}^{1}}+K},\\\
\lim_{x_{\ell}^{2}\rightarrow\infty}\frac{y_{\ell}^{2}}{x_{\ell}^{2}}&=\lim_{x_{\ell}^{2}\rightarrow\infty}(y_{\ell}^{2})^{\prime}=\frac{\sigma_{2}}{\sigma_{1}}\lim_{x_{\ell}^{2}\rightarrow\infty}\frac{g_{2}\bar{F}_{2}e^{\sigma_{2}\sqrt{T}x_{\ell}^{2}}}{g_{2}\bar{F}_{2}e^{\sigma_{2}\sqrt{T}x_{\ell}^{2}}+K},\\\
\lim_{x_{\ell}^{3}\rightarrow\infty}\frac{y_{\ell}^{3}}{x_{\ell}^{3}}&=\lim_{x_{\ell}^{3}\rightarrow\infty}(y_{\ell}^{3})^{\prime}=\frac{\sigma_{2}}{\sigma_{1}}\lim_{x_{\ell}^{3}\rightarrow\infty}\frac{\bar{F}_{2}e^{\sigma_{2}\sqrt{T}x_{\ell}^{3}}}{\bar{F}_{2}e^{\sigma_{2}\sqrt{T}x_{\ell}^{3}}+K}.\\\
\end{split}$ (47)

Inspired by these relations, We define the following three functions

$b_{1}(x)=\frac{\alpha\bar{F}_{2}e^{\sigma_{2}\sqrt{T}x}}{\alpha\bar{F}_{2}e^{\sigma_{2}\sqrt{T}x}+K},\;\;b_{2}(x)=\frac{g_{2}\bar{F}_{2}e^{\sigma_{2}\sqrt{T}x}}{g_{2}\bar{F}_{2}e^{\sigma_{2}\sqrt{T}x}+K},\;\;b_{3}(x)=\frac{\bar{F}_{2}e^{\sigma_{2}\sqrt{T}x}}{\bar{F}_{2}e^{\sigma_{2}\sqrt{T}x}+K}.$
(48)

We observe that, these functions are strictly increasing on the real line and
takes values in the interval $(0,1)$.

We remark that from (47) we see that the slope of the asymptotic line of the
curve $\mathcal{C}_{i}$ is given by the limit of
$\frac{\sigma_{2}}{\sigma_{1}}b_{i}(x)$ when $x\rightarrow+\infty$ for each
$i=1,2,3$. We would like to replace the parameter $b$ in Proposition 5.2 by
the above $b_{1},b_{2},b_{3}$ in (48) and obtain a more general closed-form
spread option price formula. We also have to specify $a_{1},a_{2},a_{3}$. We
impose the constraints $a_{1}(x)b_{1}(x)=F_{2},a_{2}(x)b_{2}(x)=F_{2},$ and
$a_{3}(x)b_{3}(x)=F_{2}$, and obtain

$\begin{split}a_{1}(x)&=F_{2}+Ke^{-\sigma_{2}\sqrt{T}x-\rho\sigma_{1}\sigma_{2}T+\frac{1}{2}\sigma_{2}^{2}T},a_{2}(x)=F_{2}+Ke^{-\sigma_{2}\sqrt{T}x-\frac{1}{2}\sigma_{2}^{2}T},\\\
a_{3}(x)&=F_{2}+Ke^{-\sigma_{2}\sqrt{T}x+\frac{1}{2}\sigma_{2}^{2}T}.\end{split}$
(49)

Now we are ready to state our main result of this paper.

######  Proposition 4.4.

The price of the spread option can be approximated by

$\begin{split}\Pi_{T}(&\lambda,\mu,\gamma)=e^{-rT}F_{1}\Phi\big{(}\frac{-\delta_{1}(\lambda)}{\sqrt{1+\kappa_{1}^{2}(\lambda)-2\rho\kappa_{1}(\lambda)}}\big{)}\\\
&-e^{-rT}F_{2}\Phi\big{(}\frac{-\delta_{2}(\mu)}{\sqrt{1+\kappa_{2}^{2}(\mu)-2\rho\kappa_{2}(\mu)}}\big{)}-e^{-rT}K\Phi\big{(}\frac{-\delta_{3}(\gamma)}{\sqrt{1+\kappa_{3}^{2}(\gamma))-2\rho\kappa_{3}(\gamma)}}\big{)},\\\
\end{split}$ (50)

for some appropriately chosen parameters $\lambda,\mu,\gamma$, where

$\kappa_{1}(\lambda)=\frac{\sigma_{2}}{\sigma_{1}}b_{1}(\lambda),\;\kappa_{2}(\mu)=\frac{\sigma_{2}}{\sigma_{1}}b_{2}(\mu),\;\kappa_{3}(\gamma)=\frac{\sigma_{2}}{\sigma_{1}}b_{3}(\gamma),$
(51)

and

$\begin{split}\delta_{1}(\lambda)&=z_{0}^{1}[\sigma_{2}b_{1}(\lambda),a_{1}(\lambda),\lambda]-\kappa_{1}(\lambda)\lambda,\\\
\delta_{2}(\mu)&=z_{0}^{2}[\sigma_{2}b_{2}(\mu),a_{2}(\mu),\mu]-\kappa_{2}(\mu)\mu,\\\
\delta_{3}(\gamma)&=z_{0}^{3}[\sigma_{2}b_{3}(\gamma),a_{3}(\gamma),\gamma]-\kappa_{3}(\gamma)\gamma,\end{split}$
(52)

and the functions $z_{0}^{i}(\cdot),i=1,2,3,$ are given as in (43), and
$b_{i}(\cdot),i=1,2,3,$ are given as in (48).

We remark that our closed form formula in the above Proposition depends on the
three parameters $\lambda,\mu,$ and $\gamma$. If we choose the following
parameter values

$\begin{split}\lambda&=(\frac{1}{2}\sigma_{2}-\rho\sigma_{1})\sqrt{T},\\\
\mu&=-\frac{1}{2}\sigma_{2}\sqrt{T},\\\
\gamma&=\frac{1}{2}\sigma_{2}\sqrt{T},\end{split}$ (53)

our formula in Proposition 4.4 above reduces to the Bjerksund and Stensland
formula. We state this fact as a Corollary below

######  Corollary 4.5.

We have

$\Pi^{BS}_{T}=\Pi_{T}((\frac{1}{2}\sigma_{2}-\rho\sigma_{1})\sqrt{T},-\frac{1}{2}\sigma_{2}\sqrt{T},\frac{1}{2}\sigma_{2}\sqrt{T}).$
(54)

##  5 The Greek parameters

The Greeks of option pricing formulas are important for risk management and
hedging issues. In this section, we calculate the Greek parameters for our
formula. First, observe that our formula (50) can also be written as follows

$\Pi_{T}(\lambda,\mu,\gamma)=e^{-rT}F_{1}\Phi(I)-e^{-rT}F_{2}\Phi(J)-e^{-rT}K\Phi(H),$
(55)

where

$\begin{split}I=&\frac{ln(\frac{F_{1}}{a_{1}})+(\frac{1}{2}\sigma_{1}^{2}-\rho\sigma_{1}\sigma_{2}b_{1}+\frac{1}{2}\sigma_{2}^{2}b_{1}^{2})T}{\bar{\sigma}_{1}\sqrt{T}},\\\
J=&\frac{ln(\frac{F_{1}}{a_{2}})+(-\frac{1}{2}\sigma_{1}^{2}+\rho\sigma_{1}\sigma_{2}+\frac{1}{2}\sigma_{2}^{2}b_{2}^{2}-\sigma_{2}^{2}b_{2})T}{\bar{\sigma}_{2}\sqrt{T}},\\\
H=&\frac{ln(\frac{F_{1}}{a_{3}})+(-\frac{1}{2}\sigma_{1}^{2}+\frac{1}{2}\sigma_{2}^{2}b_{3}^{2})T}{\bar{\sigma}_{3}\sqrt{T}},\end{split}$
(56)

with

$\begin{split}\bar{\sigma}_{1}=&\sqrt{\sigma_{1}^{2}-2\rho\sigma_{1}\sigma_{2}b_{1}+\sigma_{2}^{2}b_{1}^{2}},\\\
\bar{\sigma}_{2}=&\sqrt{\sigma_{1}^{2}-2\rho\sigma_{1}\sigma_{2}b_{2}+\sigma_{2}^{2}b_{2}^{2}},\\\
\bar{\sigma}_{3}=&\sqrt{\sigma_{1}^{2}-2\rho\sigma_{1}\sigma_{2}b_{3}+\sigma_{2}^{2}b_{3}^{2}},\\\
\end{split}$ (57)

and the functions $b_{1}(\lambda),b_{2}(\mu),b_{3}(\gamma)$ are given as in
(48).

Clearly, this formula is similar to the Bjerksund and Stensland formula (26)
above. To find the Greeks, we assume $b_{1},b_{2},b_{3}$ are constants as in
Bjerksund and Stensland [1] and $a_{1},a_{2},a_{3}$ are given as in (49).
Under this assumptions, we calculate the Greeks as follows.

The delta parameters of our formula are given as follows

$\begin{split}\frac{\partial\Pi_{T}}{\partial
F_{1}}=&e^{-rT}\Phi(I)+e^{-rT}\left\\{\frac{F_{1}\phi(I)}{\bar{\sigma}_{1}\sqrt{T}}-\frac{F_{2}\phi(J)}{\bar{\sigma}_{2}\sqrt{T}}-\frac{K\phi(H)}{\bar{\sigma}_{3}\sqrt{T}}\right\\}\frac{1}{F_{1}},\\\
\frac{\partial\Pi_{T}}{\partial
F_{2}}=&-e^{-rT}\Phi(J)-e^{-rT}\left\\{\frac{F_{1}\phi(I)}{\bar{\sigma}_{1}\sqrt{T}}\frac{a_{1}^{\prime}}{a_{1}}-\frac{F_{2}\phi(J)}{\bar{\sigma}_{2}\sqrt{T}}\frac{a_{2}^{\prime}}{a_{2}}-\frac{K\phi(H)}{\bar{\sigma}_{3}\sqrt{T}}\frac{a_{3}^{\prime}}{a_{3}}\right\\}\\\
=&-e^{-rT}\Phi(J)-e^{-rT}\left\\{\frac{F_{1}\phi(I)}{\bar{\sigma}_{1}\sqrt{T}}\frac{b_{1}}{F_{2}}-\frac{F_{2}\phi(J)}{\bar{\sigma}_{2}\sqrt{T}}\frac{b_{2}}{F_{2}}-\frac{K\phi(H)}{\bar{\sigma}_{3}\sqrt{T}}\frac{b_{3}}{F_{2}}\right\\}.\end{split}$
(58)

The Gamma parameters of our formula are given as follows

$\displaystyle\begin{split}\frac{\partial^{2}\Pi_{T}}{\partial
F_{1}^{2}}=&2e^{-rT}\phi(I)\frac{1}{\bar{\sigma}_{1}\sqrt{T}F_{1}}\\\
-&e^{-rT}\left\\{F_{1}\phi(I)I\frac{1}{\bar{\sigma}_{1}^{2}T}-F_{2}\phi(J)J\frac{1}{\bar{\sigma}_{2}^{2}T}-K\phi(H)H\frac{1}{\bar{\sigma}_{3}^{2}T}\right\\}\frac{1}{F_{1}^{2}}\\\
-&e^{-rT}\left\\{\frac{F_{1}\phi(I)}{\bar{\sigma}_{1}\sqrt{T}}-\frac{F_{2}\phi(J)}{\bar{\sigma}_{2}\sqrt{T}}-\frac{K\phi(H)}{\bar{\sigma}_{3}\sqrt{T}}\right\\}\frac{1}{F_{1}^{2}},\\\
\frac{\partial^{2}\Pi_{T}}{\partial F_{1}\partial
F_{2}}=&-e^{-rT}\phi(I)\frac{a_{1}^{\prime}}{a_{1}\bar{\sigma}_{1}\sqrt{T}}-e^{-rT}\phi(J)\frac{1}{F_{1}\bar{\sigma}_{2}\sqrt{T}}\\\
+&e^{-rT}\left\\{\frac{F_{1}\phi(I)I}{\bar{\sigma}_{1}^{2}T}\frac{a_{1}^{\prime}}{a_{1}}-\frac{F_{2}\phi(J)J}{\bar{\sigma}_{2}^{2}T}\frac{a_{2}^{\prime}}{a_{2}}-\frac{K\phi(H)H}{\bar{\sigma}_{3}^{2}T}\frac{a_{3}^{\prime}}{a_{3}}\right\\}\frac{1}{F_{1}}\\\
=&-e^{-rT}\phi(I)\frac{b_{1}}{F_{2}\bar{\sigma}_{1}\sqrt{T}}-e^{-rT}\phi(J)\frac{1}{F_{1}\bar{\sigma}_{2}\sqrt{T}}\\\
+&e^{-rT}\left\\{\frac{F_{1}\phi(I)I}{\bar{\sigma}_{1}^{2}T}\frac{b_{1}}{F_{2}}-\frac{F_{2}\phi(J)J}{\bar{\sigma}_{2}^{2}T}\frac{b_{2}}{F_{2}}-\frac{K\phi(H)H}{\bar{\sigma}_{3}^{2}T}\frac{b_{3}}{F_{2}}\right\\}\frac{1}{F_{1}},\\\
\end{split}$ (59)
$\displaystyle\begin{split}\frac{\partial^{2}\Pi_{T}}{\partial
F_{2}^{2}}=&2e^{-rT}\frac{\phi(J)}{\bar{\sigma}_{2}\sqrt{T}}\frac{a_{2}^{\prime}}{a_{2}}\\\
-e^{-rT}&\left\\{\frac{F_{1}\phi(I)I}{\bar{\sigma}_{1}^{2}T}(\frac{a_{1}^{\prime}}{a_{1}})^{2}-\frac{F_{2}\phi(J)J}{\bar{\sigma}_{2}^{2}T}(\frac{a_{2}^{\prime}}{a_{2}})^{2}-\frac{K\phi(H)H}{\bar{\sigma}_{3}^{2}T}(\frac{a_{3}^{\prime}}{a_{3}})^{2}\right\\}\\\
-e^{-rT}&\left\\{\frac{F_{1}\phi(I)}{\bar{\sigma}_{1}\sqrt{T}}\frac{a_{1}^{\prime\prime}a_{1}-(a_{1}^{\prime})^{2}}{a_{1}^{2}}-\frac{F_{2}\phi(J)}{\bar{\sigma}_{2}\sqrt{T}}\frac{a_{2}^{\prime\prime}a_{2}-(a_{2}^{\prime})^{2}}{a_{2}^{2}}-\frac{K\phi(H)}{\bar{\sigma}_{3}\sqrt{T}}\frac{a_{3}^{\prime\prime}a_{3}-(a_{3}^{\prime})^{2}}{a_{3}^{2}}\right\\}\\\
=2e^{-rT}&\frac{\phi(J)}{\bar{\sigma}_{2}\sqrt{T}}\frac{b_{2}}{F_{2}}-e^{-rT}\left\\{\frac{F_{1}\phi(I)I}{\bar{\sigma}_{1}^{2}T}(\frac{b_{1}}{F_{2}})^{2}-\frac{F_{2}\phi(J)J}{\bar{\sigma}_{2}^{2}T}(\frac{b_{2}}{F_{2}})^{2}-\frac{K\phi(H)H}{\bar{\sigma}_{3}^{2}T}(\frac{b_{3}}{F_{2}})^{2}\right\\}\\\
+&e^{-rT}\left\\{\frac{F_{1}\phi(I)}{\bar{\sigma}_{1}\sqrt{T}}(\frac{b_{1}}{F_{2}})^{2}-\frac{F_{2}\phi(J)}{\bar{\sigma}_{2}\sqrt{T}}(\frac{b_{2}}{F_{2}})^{2}-\frac{K\phi(H)}{\bar{\sigma}_{3}\sqrt{T}}(\frac{b_{3}}{F_{2}})^{2}\right\\}.\\\
\end{split}$ (60)

The Vegas are given by

$\begin{split}\frac{\partial\Pi_{T}}{\partial\sigma_{1}}=&e^{-rT}\left\\{F_{1}\phi(I)\frac{(\sigma_{1}-\rho\sigma_{2}b_{1})T}{\bar{\sigma}_{1}\sqrt{T}}+F_{2}\phi(J)\frac{(\sigma_{1}-\rho\sigma_{2})T}{\bar{\sigma}_{2}\sqrt{T}}+K\phi(H)\frac{\sigma_{1}T}{\bar{\sigma}_{3}\sqrt{T}}\right\\}\\\
-&e^{-rT}\left\\{F_{1}\phi(I)I\frac{\sigma_{1}-\rho\sigma_{2}b_{1}}{\bar{\sigma}_{1}^{2}}-F_{2}\phi(J)J\frac{\sigma_{1}-\rho\sigma_{2}b_{2}}{\bar{\sigma}_{2}^{2}}-K\phi(H)H\frac{\sigma_{1}-\rho\sigma_{2}b_{3}}{\bar{\sigma}_{3}^{2}}\right\\}.\\\
\frac{\partial\Pi_{T}}{\partial\sigma_{2}}=&\\\
e^{-rT}&\left\\{F_{1}\phi(I)\frac{(-\rho\sigma_{1}b_{1}+\sigma_{2}b_{1}^{2})T}{\bar{\sigma}_{1}\sqrt{T}}-F_{2}\phi(J)\frac{(\rho\sigma_{1}+\sigma_{2}b_{2}^{2}-2\sigma_{2}b_{2})T}{\bar{\sigma}_{2}\sqrt{T}}-K\phi(H)\frac{\sigma_{2}b_{3}^{2}T}{\bar{\sigma}_{3}\sqrt{T}}\right\\}-\\\
e^{-rT}&\left\\{F_{1}\phi(I)I\frac{-\rho\sigma_{1}b_{1}+\sigma_{2}b_{1}^{2}}{\bar{\sigma}_{1}^{2}}-F_{2}\phi(J)J\frac{-\rho\sigma_{1}b_{2}+\sigma_{2}b_{2}^{2}}{\bar{\sigma}_{2}^{2}}-K\phi(H)H\frac{-\rho\sigma_{1}b_{3}+\sigma_{2}b_{3}^{2}}{\bar{\sigma}_{3}^{2}}\right\\}.\\\
\end{split}$ (61)

The $\Theta$ of our formula is given by

$\begin{split}\frac{\partial\Pi_{T}}{\partial T}=&-r\Pi_{T}\\\
+&e^{-rT}\left\\{F_{1}\phi(I)\frac{\frac{1}{2}\sigma_{1}^{2}-\rho\sigma_{1}\sigma_{2}b_{1}+\frac{1}{2}\sigma_{2}^{2}b_{1}^{2}}{\bar{\sigma}_{1}\sqrt{T}}-F_{2}\phi(J)\frac{-\frac{1}{2}\sigma_{1}^{2}+\rho\sigma_{1}\sigma_{2}+\frac{1}{2}\sigma_{2}^{2}b_{2}^{2}-\sigma_{2}^{2}b_{2}}{\bar{\sigma}_{2}\sqrt{T}}\right.\\\
&\left.-K\phi(H)\frac{-\frac{1}{2}\sigma_{1}^{2}+\frac{1}{2}\sigma_{2}^{2}b_{3}^{2}}{\bar{\sigma}_{3}\sqrt{T}}\right\\}-e^{-rT}\\{F_{1}\phi(I)I-F_{2}\phi(J)J-K\phi(H)H\\}\frac{1}{2T}.\\\
\end{split}$ (62)

The $\rho$ is given by

$\frac{\partial\Pi_{T}}{\partial
r}=-T\\{e^{-rT}F_{1}\Phi(I)-e^{-rT}F_{2}\Phi(J)-e^{-rT}K\Phi(H)\\}=-T\Pi_{T}.$
(63)

The partial derivative w.r.t $\rho$ is given by

$\begin{split}\frac{\partial\Pi_{T}}{\partial\rho}=&e^{-rT}\left\\{-F_{1}\phi(I)\frac{b_{1}\sigma_{1}\sigma_{2}T}{\bar{\sigma}_{1}\sqrt{T}}-F_{2}\phi(J)\frac{\sigma_{1}\sigma_{2}T}{\bar{\sigma}_{2}\sqrt{T}}\right\\}\\\
+&e^{-rT}\left\\{\frac{F_{1}\phi(I)Ib_{1}}{\bar{\sigma_{1}^{2}}}-\frac{F_{2}\phi(J)Jb_{2}}{\bar{\sigma}_{2}^{2}}-\frac{K\phi(H)Hb_{3}}{\bar{\sigma}_{3}^{2}}\right\\}\sigma_{1}\sigma_{2}.\\\
\end{split}$ (64)

It can be verified that the Greeks satisfy the following partial differential
equations:

$\frac{1}{2}\sigma_{1}^{2}F_{1}^{2}\frac{\partial^{2}\Pi_{T}}{\partial
F_{1}^{2}}+\rho\sigma_{1}F_{1}\sigma_{2}F_{2}\frac{\partial^{2}\Pi_{T}}{\partial
F_{1}\partial
F_{2}}+\frac{1}{2}\sigma_{2}^{2}F_{2}^{2}\frac{\partial^{2}\Pi_{T}}{\partial
F_{2}^{2}}-\frac{\partial\Pi_{T}}{\partial T}=r\Pi_{T}.$ (65)

and

$\frac{1}{2}\sigma_{1}\frac{\partial\Pi_{T}}{\partial\sigma_{1}}+\frac{1}{2}\sigma_{2}\frac{\partial\Pi_{T}}{\partial\sigma_{2}}+r\frac{\partial\Pi_{T}}{\partial
r}=T\frac{\partial\Pi_{T}}{\partial T}.$ (66)

We remark that the above calculations are direct and similar to the
calculations of Greeks of the corresponding formula in Bjerksund and Stensland
[1]. Thus we will skip the details of calculations of them in the Appendix.

##  6 Numerical results

In this section, we present the numerical performance of our main result
Proposition 4.4 and of the discretizing method in Proposition 2.3 above. We
first compare the performance of the Proposition 2.3 with Monte Carlo(MC) in
table 1 below. The Monte Carlo obtained spread prices are used as the true
spread option prices here. In our comparison, we continue to use the same
model parameters as in the numerical examples in Carmon-Durrleman [7] and
Bjerksund-Stensland [1]. More specifically, we pick the following model
parameters $r=0.05,T=1,F_{1}=112.22,F_{2}=103.05,\sigma_{1}=0.1,$ and
$\sigma_{2}=0.15$ in our numerical tests.

As in [1], for the same combinations of K and $\rho$, and for those cases
$K<0$, an opposite spread payoff ($S_{2}-S_{1}$) was applied for a put option.
The put-call parity was used to obtain the put value. For all these set of
parameters, the spread option prices were calculated by Monte Carlo
simulation, Proposition 2.3, Kirk’s approximation, Bjerksund-Stensland
approximation, and Proposition 4.3 separately. The numerical computations for
all these were implemented in Python 3.8.8 on a Dell desktop PC with 2.30GHz
Intel Core CPU.

In the numerical tests, there is some flexibility in choosing
$\lambda,\mu,\gamma,$ in the spread prices $\Pi_{T}(\lambda,\mu,\gamma)$
presented in Proposition 4.4. For our numerical tests in this section we
picked
$\lambda=(\frac{1}{2}\sigma_{2}-\rho\sigma_{1})+\sqrt{|\sigma_{2}-\sigma_{1}|}/3$
and determined the $\mu$ and $\gamma$ from the relation
$b_{1}(\lambda)=b_{2}(\mu)=b_{3}(\gamma)$. Clearly, this is not optimal choice
for these parameters $\lambda,\mu,\gamma$. Here we only attempt to show that
our alternative formula is comparable to the spread option formula in
Bjerksund-Stensland [1].

The Table 1 below compares MC simulation results with 100000 paths to the
formula in Proposition 2.3 with $b=5$ and $N=3000$. It can be seen that the
formula in Proposition 2.3 gives highly accurate spread option prices as they
are very close to the spread prices obtained by the MC simulation. We also
observe that the speed of this method is faster than the Monte Carlo approach.
The time consumption was recorded for both of these two methods and the MC
took 67.41s and the discrete method in Proposition 2.3 took 40.58s with $b=5$
and $N=3000$. Therefore, in the following numerical tests we use the formula
in Proposition 2.3 as true spread prices.

Table 1: Comparison of Monte Carlo and Proposition 2.3

K $\rho$ | -0.99 | -0.5 | 0 | 0.3 | 0.8 | 0.99
---|---|---|---|---|---|---
-20 | 29.64274378 | 28.99479250 | 28.38110019 | 28.07014138 | 27.7700518 | 27.75380043
| 29.64274375 | 28.99479248 | 28.38110018 | 28.07014139 | 27.770518 | 27.75380040
-10 | 21.84990851 | 20.90504376 | 19.88891999 | 19.27012445 | 18.38114096 | 18.24534835
| 21.84990830 | 20.90504291 | 19.88892000 | 19.27012446 | 18.38114087 | 18.24534802
0 | 15.13321639 | 13.9180430 | 12.52373725 | 11.56183922 | 9.63252498 | 8.82120140
| 15.13321639 | 13.91800429 | 12.52373725 | 11.56183922 | 9.63252501 | 8.82120141
5 | 12.22000018 | 10.95620431 | 9.44533360 | 8.36741249 | 5.96702485 | 4.54620032
| 12.22000016 | 10.95620391 | 9.44533361 | 8.36741249 | 5.96702488 | 4.54620020
15 | 7.49782906 | 6.24223330 | 4.7444690 | 3.67981792 | 1.3424512 | 0.10249913
| 7.49782906 | 6.24223324 | 4.7444681 | 3.67981791 | 1.3424516 | 0.10249892
25 | 4.18077521 | 3.13000435 | 1.96209910 | 1.21996422 | 0.10413210 | 0.00000000
| 4.1807518 | 3.13000410 | 1.96209915 | 1.21996421 | 0.10413200 | 0.00000000

  * •

The first row corresponds to the Monte Carlo simulation method with 100000
paths, the total time consumed is 67.41s. The second row corresponds to
Proposition 2.3, the total time consumed is 40.58s.

The following tables compare the performance of the Kirk’s formula, the
Bjerksund-Stensland’s formula, and the Proposition 4.3 above. In the first
Table 2, we use the same model parameters as in Bjerksund-Stensland [1]. For
this set of parameters, Bjerksund-Stensland’s formula gives a mean relative
error of 0.000522 and our Proposition 4.3 gives a mean relative error of
0.000549.

Table 2: The same model parameters as in [1]

K $\rho$ | -0.99 | -0.5 | 0 | 0.3 | 0.8 | 0.99
---|---|---|---|---|---|---
-20 | 29.6427 | 28.9948 | 28.3811 | 28.0701 | 27.7701 | 27.7538
| 29.6616 | 29.0056 | 28.3848 | 28.0709 | 27.7704 | 27.7539
| 29.6427 | 28.9948 | 28.3811 | 28.0701 | 27.7701 | 27.7538
| 29.6427 | 28.9948 | 28.3811 | 28.0701 | 27.7701 | 27.7538
-10 | 21.8499 | 20.9050 | 19.8889 | 19.2701 | 18.3811 | 18.2453
| 21.8600 | 20.9114 | 19.8917 | 19.2710 | 18.3816 | 18.2460
| 21.8499 | 20.9049 | 19.8888 | 19.2701 | 18.3811 | 18.2453
| 21.8499 | 20.9049 | 19.8888 | 19.2701 | 18.3811 | 18.2453
0 | 15.1332 | 13.9180 | 12.5237 | 11.5618 | 9.6325 | 8.8212
| 15.1332 | 13.9180 | 12.5237 | 11.5618 | 9.6325 | 8.8212
| 15.1332 | 13.9180 | 12.5237 | 11.5618 | 9.6325 | 8.8212
| 15.1332 | 13.9180 | 12.5237 | 11.5618 | 9.6325 | 8.8212
5 | 12.2200 | 10.9562 | 9.4453 | 8.3674 | 5.9670 | 4.5462
| 12.2183 | 10.9543 | 9.4431 | 8.3649 | 5.9628 | 4.5353
| 12.2200 | 10.9562 | 9.4453 | 8.3674 | 5.9670 | 4.5462
| 12.2200 | 10.9562 | 9.4453 | 8.3674 | 5.9670 | 4.5462
15 | 7.4978 | 6.2422 | 4.7445 | 3.6798 | 1.3425 | 0.1025
| 7.5135 | 6.2559 | 4.7562 | 3.6907 | 1.3545 | 0.1257
| 7.4977 | 6.2421 | 4.7443 | 3.6796 | 1.3421 | 0.1017
| 7.4977 | 6.2421 | 4.7443 | 3.6796 | 1.3421 | 0.1016
25 | 4.1808 | 3.1300 | 1.9621 | 1.2200 | 0.1041 | 0.0000
| 4.2268 | 3.1686 | 1.9923 | 1.2441 | 0.1124 | 0.0000
| 4.1807 | 3.1298 | 1.9617 | 1.2194 | 0.1032 | 0.0000
| 4.1807 | 3.1298 | 1.9617 | 1.2194 | 0.1032 | 0.0000

  * •

The first row corresponds to the formula in Proposition 2.3 above

  * •

The second row corresponds to Kirk’s formula

  * •

The third row corresponds to Bjerksund-Stensland’s formula

  * •

The fourth row corresponds to Proportion 4.3 above.

In our next numerical test, which is Table 3 below, we use the same model
parameters as in Bjerksund-Stensland [1] except that now we take
$\sigma_{2}=0.9$ keeping the other parameters the same. For this set of model
parameters, Bjerksund-Stensland’s formula gives a relative error of 0.00125
and our Proposition 4.3 gives a relative error of 0.000017. So in this case,
our formula improves the performance of the Bjerksund-Stensland’s formula.

Table 3: The same parameters as in [1] except now $\sigma_{2}=0.9$

K $\rho$ | -0.99 | -0.5 | 0 | 0.3 | 0.8 | 0.99
---|---|---|---|---|---|---
-20 | 57.3140 | 55.8463 | 54.2664 | 53.2739 | 51.5357 | 50.8447
| 57.3316 | 55.8520 | 54.2665 | 53.2747 | 51.5462 | 50.8623
| 57.3140 | 55.8463 | 54.2664 | 53.2739 | 51.5357 | 50.8447
| 57.3140 | 55.8463 | 54.2664 | 53.2739 | 51.5357 | 50.8447
-10 | 50.3507 | 48.8007 | 47.1273 | 46.0732 | 44.2212 | 43.4827
| 50.3596 | 48.8036 | 47.1274 | 46.0735 | 44.2264 | 43.4916
| 50.3507 | 48.8007 | 47.1273 | 46.0732 | 44.2212 | 43.4827
| 50.3507 | 48.8007 | 47.1273 | 46.0732 | 44.2212 | 43.4827
0 | 43.6688 | 42.0392 | 40.2738 | 39.1582 | 37.1908 | 36.4034
| 43.6688 | 42.0392 | 40.2738 | 39.1582 | 37.1908 | 36.4034
| 43.6688 | 42.0392 | 40.2738 | 39.1582 | 37.1908 | 36.4034
| 43.6688 | 42.0392 | 40.2738 | 39.1582 | 37.1908 | 36.4034
5 | 40.4445 | 38.7772 | 36.9676 | 35.8220 | 33.7973 | 32.9853
| 40.5532 | 38.8694 | 37.0419 | 35.8848 | 33.8393 | 33.0186
| 40.4384 | 38.7708 | 36.9608 | 35.8150 | 33.7897 | 32.9773
| 40.4365 | 38.7699 | 36.9608 | 35.8155 | 33.7912 | 32.9793
15 | 34.2540 | 32.5183 | 30.6268 | 29.4246 | 27.2897 | 26.4293
| 34.7191 | 32.9457 | 31.0149 | 29.7885 | 27.6115 | 26.7342
| 34.2116 | 32.4737 | 30.5791 | 29.3744 | 27.2339 | 26.3706
| 34.1967 | 32.4662 | 30.5791 | 29.3790 | 27.2465 | 26.3866
25 | 28.4438 | 26.6541 | 24.6953 | 23.4450 | 21.2123 | 20.3073
| 29.4191 | 27.5847 | 25.5840 | 24.3110 | 22.0454 | 21.1300
| 28.3588 | 26.5640 | 24.5981 | 23.3423 | 21.0965 | 20.1847
| 28.3241 | 26.5466 | 24.5981 | 23.3529 | 21.1263 | 20.2226

  * •

The first row corresponds to the formula in Proposition 2.3 above.

  * •

The second row corresponds to Kirk’s formula

  * •

The third row corresponds to the Bjerksund-Stensland’s formula

  * •

The fourth row corresponds to our Proportion 4.3

##  7 Conclusion

In this paper, we present a closed form formula for the prices of spread
options under log normal models. Our formula involves three parameters and it
reduces to the Bjerksund-Stensland formula for certain values of these
parameters. Our approach in deriving this formula is different from the method
used in the paper of Bjerksund and Stensland. We made use the slope of the
asymptotic lines of the exercise boundary curves to construct a closed form
formula for spread option prices. Our numerical tests show that our formula
performs better for a certain range of model parameters than the Bjerksund-
Stensland formula.

In our proposition 2.3, we provide a formula that uses numerical procedure in
computing spread option prices. Numerical tests show that this method is
highly accurate and much faster than the MC method in computing spread option
prices.

##  8 Appendix

Proof of Proposition 2.1: We have

$\begin{split}&E[(F_{1}e^{-\frac{1}{2}\sigma^{2}_{1}T+\sigma_{1}\sqrt{T}Y}-F_{2}e^{-\frac{1}{2}\sigma^{2}_{2}T+\sigma_{2}\sqrt{T}X}-K)^{+}]\\\
=&F_{1}E\left[e^{-\frac{1}{2}\sigma^{2}_{1}T+\sigma_{1}\sqrt{T}Y}\cdot
1_{\\{F_{1}e^{-\frac{1}{2}\sigma^{2}_{1}T+\sigma_{1}\sqrt{T}Y}-F_{2}e^{-\frac{1}{2}\sigma^{2}_{2}T+\sigma_{2}\sqrt{T}X}-K\geq
0\\}}\right]\\\
-&F_{2}E\left[e^{-\frac{1}{2}\sigma^{2}_{2}T+\sigma_{2}\sqrt{T}X}\cdot
1_{\\{F_{1}e^{-\frac{1}{2}\sigma^{2}_{1}T+\sigma_{1}\sqrt{T}Y}-F_{2}e^{-\frac{1}{2}\sigma^{2}_{2}T+\sigma_{2}\sqrt{T}X}-K\geq
0\\}}\right]\\\
-&KQ\left(F_{1}e^{-\frac{1}{2}\sigma^{2}_{1}T+\sigma_{1}\sqrt{T}Y}-F_{2}e^{-\frac{1}{2}\sigma^{2}_{2}T+\sigma_{2}\sqrt{T}X}-K\geq
0\right).\end{split}$ (67)

The first two terms of the right-hand-side of (67) can be calculated as
follows

$\begin{split}&F_{1}E\left[e^{-\frac{1}{2}\sigma^{2}_{1}T+\sigma_{1}\sqrt{T}Y}\cdot
1_{\\{F_{1}e^{-\frac{1}{2}\sigma^{2}_{1}T+\sigma_{1}\sqrt{T}Y}-F_{2}e^{-\frac{1}{2}\sigma^{2}_{2}T+\sigma_{2}\sqrt{T}X}-K\geq
0\\}}\right]\\\
=&F_{1}E\left[1_{\\{F_{1}e^{-\frac{1}{2}\sigma^{2}_{1}T+\sigma_{1}\sqrt{T}(Y+\sigma_{1}\sqrt{T})}-F_{2}e^{-\frac{1}{2}\sigma^{2}_{2}T+\sigma_{2}\sqrt{T}(X+\rho\sigma_{1}\sqrt{T})}-K\geq
0\\}}\right]\\\
=&F_{1}Q\left(F_{1}e^{\frac{1}{2}\sigma_{1}^{2}T+\sigma_{1}\sqrt{T}Y}-F_{2}e^{-\frac{1}{2}\sigma_{2}^{2}T+\rho\sigma_{1}\sigma_{2}T+\sigma_{2}\sqrt{T}X}-K\geq
0\right)\\\
=&F_{1}Q\left(g_{1}\bar{F}_{1}e^{\sigma_{1}\sqrt{T}Y}-\alpha\bar{F}_{2}e^{\sigma_{2}\sqrt{T}X}-K\geq
0\right),\end{split}$ (68)

and

$\begin{split}&F_{2}E\left[e^{-\frac{1}{2}\sigma^{2}_{2}T+\sigma_{2}\sqrt{T}X}\cdot
1_{\\{F_{1}e^{-\frac{1}{2}\sigma^{2}_{1}T+\sigma_{1}\sqrt{T}Y}-F_{2}e^{-\frac{1}{2}\sigma^{2}_{2}T+\sigma_{2}\sqrt{T}X}-K\geq
0\\}}\right]\\\
=&F_{2}E\left[1_{\\{F_{1}e^{-\frac{1}{2}\sigma^{2}_{1}T+\sigma_{1}\sqrt{T}(Y+\rho\sigma_{2}\sqrt{T})}-F_{2}e^{-\frac{1}{2}\sigma^{2}_{2}T+\sigma_{2}\sqrt{T}(X+\sigma_{2}\sqrt{T})}-K\geq
0\\}}\right]\\\
=&F_{2}Q\left(F_{1}e^{-\frac{1}{2}\sigma_{1}^{2}T+\rho\sigma_{1}\sigma_{2}\sqrt{T}+\sigma_{1}\sqrt{T}Y}-F_{2}e^{\frac{1}{2}\sigma_{2}^{2}T+\sigma_{2}\sqrt{T}X}-K\geq
0\right)\\\
=&F_{2}Q\left(\alpha\bar{F}_{1}e^{\sigma_{1}\sqrt{T}Y}-g_{2}\bar{F}_{2}e^{\sigma_{2}\sqrt{T}X}-K\geq
0\right).\end{split}$ (69)

The last term of the right-hand-side of (67) equals to

$\begin{split}&KQ\left(F_{1}e^{-\frac{1}{2}\sigma^{2}_{1}T+\sigma_{1}\sqrt{T}Y}-F_{2}e^{-\frac{1}{2}\sigma^{2}_{2}T+\sigma_{2}\sqrt{T}X}-K\geq
0\right)\\\
=&KQ\left(\bar{F}_{1}e^{\sigma_{1}\sqrt{T}Y}-\bar{F}_{2}e^{\sigma_{2}\sqrt{T}X}-k\geq
0\right).\end{split}$ (70)

Proof of Corollary 2.2: From Proposition 2.1 we have

$\Pi_{T}=e^{-rT}F_{1}C_{D}^{1}-e^{-rT}F_{2}C_{D}^{2}-e^{-rT}KC_{D}^{3},$ (71)

where

$\begin{split}C_{D}^{1}&=Q\left(g_{1}\bar{F}_{1}e^{\sigma_{1}\sqrt{T}Y}-\alpha\bar{F}_{2}e^{\sigma_{2}\sqrt{T}X}-K\geq
0\right),\\\
C_{D}^{2}&=Q\left(\alpha\bar{F}_{1}e^{\sigma_{1}\sqrt{T}Y}-g_{2}\bar{F}_{2}e^{\sigma_{2}\sqrt{T}X}-K\geq
0\right),\\\
C_{D}^{3}&=Q\left(\bar{F}_{1}e^{\sigma_{1}\sqrt{T}Y}-\bar{F}_{2}e^{\sigma_{2}\sqrt{T}X}-K\geq
0\right).\\\ \end{split}$

When $K=0$, the third term of the right-hand-side of (71) equals to zero, and
the other two terms are calculated as follows separately

$\begin{split}C_{D}^{1}&=Q\left(g_{1}\bar{F}_{1}e^{\sigma_{1}\sqrt{T}Y}-\alpha\bar{F}_{2}e^{\sigma_{2}\sqrt{T}X}\geq
0\right)\\\
&=Q\left(\sigma_{1}\sqrt{T}Y-\sigma_{2}\sqrt{T}X\geq\ln\left(\frac{\alpha\bar{F}_{2}}{g_{1}\bar{F}_{1}}\right)\right)\\\
&=Q\left(\epsilon\geq\frac{\ln\left(\frac{\alpha\bar{F}_{2}}{g_{1}\bar{F}_{1}}\right)}{\sqrt{T}\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}-2\rho\sigma_{1}\sigma_{2}}}\right)\\\
&=\Phi\left(\frac{\ln\left(\frac{g_{1}\bar{F}_{1}}{\alpha\bar{F}_{2}}\right)}{\sqrt{T}\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}-2\rho\sigma_{1}\sigma_{2}}}\right),\end{split}$

and

$\begin{split}C_{D}^{2}&=Q\left(\alpha\bar{F}_{1}e^{\sigma_{1}\sqrt{T}Y}-g_{2}\bar{F}_{2}e^{\sigma_{2}\sqrt{T}X}\geq
0\right)\\\
&=Q\left(\sigma\sqrt{T}Y-\sigma_{2}\sqrt{T}X\geq\ln\left(\frac{g_{2}\bar{F}_{2}}{\alpha\bar{F}_{1}}\right)\right)\\\
&=Q\left(\epsilon\geq\frac{\ln\left(\frac{g_{2}\bar{F}_{2}}{\alpha\bar{F}_{1}}\right)}{\sqrt{T}\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}-2\rho\sigma_{1}\sigma_{2}}}\right)\\\
&=\Phi\left(\frac{\ln\left(\frac{\alpha\bar{F}_{1}}{g_{2}\bar{F}_{2}}\right)}{\sqrt{T}\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}-2\rho\sigma_{1}\sigma_{2}}}\right),\end{split}$

where $\epsilon\sim N(0,1)$.

Proof of Proposition 2.3: From Proposition 2.1 we have

$\Pi_{T}=e^{-rT}F_{1}C_{D}^{1}-e^{-rT}F_{2}C_{D}^{2}-e^{-rT}KC_{D}^{3},$ (72)

where

$\begin{split}C_{D}^{1}&=Q\left(g_{1}\bar{F}_{1}e^{\sigma_{1}\sqrt{T}Y}-\alpha\bar{F}_{2}e^{\sigma_{2}\sqrt{T}X}-K\geq
0\right),\\\
C_{D}^{2}&=Q\left(\alpha\bar{F}_{1}e^{\sigma_{1}\sqrt{T}Y}-g_{2}\bar{F}_{2}e^{\sigma_{2}\sqrt{T}X}-K\geq
0\right),\\\
C_{D}^{3}&=Q\left(\bar{F}_{1}e^{\sigma_{1}\sqrt{T}Y}-\bar{F}_{2}e^{\sigma_{2}\sqrt{T}X}-K\geq
0\right).\\\ \end{split}$

Then we can approximate each $C_{D}^{1},C_{D}^{2},C_{D}^{3},$ as follows

$\begin{split}C_{D}^{1}=&Q\left(g_{1}\bar{F}_{1}e^{\sigma_{1}\sqrt{T}Y}-\alpha\bar{F}_{2}e^{\sigma_{2}\sqrt{T}X}-K\geq
0\right)\\\
=&Q\left(Y\geq\frac{\ln\left(\frac{\alpha\bar{F}_{2}}{g_{1}\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}X}+\frac{K}{g_{1}\bar{F}_{1}}\right)}{\sigma_{1}\sqrt{T}}\right)\\\
\approx&\sum_{i=0}^{N-1}Q\left(Y\geq\frac{\ln\left(\frac{\alpha\bar{F}_{2}}{g_{1}\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}X}+\frac{K}{g_{1}\bar{F}_{1}}\right)}{\sigma_{1}\sqrt{T}}\bigg{|}a_{i}<X\leq
a_{i+1}\right)Q\left(a_{i}<X\leq a_{i+1}\right)\\\
\approx&\sum_{i=0}^{N-1}Q\left(Y\geq\frac{\ln\left(\frac{\alpha\bar{F}_{2}}{g_{1}\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}\theta_{i}}+\frac{K}{g_{1}\bar{F}_{1}}\right)}{\sigma_{1}\sqrt{T}}\right)Q\left(a_{i}<X\leq
a_{i+1}\right)\\\
=&\sum_{i=0}^{N-1}\Phi\left(\frac{-\ln\left(\frac{\alpha\bar{F}_{2}}{g_{1}\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}\theta_{i}}+\frac{K}{g_{1}\bar{F}_{1}}\right)+\sigma_{1}\rho\sqrt{T}\theta_{i}}{\sigma_{1}\sqrt{T}\sqrt{1-\rho^{2}}}\right)\left(\Phi(a_{i+1})-\Phi(a_{i})\right),\end{split}$
$\begin{split}C_{D}^{2}=&Q\left(\alpha\bar{F}_{1}e^{\sigma_{1}\sqrt{T}Y}-g_{2}\bar{F}_{2}e^{\sigma_{2}\sqrt{T}X}-K\geq
0\right)\\\
=&Q\left(Y\geq\frac{\ln\left(\frac{g_{2}\bar{F}_{2}}{\alpha\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}X}+\frac{K}{\alpha\bar{F}_{1}}\right)}{\sigma_{1}\sqrt{T}}\right)\\\
\approx&\sum_{i=0}^{N-1}Q\left(Y\geq\frac{\ln\left(\frac{g_{2}\bar{F}_{2}}{\alpha\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}X}+\frac{K}{\alpha\bar{F}_{1}}\right)}{\sigma_{1}\sqrt{T}}\bigg{|}a_{i}<X\leq
a_{i+1}\right)Q\left(a_{i}<X\leq a_{i+1}\right)\\\
\approx&\sum_{i=0}^{N-1}Q\left(Y\geq\frac{\ln\left(\frac{g_{2}\bar{F}_{2}}{\alpha\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}\theta_{i}}+\frac{K}{\alpha\bar{F}_{1}}\right)}{\sigma_{1}\sqrt{T}}\right)Q\left(a_{i}<X\leq
a_{i+1}\right)\\\
=&\sum_{i=0}^{N-1}\Phi\left(\frac{-\ln\left(\frac{g_{2}\bar{F}_{2}}{\alpha\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}\theta_{i}}+\frac{K}{\alpha\bar{F}_{1}}\right)+\sigma_{1}\rho\sqrt{T}\theta_{i}}{\sigma_{1}\sqrt{T}\sqrt{1-\rho^{2}}}\right)\left(\Phi(a_{i+1})-\Phi(a_{i})\right),\end{split}$

and

$\begin{split}C_{D}^{3}=&Q\left(\bar{F}_{1}e^{\sigma_{1}\sqrt{T}Y}-\bar{F}_{2}e^{\sigma_{2}\sqrt{T}X}-K\geq
0\right)\\\
=&Q\left(Y\geq\frac{\ln\left(\frac{\bar{F}_{2}}{\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}X}+\frac{K}{\bar{F}_{1}}\right)}{\sigma_{1}\sqrt{T}}\right)\\\
\approx&\sum_{i=0}^{N-1}Q\left(Y\geq\frac{\ln\left(\frac{\bar{F}_{2}}{\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}X}+\frac{K}{\bar{F}_{1}}\right)}{\sigma_{1}\sqrt{T}}\bigg{|}a_{i}<X\leq
a_{i+1}\right)Q\left(a_{i}<X\leq a_{i+1}\right)\\\
\approx&\sum_{i=0}^{N-1}Q\left(Y\geq\frac{\ln\left(\frac{\bar{F}_{2}}{\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}\theta_{i}}+\frac{K}{\bar{F}_{1}}\right)}{\sigma_{1}\sqrt{T}}\right)Q\left(a_{i}<X\leq
a_{i+1}\right)\\\
=&\sum_{i=0}^{N-1}\Phi\left(\frac{-\ln\left(\frac{\bar{F}_{2}}{\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}\theta_{i}}+\frac{K}{\bar{F}_{1}}\right)+\sigma_{1}\rho\sqrt{T}\theta_{i}}{\sigma_{1}\sqrt{T}\sqrt{1-\rho^{2}}}\right)\left(\Phi(a_{i+1})-\Phi(a_{i})\right).\end{split}$

Proof of Proposition 3.2: With the $\theta_{0}$ in (29) we have

$\begin{split}Y_{\theta_{0}}&=Z_{1}\sin\theta_{0}-Z_{2}\cos\theta_{0}\\\
&=\frac{-\sigma_{1}Z_{1}sin\phi}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}b^{2}-2\sigma_{1}\sigma_{2}bZ_{2}cos\phi}}-\frac{\sigma_{2}bZ_{2}-\sigma_{1}Z_{2}cos\phi}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}b^{2}-2\sigma_{1}\sigma_{2}bcos\phi}}\\\
&=\frac{\sigma_{1}(-Z_{1}\sin\phi+Z_{2}\cos\phi)-\sigma_{2}bZ_{2}}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}b^{2}-2\sigma_{1}\sigma_{2}bcos\phi}}.\end{split}$

Hence, with the definition of $d_{0}$ in (30), we have

$\begin{split}\Big{\\{}Y_{\theta_{0}}\leq
d_{0}\Big{\\}}=\Big{\\{}-\sigma_{1}\sqrt{T}\epsilon_{1}+\sigma_{2}\sqrt{T}b\epsilon_{2}\leq\ln\left(\frac{F_{1}}{a}\right)-\frac{1}{2}\sigma_{1}T+\frac{1}{2}b\sigma_{2}^{2}T\Big{\\}},\end{split}$
(73)

with $\rho(\epsilon_{1},\;\epsilon_{2})=\cos\phi,\epsilon_{1}=Z_{1}\;\sin\phi-
Z_{4}\cos\phi,\epsilon_{2}=-Z_{2}$. One the other hand, since

$\begin{split}\frac{a(S_{2}(T))^{b}}{E[S_{2}(T)]^{b}}=\frac{aF_{2}^{b}\exp\\{-\frac{1}{2}\sigma_{2}^{2}bT+\sigma_{2}\sqrt{T}b\epsilon_{2}\\}}{E[F_{2}^{b}\exp\\{-\frac{1}{2}\sigma_{2}^{2}bT+\sigma_{2}\sqrt{T}b\epsilon_{2}\\}]}=a\exp\\{-\frac{1}{2}\sigma_{2}^{2}bT+\sigma_{2}\sqrt{T}b\epsilon_{2}\\},\end{split}$

we have

$\begin{split}\left\\{S_{1}\geq\frac{a(S_{2}(T))^{b}}{E[S_{2}(T)]^{b}}\right\\}&=\left\\{\exp\\{-\frac{1}{2}\sigma_{1}^{2}T+\sigma_{1}\sqrt{T}\epsilon_{1}\\}\geq\frac{a}{F_{1}}\exp\\{-\frac{1}{2}\sigma_{2}^{2}bT+\sigma_{2}\sqrt{T}b\epsilon_{2}\\}\right\\}\\\
&=\left\\{-\frac{1}{2}\sigma_{1}^{2}T+\sigma_{1}\sqrt{T}\epsilon_{1}\geq\ln\left(\frac{a}{F_{1}}\right)-\frac{1}{2}\sigma_{2}^{2}bT+\sigma_{2}\sqrt{T}b\epsilon_{2}\right\\}\\\
&=\left\\{-\sigma_{1}\sqrt{T}\epsilon_{1}+\sigma_{2}\sqrt{T}b\epsilon_{2}\leq\ln\left(\frac{F_{1}}{a}\right)-\frac{1}{2}\sigma_{1}T+\frac{1}{2}b\sigma_{2}^{2}T\right\\}.\\\
\end{split}$ (74)

The right-side-hands of (73) and (74) are equal, and thus we have

$\begin{split}\\{Y_{\theta_{0}}\leq
d_{0}\\}=\left\\{S_{1}\geq\frac{a(S_{2}(T))^{b}}{E[S_{2}(T)]^{b}}\right\\}.\end{split}$

This completes the proof.

Proof of Lemma 4.1: Since $X$ and $Y$ are two standard normal random variables
with $Cov(Y,X)=\rho$, we can write $Y$ as $Y=\rho
X+\sqrt{1-\rho^{2}}\tilde{\epsilon}$ with $\tilde{\epsilon}\sim N(0,1)$ and
$Cov(\tilde{\epsilon},X)=0$. We have

$\begin{split}mY-nX&=(m\rho-n)X+m\sqrt{1-\rho^{2}}\tilde{\epsilon}\\\ &\sim
N(0,m^{2}+n^{2}-2\rho mn),\end{split}$

therefore

$\begin{split}Q(mY-nX\geq l)&=Q\left(\frac{mY-nX}{\sqrt{m^{2}+n^{2}-2\rho
mn}}\geq\frac{l}{\sqrt{m^{2}+n^{2}-2\rho mn}}\right)\\\
&=\Phi\left(\frac{-l}{\sqrt{m^{2}+n^{2}-2\rho mn}}\right).\end{split}$

Proof of Lemma 4.2: We can rewrite the three curves
$\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}$ as

$\begin{split}\mathcal{C}_{1}:\;&y=\frac{1}{\sigma_{1}\sqrt{T}}\ln\left(\frac{\alpha\bar{F}_{2}}{g_{1}\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}x}+\frac{K}{g_{1}\bar{F}_{1}}\right),\\\
\mathcal{C}_{2}:\;&y=\frac{1}{\sigma_{1}\sqrt{T}}\ln\left(\frac{g_{2}\bar{F}_{2}}{\alpha\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}x}+\frac{K}{\alpha\bar{F}_{1}}\right),\\\
\mathcal{C}_{3}:\;&y=\frac{1}{\sigma_{1}\sqrt{T}}\ln\left(\frac{\bar{F}_{2}}{\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}x}+\frac{K}{\bar{F}_{1}}\right).\end{split}$

For curve $\mathcal{C}_{1}$, we have

$\begin{split}\lim_{x\rightarrow+\infty}\frac{y}{x}&=\frac{1}{\sigma_{1}\sqrt{T}}\lim_{x\rightarrow+\infty}\frac{\ln\left(\frac{\alpha\bar{F}_{2}}{g_{1}\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}x}+\frac{K}{g_{1}\bar{F}_{1}}\right)}{x}\\\
&=\frac{1}{\sigma_{1}\sqrt{T}}\lim_{x\rightarrow+\infty}\frac{\frac{\alpha\bar{F}_{2}}{g_{1}\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}x}\sigma_{2}\sqrt{T}}{\frac{\alpha\bar{F}_{2}}{g_{1}\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}x}+\frac{K}{g_{1}\bar{F}_{1}}}=\frac{\sigma_{2}}{\sigma_{1}},\\\
\lim_{x\rightarrow-\infty}y&=\frac{1}{\sigma_{1}\sqrt{T}}\ln\left(\frac{K}{g_{1}\bar{F}_{1}}\right)=\frac{1}{\sigma_{1}\sqrt{T}}\ln\left(\frac{K}{\bar{F}_{1}}\right)-\sigma_{1}\sqrt{T}.\end{split}$

For curve $\mathcal{C}_{2}$, we have

$\begin{split}\lim_{x\rightarrow+\infty}\frac{y}{x}&=\frac{1}{\sigma_{1}\sqrt{T}}\lim_{x\rightarrow+\infty}\frac{\ln\left(\frac{g_{2}\bar{F}_{2}}{\alpha\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}x}+\frac{K}{\alpha\bar{F}_{1}}\right)}{x}\\\
&=\frac{1}{\sigma_{1}\sqrt{T}}\lim_{x\rightarrow+\infty}\frac{\frac{g_{2}\bar{F}_{2}}{\alpha\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}x}\sigma_{2}\sqrt{T}}{\frac{g_{2}\bar{F}_{2}}{\alpha\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}x}+\frac{K}{g_{1}\bar{F}_{1}}}=\frac{\sigma_{2}}{\sigma_{1}},\\\
\lim_{x\rightarrow-\infty}y&=\frac{1}{\sigma_{1}\sqrt{T}}\ln\left(\frac{K}{\alpha\bar{F}_{1}}\right)=\frac{1}{\sigma_{1}\sqrt{T}}\ln\left(\frac{K}{\bar{F}_{1}}\right)-\rho\sigma_{2}\sqrt{T}.\end{split}$

For curve $\mathcal{C}_{3}$, we have

$\begin{split}\lim_{x\rightarrow+\infty}\frac{y}{x}&=\frac{1}{\sigma_{1}\sqrt{T}}\lim_{x\rightarrow+\infty}\frac{\ln\left(\frac{\bar{F}_{2}}{\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}x}+\frac{K}{\bar{F}_{1}}\right)}{x}\\\
&=\frac{1}{\sigma_{1}\sqrt{T}}\lim_{x\rightarrow+\infty}\frac{\frac{\bar{F}_{2}}{\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}x}\sigma_{2}\sqrt{T}}{\frac{\bar{F}_{2}}{\bar{F}_{1}}e^{\sigma_{2}\sqrt{T}x}+\frac{K}{\bar{F}_{1}}}=\frac{\sigma_{2}}{\sigma_{1}},\\\
\lim_{x\rightarrow-\infty}y&=\frac{1}{\sigma_{1}\sqrt{T}}\ln\left(\frac{K}{\bar{F}_{1}}\right).\end{split}$
(75)

Proof of Proposition 4.3: We need to approximate
$C_{D}^{1},C_{D}^{2},C_{D}^{3},$ in Proposition 2.1. For this, we replace
$\mathcal{C}_{i}$ by lines $\ell_{i}$ for each $i=1,2,3$. With these, we have

$\begin{split}C_{D}^{i}\approx
Q\left(y^{i}-b\frac{\sigma_{2}}{\sigma_{1}}x^{i}\geq
y_{0}^{i}-b\frac{\sigma_{2}}{\sigma_{1}}x_{0}^{i}\right).\end{split}$

Next, we substitute the $y_{0}^{i}$ with $z_{0}^{i}(b\sigma_{2},a,x_{0}^{i})$,
for each $i=1,2,3$ and use Lemma 4.1 to obtain

$\begin{split}&Q\left(y^{i}-b\frac{\sigma_{2}}{\sigma_{1}}x^{i}\geq
y_{0}^{i}-b\frac{\sigma_{2}}{\sigma_{1}}x_{0}^{i}\right)\\\
\approx&Q\left(y^{i}-b\frac{\sigma_{2}}{\sigma_{1}}x^{i}\geq
z_{0}^{i}(b\sigma_{2},a,x_{0}^{i})-b\frac{\sigma_{2}}{\sigma_{1}}x_{0}^{i}\right)\\\
=&Q\left(\hat{\epsilon}_{i}\geq\frac{z_{0}^{i}(b\sigma_{2},a,x_{0}^{i})-b\frac{\sigma_{2}}{\sigma_{1}}x_{0}^{i}}{\sqrt{1+\left(b\frac{\sigma_{2}}{\sigma_{1}}\right)^{2}-2\rho
b\frac{\sigma_{2}}{\sigma_{1}}}}\right)\\\
=&\Phi\left(\frac{-\delta_{i}}{1+\kappa^{2}-2\rho\kappa}\right).\end{split}$

Proof of Proposition 4.4: If we substitute $a$ and $b$ by $a_{1}(\lambda)$,
$a_{2}(\mu)$, $a_{3}(\gamma)$ and $b_{1}(\lambda)$, $b_{2}(\mu)$,
$b_{3}(\gamma)$ respectively in Proposition 4.3 and follow the same steps in
this Proposition, we can easily obtain the result.

Proof of Corollary 4.5: When
$\lambda=(\frac{1}{2}\sigma_{2}-\rho\sigma_{1})\sqrt{T}$, we have

$\begin{split}a_{1}(\lambda)&=a_{1}\left((\frac{1}{2}\sigma_{2}-\rho\sigma_{1})\sqrt{T}\right)=F_{2}+Ke^{-\sigma_{2}\sqrt{T}(\frac{1}{2}\sigma_{2}-\rho\sigma_{1})\sqrt{T}-\rho\sigma_{1}\sigma_{2}T+\frac{1}{2}\sigma_{2}^{2}T}=F_{2}+K=a,\\\
b_{1}(\lambda)&=b_{1}\left((\frac{1}{2}\sigma_{2}-\rho\sigma_{1})\sqrt{T}\right)=\frac{e^{\rho\sigma_{1}\sigma_{2}T}F_{2}e^{-\frac{1}{2}\sigma_{2}^{2}T}e^{\sigma_{2}\sqrt{T}(\frac{1}{2}\sigma_{2}-\rho\sigma_{1})\sqrt{T}}}{e^{\rho\sigma_{1}\sigma_{2}T}F_{2}e^{-\frac{1}{2}\sigma_{2}^{2}T}e^{\sigma_{2}\sqrt{T}(\frac{1}{2}\sigma_{2}-\rho\sigma_{1})\sqrt{T}}+K}=\frac{F_{2}}{F_{2}+K}=b,\\\
\kappa_{1}(\lambda)&=\kappa_{1}\left((\frac{1}{2}\sigma_{2}-\rho\sigma_{1})\sqrt{T}\right)=\frac{\sigma_{2}}{\sigma_{1}}b,\end{split}$
(76)

and

$\begin{split}z_{0}^{1}(\sigma_{2}b_{1}(\lambda),a_{1}(\lambda),\lambda)&=\frac{1}{\sigma_{1}\sqrt{T}}\ln\left(\frac{a}{F_{1}}\right)+\frac{1}{\sigma_{1}\sqrt{T}}\left(\rho\sigma_{1}\sigma_{2}bT+\sigma_{2}b\sqrt{T}(\frac{1}{2}\sigma_{2}-\rho\sigma_{1})\sqrt{T}-\frac{1}{2}\sigma_{2}^{2}b^{2}T\right)\\\
-\frac{1}{2}\sigma_{1}\sqrt{T}&=\frac{1}{\sigma_{1}\sqrt{T}}\left(\ln\left(\frac{a}{F_{1}}\right)+\frac{1}{2}\sigma_{2}^{2}bT-\frac{1}{2}\sigma_{2}^{2}b^{2}T-\frac{1}{2}\sigma_{1}^{2}T\right).\end{split}$
(77)

Hence

$\begin{split}&\frac{-\delta_{1}(\lambda)}{\sqrt{1+\kappa_{1}^{2}(\lambda)-2\rho\kappa_{1}(\lambda)}}\\\
=&\frac{-z_{0}^{1}(\sigma_{2}b_{1}(\lambda),a_{1}(\lambda),\lambda)+\kappa_{1}(\lambda)\lambda}{\sqrt{1+\kappa_{1}^{2}(\lambda)-2\rho\kappa_{1}(\lambda)}}\\\
=&\frac{-\frac{1}{\sigma_{1}\sqrt{T}}\left(\ln\left(\frac{a}{F_{1}}\right)+\frac{1}{2}\sigma_{2}^{2}bT-\frac{1}{2}\sigma_{2}^{2}b^{2}T-\frac{1}{2}\sigma_{1}^{2}T\right)+\frac{1}{\sigma_{1}\sqrt{T}}\left(\frac{1}{2}\sigma_{2}^{2}bT-\rho\sigma_{1}\sigma_{2}bT\right)}{\sqrt{1+\left(\frac{\sigma_{2}}{\sigma_{1}}b\right)^{2}-2\rho\frac{\sigma_{2}}{\sigma_{1}}b}}\\\
=&\frac{\ln\left(\frac{F_{1}}{a}\right)+\frac{1}{2}\sigma_{1}^{2}T+\frac{1}{2}\sigma_{2}^{2}b^{2}T-\rho\sigma_{1}\sigma_{2}bT}{\sqrt{T}\sqrt{\sigma_{1}^{2}+b^{2}\sigma_{2}^{2}-2\rho\sigma_{1}\sigma_{2}b}}.\end{split}$
(78)

When $\mu=-\frac{1}{2}\sigma_{2}\sqrt{T}$, we have

$\begin{split}a_{2}(\mu)&=a_{2}\left(-\frac{1}{2}\sigma_{2}\sqrt{T}\right)=F_{2}+Ke^{-\sigma_{2}\sqrt{T}(-\frac{1}{2}\sigma_{2}\sqrt{T})-\frac{1}{2}\sigma_{2}^{2}T}=F_{2}+K=a,\\\
b_{2}(\mu)&=b_{2}\left(-\frac{1}{2}\sigma_{2}\sqrt{T}\right)=\frac{e^{\sigma_{2}^{2}T}F_{2}e^{-\frac{1}{2}\sigma_{2}^{2}T}e^{\sigma_{2}\sqrt{T}(-\frac{1}{2}\sigma_{2}\sqrt{T})}}{e^{\sigma_{2}^{2}T}F_{2}e^{-\frac{1}{2}\sigma_{2}^{2}T}e^{\sigma_{2}\sqrt{T}(-\frac{1}{2}\sigma_{2}\sqrt{T})}+K}=\frac{F_{2}}{F_{2}+K}=b,\\\
\kappa_{2}(\mu)&=\kappa_{2}\left(-\frac{1}{2}\sigma_{2}\sqrt{T}\right)=\frac{\sigma_{2}}{\sigma_{1}}b,\end{split}$
(79)

and

$\begin{split}z_{0}^{2}(\sigma_{2}b_{2}(\mu),a_{2}(\mu),\mu)&=\frac{1}{\sigma_{1}\sqrt{T}}\ln\left(\frac{a}{F_{1}}\right)+\frac{1}{\sigma_{1}\sqrt{T}}\left(\sigma_{2}^{2}bT-\frac{1}{2}\sigma_{2}^{2}bT-\frac{1}{2}\sigma_{2}^{2}b^{2}T\right)+\left(\frac{1}{2}\sigma_{1}-\rho\sigma_{2}\right)\sqrt{T}\\\
&=\frac{1}{\sigma_{1}\sqrt{T}}\left(\ln\left(\frac{a}{F_{1}}\right)+\frac{1}{2}\sigma_{2}^{2}bT-\frac{1}{2}\sigma_{2}^{2}b^{2}T+\frac{1}{2}\sigma_{1}^{2}T-\rho\sigma_{1}\sigma_{2}T\right).\end{split}$
(80)

Hence

$\begin{split}&\frac{-\delta_{2}(\mu)}{\sqrt{1+\kappa_{2}^{2}(\mu)-2\rho\kappa_{2}(\mu)}}\\\
=&\frac{-z_{0}^{2}(\sigma_{2}b_{2}(\mu),a_{2}(\mu),\mu)+\kappa_{2}(\mu)\mu}{\sqrt{1+\kappa_{2}^{2}(\mu)-2\rho\kappa_{2}(\mu)}}\\\
=&\frac{-\frac{1}{\sigma_{1}\sqrt{T}}\left(\ln\left(\frac{a}{F_{1}}\right)+\frac{1}{2}\sigma_{2}^{2}bT-\frac{1}{2}\sigma_{2}^{2}b^{2}T+\frac{1}{2}\sigma_{1}^{2}T-\rho\sigma_{1}\sigma_{2}T\right)+\frac{1}{\sigma_{1}\sqrt{T}}\left(-\frac{1}{2}\sigma_{2}^{2}bT\right)}{\sqrt{1+\left(\frac{\sigma_{2}}{\sigma_{1}}b\right)^{2}-2\rho\frac{\sigma_{2}}{\sigma_{1}}b}}\\\
=&\frac{\ln\left(\frac{F_{1}}{a}\right)-\frac{1}{2}\sigma_{1}^{2}T-\sigma_{2}^{2}bT+\frac{1}{2}\sigma_{2}^{2}b^{2}T+\rho\sigma_{1}\sigma_{2}T}{\sqrt{T}\sqrt{\sigma_{1}^{2}+b^{2}\sigma_{2}^{2}-2\rho\sigma_{1}\sigma_{2}b}}.\end{split}$
(81)

When $\gamma=\frac{1}{2}\sigma_{2}\sqrt{T}$, we have

$\begin{split}a_{3}(\gamma)&=a_{3}\left(\frac{1}{2}\sigma_{2}\sqrt{T}\right)=F_{2}+Ke^{-\sigma_{2}\sqrt{T}\frac{1}{2}\sigma_{2}\sqrt{T}+\frac{1}{2}\sigma_{2}^{2}T}=F_{2}+K=a,\\\
b_{3}(\gamma)&=b_{3}\left(\frac{1}{2}\sigma_{2}\sqrt{T}\right)=\frac{F_{2}e^{-\frac{1}{2}\sigma_{2}^{2}T}e^{\sigma_{2}\sqrt{T}\frac{1}{2}\sigma_{2}\sqrt{T}}}{F_{2}e^{-\frac{1}{2}\sigma_{2}^{2}T}e^{\sigma_{2}\sqrt{T}\frac{1}{2}\sigma_{2}\sqrt{T}}+K}=\frac{F_{2}}{F_{2}+K}=b,\\\
\kappa_{3}(\gamma)&=\kappa_{3}\left(\frac{1}{2}\sigma_{2}\sqrt{T}\right)=\frac{\sigma_{2}}{\sigma_{1}}b,\end{split}$
(82)

and

$\begin{split}z_{0}^{3}(\sigma_{2}b_{3}(\gamma),a_{3}(\gamma),\gamma)&=\frac{1}{\sigma_{1}\sqrt{T}}\ln\left(\frac{a}{F_{1}}\right)+\frac{1}{\sigma_{1}\sqrt{T}}\left(\frac{1}{2}\sigma_{2}^{2}bT-\frac{1}{2}\sigma_{2}^{2}b^{2}T\right)+\frac{\sigma_{1}\sqrt{T}}{2}\\\
&=\frac{1}{\sigma_{1}\sqrt{T}}\left(\ln\left(\frac{a}{F_{1}}\right)+\frac{1}{2}\sigma_{2}^{2}bT-\frac{1}{2}\sigma_{2}^{2}b^{2}T+\frac{1}{2}\sigma_{1}^{2}T\right).\end{split}$
(83)

Hence

$\begin{split}&\frac{-\delta_{3}(\gamma)}{\sqrt{1+\kappa_{3}^{2}(\gamma)-2\rho\kappa_{3}(\gamma)}}\\\
=&\frac{-z_{0}^{3}(\sigma_{2}b_{3}(\gamma),a_{3}(\gamma),\gamma)+\kappa_{3}(\gamma)\gamma}{\sqrt{1+\kappa_{3}^{2}(\gamma)-2\rho\kappa_{3}(\gamma)}}\\\
=&\frac{-\frac{1}{\sigma_{1}\sqrt{T}}\left(\ln\left(\frac{a}{F_{1}}\right)+\frac{1}{2}\sigma_{2}^{2}bT-\frac{1}{2}\sigma_{2}^{2}b^{2}T+\frac{1}{2}\sigma_{1}^{2}T\right)+\frac{1}{\sigma_{1}\sqrt{T}}\left(\frac{1}{2}\sigma_{2}^{2}bT\right)}{\sqrt{1+\left(\frac{\sigma_{2}}{\sigma_{1}}b\right)^{2}-2\rho\frac{\sigma_{2}}{\sigma_{1}}b}}\\\
=&\frac{\ln\left(\frac{F_{1}}{a}\right)-\frac{1}{2}\sigma_{1}^{2}T+\frac{1}{2}\sigma_{2}^{2}b^{2}T}{\sqrt{T}\sqrt{\sigma_{1}^{2}+b^{2}\sigma_{2}^{2}-2\rho\sigma_{1}\sigma_{2}b}}.\end{split}$
(84)

This completes the proof.

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