Sector Formula for Approximation of Spread Option Value & Greeks and Its Applications

Abstract

The goal of this paper is to derive closed-form approximation formulas for the spread option value and Greeks by using double integration and investigating the exercise boundary. We have found that the straight-line approximation suggested in previous research does not perform well for curved exercise boundaries. We propose a novel approach: to integrate in a sector and find a closed-form formula expressed in terms of the bivariate normal CDF. We call it the sector formula. Numerical tests show the good accuracy of our sector formula. We demonstrate applications of the formula to the market data of calendar spread options for three major commodities, WTI, Natural Gas, and Corn, listed on the CME site as of May, April, and June 2024.

1. Introduction

The ubiquitous role of spread options in all markets is well known. Examples include foreign exchange markets where spreads involve interest rates in different countries or fixed-income markets with spreads between rates on different maturities or spreads between quality levels; see the overview of the “zoology” of spread options in [1].

Spread options are especially important in commodity markets, as many energy assets have spread options embedded in them. For example, the evaluation of power plants can be represented as a spark spread option: the spread between power and fuel prices, as reviewed in [2], for example. Another application involves the evaluation of storage, whose value can be replicated through calendar spread options. This subject is discussed in numerous sources (see a chapter in the recent book [3]). Transmission contracts can be modeled by options on a locational spread, the difference between prices of power or gas at two liquid points. In electricity markets, an interconnector gives the owner the option to transmit electricity between two locations and can be valued as a strip of real options written on the spread between power prices, as considered in [4]. Other examples include the evaluation of refineries through crack spreads, crush spreads in agricultural markets, and many others. Therefore, an accurate evaluation of spread options and their Greeks is utterly important for commodity trading and risk management.

In this paper, we study a classical case of a commodity spread option on two futures, $F_1$ and $F_2$, with the payoff

$$max(F_1\left(T\right)-F_2\left(T\right)-K,0)$$

(1)

where T is the option expiry, and K is the strike. Using futures simplifies calculations. We also choose futures since the majority of trading in commodities happens in futures markets; applications to commodity markets are our main interest. For the same reason, we only consider the case of positive correlations. We assume the standard setup of the joint normal distribution of futures returns:

$$\begin{array}{cc}\hfill & d{F}_1\left(t\right)=\sqrt{1-{\rho }^2}{\sigma }_1{F}_1\left(t\right)d{W}_1\left(t\right)+\rho {\sigma }_1{F}_2\left(t\right)d{W}_2\left(t\right)\hfill \\ \hfill & d{F}_2\left(t\right)={\sigma }_2{F}_2\left(t\right)d{W}_2\left(t\right)\hfill \end{array}$$

(2)

where $W_1\left(t\right)$ and $W_2\left(t\right)$ are two independent standard Brownian motions under the risk-neutral measure, ${\sigma }_1$ and ${\sigma }_2$ are the volatilities, and $\rho$ is the correlation between the two Brownian motions. Let $F_1\left(0\right)=F_1$ and $F_2\left(0\right)=F_2$. The futures’ prices at the time of expiry T can be written as

$$\begin{array}{cc}\hfill & F_1\left(T\right)=F_1\exp (-{\displaystyle \frac{1}{2}}{\sigma }_1^{2}T+\sqrt{1-{\rho }^2}{\sigma }_1\sqrt{T}X+\rho {\sigma }_1\sqrt{T}Y)\hfill \\ \hfill & F_2\left(T\right)=F_2\exp (-{\displaystyle \frac{1}{2}}{\sigma }_2^{2}T+{\sigma }_2\sqrt{T}Y)\hfill \end{array}$$

(3)

where X and Y are two independent standard normal random variables.

The spread call price is the discounted expected payoff under the risk-neutral measure:

$$c(K,T)=e^{-rT}E\left[(F_1\left(T\right)-F_2\left(T\right)-K)I_{[F_1\left(T\right)-F_2\left(T\right)-K\ge 0]}\right]$$

(4)

We call the boundary of the domain

$$F_1\left(T\right)-F_2\left(T\right)-K\ge 0$$

the exercise boundary. A more precise definition will be provided later.

In this paper, we consider only the cases of non-negative strikes, $K\ge 0$. A spread option with a negative strike, $K_1<0$, can be transformed into a spread option with a positive strike, $K=-K_1$, as follows: the payoff of a call spread option with a negative strike is equivalent to a put spread option with strike K and a reverse spread:

$${\left(F_1\left(T\right)-F_2\left(T\right)-K_1\right)}^{+}={\left(K-(F_2\left(T\right)-F_1\left(T\right))\right)}^{+}$$

Using the standard put–call parity and no-arbitrage arguments, we derive the following relationship:

$$c\left(K_1,T,F_1,F_2\right)=c\left(K,T,F_2,F_1\right)+K{e}^{-rT}+(F_1-F_2)e^{-rT}$$

(5)

The inputs to a spread option evaluation include commodity future prices $F_1$ and $F_2$, volatilities ${\sigma }_1$ and ${\sigma }_2$, correlation $\rho$, contractual strike K, and expiration T of the option. For tradable commodity locations, future prices and volatilities are observable from the market; however, a correlation is rarely perceivable. There are different methods to deal with this problem. One method is to employ historical correlations. Another way is to use the implied correlation, which is implied by correlation products, spread options in particular. The concept is similar to the implied volatility “implied” by a traded option. The subject of implied correlation is discussed in detail in [5]. The authors developed a general framework employing a Gaussian copula, in which an implied correlation can be rigorously defined for a class of derivative contracts written on two assets.

There is a vast amount of research devoted to the problem of the evaluation of spread options. For the case of a zero strike, $K=0$ (exchange option), the value is given by the well-known Margrabe formula [6]. In the case of a spread between two commodity futures, it is given by the following formula:

$$C_S=e^{-rT}\left(F_{1}N\left(d_1\right)-F_{2}N\left(d_2\right)\right),d_1={\displaystyle \frac{ln\left(\frac{F_1}{F_2}\right)+1/2{\sigma }^{2}T}{\sigma \sqrt{T}}},d_2=d_1-\sigma \sqrt{T}$$

(6)

where $\sigma =\sqrt{{\sigma }_1^2-2\rho {\sigma }_1{\sigma }_2+{\sigma }_2^2}$.

The existing methods to price non-zero-strike spread options involve numerical methods, which include Monte Carlo, tree methods [7], numerical integration [8], and analytical approximations.

Among practitioners, the most popular analytical approximation method for a commodity spread option value is the Kirk formula, ref. [9], due to its simplicity: it is a modified version of the Margrabe formula with adjusted volatility:

$$C(K,T)=e^{-rT}\left(F_{1}N\left({\widehat{d}}_1\right)-(F_2+K)N\left({\widehat{d}}_2\right)\right),{\widehat{d}}_1={\displaystyle \frac{ln\left(\frac{F_1}{F_2+K}\right)+1/2{\widehat{\sigma }}^{2}T}{\widehat{\sigma }\sqrt{T}}},{\widehat{d}}_2={\widehat{d}}_1-\widehat{\sigma }\sqrt{T}$$

(7)

$$\widehat{\sigma }=\sqrt{{\sigma }_1^2-2\rho {\sigma }_1\widehat{{\sigma }_2}+{\left({\widehat{\sigma }}_2\right)}^2},{\widehat{\sigma }}_2={\displaystyle \frac{{\sigma }_2{F}_1}{F_2+K}}$$

Carmona and Durrleman [10] modeled correlations by a trigonometric function and developed accurate lower bounds for the spread option value by solving a complicated optimization problem.

Bjerksund and Stensland [11] calculated the approximate spread option value by integrating the payoff over the following type of exercise boundary:

$$\{F_1\left(T\right)\ge a{F}_2{\left(T\right)}^b\}$$

$$C_B=e^{-rt}E\left[(F_1\left(T\right)-F_2\left(T\right)-K)I_{\{F_1\left(T\right)\ge a{F}_2{\left(T\right)}^b\}}\right]$$

(8)

For any given $a,b$, they give a general formula for the integral and verify that for $a=F_2+K$ and $b={\displaystyle \frac{F_2}{F_2+K}}$, the above formula gives a good approximation. Their approximated boundary $\{F_1\left(T\right)\ge a{F}_2{\left(T\right)}^b\}$ is a straight line in the $(x,y)$ plane, which implies that the formula works well only if the curvature of the real boundary is small.

Caldana and Fusai [12] extended the lower bound approximation of Bjerksund and Stensland beyond the classical Black–Scholes framework to models such as jump-diffusion models with different jump size distributions, multivariate stochastic volatility models, and mixtures of variance gamma models. Another recent paper [13] on pricing spread options under a non-Gaussian setup exploits the flexibility of the copula function in capturing the dependence between marginals. The authors consider variance gamma, Heston’s stochastic volatility, and affine Heston–Nandi GARCH(1,1) models.

Among practitioners, the pricing of the spread option under the assumption of lognormal prices remains the most popular due to its simplicity and ease of implementation. Thus, we chose the classical GBM setup and looked for an approximation formula.

The main contribution of our study to existing research on the problem is as follows: We derived yet another new closed-form approximation of the spread option value and Greeks by using double integration and analyzing the exercise boundary. We identified situations in the previous integration formulas that caused large errors. In such cases, we show the superiority of our formula over the previous methods and its good overall accuracy. We demonstrate the application of our sector formula to the market data of calendar spread options for major commodities.

Our approach, in some respects, is similar to the method considered in [14], where the authors propose a quadratic approximation of the exercise boundary. To overcome the difficulty of integration over a domain with a parabolic boundary, they approximate the integration using the assumption that the curvature of the real boundary near the origin is small. As a result, their method is accurate when the assumption holds. The formula is expressed in terms of univariate normal CDF functions whose arguments are defined by the coefficients of the Taylor expansion of the exercise boundary. Ultimately, the ingredients of the formula result from integration over a straight line in the two-dimensional plane.

We tackle the problem in a different way: We carefully analyzed the shape of the exercise boundary, which is defined by the sign of ${\sigma }_2-\rho {\sigma }_1$. We discovered that the straight-line approximation does not perform well for curved exercise boundaries and found explicit conditions of the inputs resulting in a big curvature and, therefore, the situations where [14]’s formula can produce errors. In our proposed approach, we integrate in a sector, thoroughly investigate the optimal choice of the sector, and find a closed-form formula expressed in terms of the bivariate normal CDF. This is a new result and can be of interest in itself. The formula is simple and can be easily implemented in Excel.

As in the other papers, we worked out Greeks, such as Delta and sensitivity to correlation (we call it “Rhoza”; see the prehistory of the name in [15]). Since a correlation is often an unknown input, this sensitivity is important for assessing model risk. The sensitivity to correlation is very useful in the calculation of the second-order Greeks of derivatives valued by Monte Carlo, such as gamma and cross-gamma (see [15]). This sensitivity is also discussed in [5], where the authors use the numerical integration formula in a more general setting. Unsurprisingly, the formulas for Greeks are fairly complicated. We identified the main terms that give reasonable estimates of Greeks in most situations.

We compare our results in terms of accuracy and speed with those obtained by three methods, refs. [9,11,14], in different types of scenarios. Our method shows great accuracy in all cases and superiority in scenarios where the curvature near the origin is not small. To compare the results, we randomized the inputs, ran the evaluation, and obtained statistics on the errors and the speed.

To verify the accuracy of closed-form pricing and Greek formulas, as a benchmark, we picked a Monte Carlo simulation method with 10,000,000 paths. We chose the Monte Carlo method because it is a standard industry procedure. To ensure the efficiency and accuracy of MC results, we applied antithetic variables and control variate techniques. Another alternative for a benchmark is to use semi-analytical techniques based on transforming the two-dimensional problem into one-dimensional integration; see [8,14]. Hereinafter, the second benchmark is referred to as “numerical integration”. We demonstrate that the two benchmark values are very close and selected the Monte Carlo method for the reasons mentioned above.

To illustrate the details of the sector formula, we give two examples and compare values and Greeks with the MC and numerical integration benchmarks.

We apply our sector formula to the market data of commodity spread options, specifically calendar spread options, for three major commodity futures: WTI, Natural Gas, and Corn. Calendar spread options provide operators of energy assets and professional traders with opportunities to hedge or benefit from moves in commodity futures spreads. These data were listed on the CME site as of 30 April, 7 May, 5 June, and 6 June 2024.

We calculated the implied correlations using the appropriate market inputs, including actively traded calendar spread options. We also applied the numerical integration formula with the same inputs to back up the implied correlations. The results are very close. The advantage of the sector formula is that it does not require any special tools and can be implemented in Excel, including for the approximated Greeks. We also illustrate how the analytical approximation of Rhoza can be utilized to calculate a model valuation adjustment to a CSO value due to the uncertainty in the correlation input. This is a standard practice for arriving at a fair value of the derivative.

The rest of this paper is organized as follows. In Section 2, we discuss the setup and analyze and approximate the exercise boundary. We derive two closed-form approximation formulas for the spread option value based on the straight-line boundary and the sector boundary, the sector formula. We also present closed-form Greeks, Deltas, and Rhoza. In Section 3, we describe the Monte Carlo procedure to arrive at the benchmark value and provide a comparison of the accuracy and speed of our method vs. the three aforementioned methods. In Section 4, we apply the sector formula to the market data of commodity spread options, specifically calendar spread options, taken from the CME site. Section 5 concludes the paper. A few of the details and proofs are provided in the Appendix sections.

2. Analytical Results of Spread Option Evaluation

2.1. Setup and Assumptions

2.1.1. Call Spread Option

Consider the spread call with maturity T and strike K, which has a payoff (1) and the value given by the risk-neutral evaluation (4).

Based on Equation (3), this value can be expressed by the following integral:

$$c(K,T)=e^{-rT}\int {\int }_{x\ge h\left(y\right)}(F_1{e}^{-{\displaystyle \frac{1}{2}}{\sigma }_1^{2}T+\sqrt{1-{\rho }^2}{\sigma }_1\sqrt{T}x+\rho {\sigma }_1\sqrt{T}y}-F_2{e}^{-{\displaystyle \frac{1}{2}}{\sigma }_2^{2}T+{\sigma }_2\sqrt{T}y}-K)\varphi \left(x\right)\varphi \left(y\right)dxdy$$

(9)

where the exercise boundary is

$$h\left(y\right)={\displaystyle \frac{ln(F_2\exp (-\frac{1}{2}{\sigma }_2^{2}T+{\sigma }_2\sqrt{T}y)+K)-\rho {\sigma }_1\sqrt{T}y}{\sqrt{1-{\rho }^2}{\sigma }_1\sqrt{T}}}+{\displaystyle \frac{\frac{1}{2}{\sigma }_1^{2}T-ln{F}_1}{\sqrt{1-{\rho }^2}{\sigma }_1\sqrt{T}}}$$

(10)

2.1.2. Exercise Boundary

Performing the integration of the payoff over such a boundary is very complicated. Therefore, we can start by investigating the boundary function $h\left(y\right)$.

The boundary slope is

$${\displaystyle \frac{\mathsf{\partial }h\left(y\right)}{\mathsf{\partial }y}}={\displaystyle \frac{1}{\sqrt{1-{\rho }^2}{\sigma }_1\sqrt{T}}}({\displaystyle \frac{F_2{\sigma }_2\sqrt{T}{e}^{-\frac{1}{2}{\sigma }_2^{2}T+{\sigma }_2\sqrt{T}y}}{F_2{e}^{-\frac{1}{2}{\sigma }_2^{2}T+{\sigma }_2\sqrt{T}y}+K}}-\rho {\sigma }_1\sqrt{T})$$

(11)

which converges asymptotically to constant values:

$$\begin{array}{ccc}\hfill & \underset{y\to +\infty }{lim}{\displaystyle \frac{\mathsf{\partial }h\left(y\right)}{\mathsf{\partial }y}}={\displaystyle \frac{{\sigma }_2-\rho {\sigma }_1}{\sqrt{1-{\rho }^2}{\sigma }_1}},\hfill & \hfill \underset{y\to -\infty }{lim}{\displaystyle \frac{\mathsf{\partial }h\left(y\right)}{\mathsf{\partial }y}}=-{\displaystyle \frac{\rho }{\sqrt{1-{\rho }^2}}}\end{array}$$

(12)

Based on these observations, the spread option evaluation can be divided into two cases, depending on the values of ${\sigma }_1$, ${\sigma }_2$, and $\rho$:

• In the first case, ${\sigma }_2\le \rho {\sigma }_1$, and the asymptotic slopes have the same sign.
• In the second case, ${\sigma }_2>\rho {\sigma }_1$, and the asymptotic slopes have different signs.

We illustrate these two cases in the following Figure 1:

2.1.3. Boundary Approximation

To approximate the boundary $\left\{\right(x,y):x\ge h(y\left)\right\}$, we propose two types of approximations.

• The first type is a straight-line boundary:

  $$A(a,b)=\left\{\right(x,y):x-by\ge a\}$$
• The second type is a sector boundary:

  $$A(a,b,c,d)=\left\{\right(x,y):x-by\ge a,x-dy\ge c\}$$

  with $b-d\ne 0$.

The integration of the spread call payoff over the straight-line-type boundary can be calculated as

$$I(a,b)\stackrel{\Delta }{=}{e}^{-rT}\int {\int }_{A(a,b)}(F_1{e}^{-{\displaystyle \frac{1}{2}}{\sigma }_1^{2}T+\sqrt{1-{\rho }^2}{\sigma }_1\sqrt{T}x+\rho {\sigma }_1\sqrt{T}y}-F_2{e}^{-{\displaystyle \frac{1}{2}}{\sigma }_2^{2}T+{\sigma }_2\sqrt{T}y}-K)\varphi \left(x\right)\varphi \left(y\right)dxdy$$

(13)

Correspondingly, the integration of the call spread payoff over the sector-type boundary can be calculated as

$$J(a,b,c,d)\stackrel{\Delta }{=}{e}^{-rT}\int {\int }_{A(a,b,c,d)}(F_1{e}^{-{\displaystyle \frac{1}{2}}{\sigma }_1^{2}T+\sqrt{1-{\rho }^2}{\sigma }_1\sqrt{T}x+\rho {\sigma }_1\sqrt{T}y}-F_2{e}^{-{\displaystyle \frac{1}{2}}{\sigma }_2^{2}T+{\sigma }_2\sqrt{T}y}-K)\varphi \left(x\right)\varphi \left(y\right)dxdy$$

(14)

Lemma 1. 

$$I(a,b)=e^{-rT}(F_{1}N\left(d_1\right)-F_{2}N\left(d_2\right)-KN\left(d_3\right))$$

(15)

where

$$d_1=-{\displaystyle \frac{a+(b\rho -\sqrt{1-{\rho }^2}){\sigma }_1\sqrt{T}}{\sqrt{1+b^2}}},d_2=-{\displaystyle \frac{a+b{\sigma }_2\sqrt{T}}{\sqrt{1+b^2}}},d_3=-{\displaystyle \frac{a}{\sqrt{1+b^2}}}$$

Proof. 

See Appendix A. □

Lemma 2. 

$$J(a,b,c,d)=e^{-rT}(F_{1}M(d_{11},d_{12},\tilde{\rho })-F_{2}M(d_{21},d_{22},\tilde{\rho })-KM(d_{31},d_{32},\tilde{\rho }))$$

(16)

where

$$\tilde{\rho }={\displaystyle \frac{bd+1}{\sqrt{(1+b^2)(1+d^2)}}}$$

$$M(a,b,\tilde{\rho })=\int {\int }_{\{x<a,y<b\}}f(x,y,\tilde{\rho })dxdy,f(x,y,\tilde{\rho })={\displaystyle \frac{1}{2\pi \sqrt{1-{\tilde{\rho }}^2}}}{e}^{-{\displaystyle \frac{x^2+y^2-2\tilde{\rho }xy}{2(1-{\tilde{\rho }}^2)}}}$$

$$d_{11}=-{\displaystyle \frac{a+(b\rho -\sqrt{1-{\rho }^2}){\sigma }_1\sqrt{T}}{\sqrt{1+b^2}}},d_{21}=-{\displaystyle \frac{a+b{\sigma }_2\sqrt{T}}{\sqrt{1+b^2}}},d_{31}=-{\displaystyle \frac{a}{\sqrt{1+b^2}}}$$

$$d_{12}=-{\displaystyle \frac{c+(d\rho -\sqrt{1-{\rho }^2}){\sigma }_1\sqrt{T}}{\sqrt{1+d^2}}},d_{22}=-{\displaystyle \frac{c+d{\sigma }_2\sqrt{T}}{\sqrt{1+d^2}}},d_{32}=-{\displaystyle \frac{c}{\sqrt{1+d^2}}}$$

Proof. 

See Appendix B. □

2.2. Domain of Integration in Prices and Returns

The expression $J(a,b,c,d)$ describes the result of the integration of the spread payoff over any sector region in the x-y plane. To clarify its meaning, we can transform $x,y$ into future prices and returns at maturity.

Corollary 1. 

For any given (a,b,c,d) with $b-d\ne 0$, if $\rho <1$, we have the presentation of the integration $J(a,b,c,d)$ as future prices at maturity.

$$J(a,b,c,d)=e^{-rT}E[(F_1\left(T\right)-F_2\left(T\right)-K)I_{\{F_{1T}\ge {\alpha }_1{F}_2{\left(T\right)}^{{\beta }_1}\}\cap \{F_1\left(T\right)\ge {\alpha }_2{F}_2{\left(T\right)}^{{\beta }_2}\}}]$$

(17)

where

$${\alpha }_1={\displaystyle \frac{F_1{e}^{a\sqrt{1-{\rho }^2}{\sigma }_1\sqrt{T}-\frac{1}{2}({\sigma }_1^2-{\sigma }_2^2{\beta }_1)T}}{F_2^{{\beta }_1}}}$$

$${\beta }_1={\displaystyle \frac{b\sqrt{1-{\rho }^2}{\sigma }_1\sqrt{T}+\rho {\sigma }_1}{{\sigma }_2}}$$

$${\alpha }_2={\displaystyle \frac{F_1{e}^{c\sqrt{1-{\rho }^2}{\sigma }_1\sqrt{T}-\frac{1}{2}({\sigma }_1^2-{\sigma }_2^2{\beta }_2)T}}{F_2^{{\beta }_2}}}$$

$${\beta }_2={\displaystyle \frac{d\sqrt{1-{\rho }^2}{\sigma }_1\sqrt{T}+\rho {\sigma }_1}{{\sigma }_2}}$$

Proof. 

See Appendix B. □

Let $R_{1T}=ln{\displaystyle \frac{F_1\left(T\right)}{F_1\left(0\right)}}$ and $R_{2T}=ln{\displaystyle \frac{F_2\left(T\right)}{F_2\left(0\right)}}$ be the returns of the two assets at maturity. Then, we can easily obtain

$$J(a,b,c,d)=e^{-rT}E[(F_1\left(T\right)-F_2\left(T\right)-K)I_{\{R_{1T}-{\beta }_1{R}_{2T}\ge ln({\alpha }_1{\displaystyle \frac{F_2^{{\beta }_1}}{F_1}})\}\cap \{R_{1T}-{\beta }_2{R}_{2T}\ge ln({\alpha }_2{\displaystyle \frac{F_2^{{\beta }_2}}{F_1}})\}}]$$

(18)

This representation shows that the integration region is also a sector shape region in the returns plane, where the sector is defined by $({\alpha }_1,{\beta }_1,{\alpha }_2,{\beta }_2)$. The presented domain of integration generalizes the domain suggested in [11].

2.3. Closed-Form Spread Option Formulas

The shape of the exercise boundary depends on the parameters ${\sigma }_1$, ${\sigma }_2$, $\rho$, and moneyness.

For example, when ${\sigma }_2-\rho {\sigma }_1\le 0$, the boundary can be approximated reasonably well by a straight line. This is illustrated by Figure 2, where we plot the exercise boundary for the chosen $F_1$, $F_2$, ${\sigma }_1$, ${\sigma }_2$, and T and different choices of correlation $\rho$ and strike K. The ATM strike is defined by $K_{ATM}=F_1-F_2$.

When ${\sigma }_2-\rho {\sigma }_1>0$, there is a turning point on the boundary, where the derivative $h^{\prime }\left(y\right)=0$. If the strike is small, the turning point is far away from the origin, and the curvature near the origin is small since the boundary $h\left(y\right)$ converges to a straight line as $y\to \infty$. However, if K becomes larger, the turning point moves closer to the origin, which leads to a larger curvature near the origin. This is shown in Figure 3.

Other situations that create a bigger curvature are high correlations, differences in ${\sigma }_1$ and ${\sigma }_2$, and longer horizons; more details are provided in Appendix C. In some of the scenarios, previous approximation methods can cause large errors. This will be illustrated by the numerical tests.

We propose two different closed-form approximation formulas based on the straight-line boundary and the sector boundary. For the straight-line-type boundary, we approximate the real boundary by the tangent line of the boundary $h\left(y\right)$ at $y=0$ and obtain our first closed-form approximation formula (see Figure 4 and Figure 5, red lines).

Proposition 1.

Let $K\ge 0$ and $\left|\rho \right|<1$. Under the GBM setup and using a straight-line boundary, the approximated spread call value is given by the following formula:

$$\tilde{c}(F_1,F_2,T,K)=e^{-rT}(F_{1}N\left(d_1\right)-F_{2}N\left(d_2\right)-KN\left(d_3\right))$$

(19)

where

$$d_1=-{\displaystyle \frac{a+(b\rho -\sqrt{1-{\rho }^2}){\sigma }_1\sqrt{T}}{\sqrt{1+b^2}}},d_2=-{\displaystyle \frac{a+b{\sigma }_2\sqrt{T}}{\sqrt{1+b^2}}},d_3=-{\displaystyle \frac{a}{\sqrt{1+b^2}}}$$

$$a={\displaystyle \frac{ln(F_2\exp (-\frac{1}{2}{\sigma }_2^{2}T)+K)+\frac{1}{2}{\sigma }_1^{2}T-ln{F}_1}{\sqrt{1-{\rho }^2}{\sigma }_1\sqrt{T}}}$$

$$b={\displaystyle \frac{\frac{F_2\exp (-\frac{1}{2}{\sigma }_2^{2}T){\sigma }_2}{F_2\exp (-\frac{1}{2}{\sigma }_2^{2}T)+K}-\rho {\sigma }_1}{\sqrt{1-{\rho }^2}{\sigma }_1}}$$

Proof. 

See Appendix C. □

For a U-shape type of boundary, we use the sector determined by two tangent lines near the origin to approximate the real boundary (see Figure 5, blue lines). We develop the second closed-form approximation formula, expressed in terms of the bivariate normal CDF. This is the main result of this paper. We call it the sector formula.

Proposition 2.

Let $K\ge 0$ and $\left|\rho \right|<1$. Under the geometric Brownian motion’s setup and using a sector boundary, the approximated spread call value is given by the following formula:

$$\widehat{c}(F_1,F_2,T,K)=e^{-rT}(F_{1}M(d_{11},d_{12},\tilde{\rho })-F_{2}M(d_{21},d_{22},\tilde{\rho })-KM(d_{31},d_{32},\tilde{\rho }))$$

(20)

where

$$\tilde{\rho }={\displaystyle \frac{bd+1}{\sqrt{(1+b^2)(1+d^2)}}}$$

$$M(a,b,\tilde{\rho })=\int {\int }_{\{x<a,y<b\}}f(x,y,\tilde{\rho })dxdy,f(x,y,\tilde{\rho })={\displaystyle \frac{1}{2\pi \sqrt{1-{\tilde{\rho }}^2}}}{e}^{-{\displaystyle \frac{x^2+y^2-2\tilde{\rho }xy}{2(1-{\tilde{\rho }}^2)}}}$$

$$d_{11}=-{\displaystyle \frac{a+(b\rho -\sqrt{1-{\rho }^2}){\sigma }_1\sqrt{T}}{\sqrt{1+b^2}}},d_{21}=-{\displaystyle \frac{a+b{\sigma }_2\sqrt{T}}{\sqrt{1+b^2}}},d_{31}=-{\displaystyle \frac{a}{\sqrt{1+b^2}}}$$

$$d_{12}=-{\displaystyle \frac{c+(d\rho -\sqrt{1-{\rho }^2}){\sigma }_1\sqrt{T}}{\sqrt{1+d^2}}},d_{22}=-{\displaystyle \frac{c+d{\sigma }_2\sqrt{T}}{\sqrt{1+d^2}}},d_{32}=-{\displaystyle \frac{c}{\sqrt{1+d^2}}}$$

$$a=x_1-b·y_1,b=h^{\prime }\left(y_1\right),c=x_2-d·y_2,d=h^{\prime }\left(y_2\right)$$

$$x_1=h\left(y_1\right),y_1={\displaystyle \frac{-uv+\sqrt{R^2(u^2+1)-v^2}}{u^2+1}}$$

$$x_2=h\left(y_2\right),y_2={\displaystyle \frac{-uv-\sqrt{R^2(u^2+1)-v^2}}{u^2+1}}$$

$$u={\displaystyle \frac{{\sigma }_2-\rho {\sigma }_1-\frac{K{\sigma }_2}{F_2{e}^{-\frac{1}{2}{\sigma }_2^{2}T}+K}}{\sqrt{1-{\rho }^2}{\sigma }_1}},v={\displaystyle \frac{ln\frac{F_2{e}^{-\frac{1}{2}{\sigma }_2^{2}T}+K}{F_1}+\frac{1}{2}{\sigma }_1^{2}T}{\sqrt{1-{\rho }^2}{\sigma }_1\sqrt{T}}},R=max(1,{\displaystyle \frac{6}{5}}\left|h\left(0\right)\right|)$$

Proof. 

See Appendix D. □

Both formulas can be used to value spread call pricing. The tangent (red) line formula (Proposition 1) is more suitable for scenarios where the curvature is small near the origin. By contrast, the sector boundary (blue lines) is universal, and it works in all scenarios. This is illustrated in Figure 4 and Figure 5. Using numerical tests, we will show that the sector formula gives better precision. This is not surprising, as by choosing an appropriate sector, we can better approximate the exercise boundary compared with a straight line. We kept the same choices of ${\sigma }_1$, ${\sigma }_2$, and T as before.

2.4. Closed-Form Greeks

For the sector formula in Proposition 2, we find closed-form Deltas with respect to $F_1$ and $F_2$, as well as a closed-form Rhoza, i.e., the first-order derivative of the call spread option price with respect to correlation $\rho$. Recall the sector formula:

$$\widehat{c}(F_1,F_2,T,K)=e^{-rT}(F_{1}M(d_{11},d_{12},\tilde{\rho })-F_{2}M(d_{21},d_{22},\tilde{\rho })-KM(d_{31},d_{32},\tilde{\rho }))$$

(21)

Proposition 3.

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathsf{\partial }\widehat{c}(F_1,F_2,T,K)}{\mathsf{\partial }{F}_1}}& =& {\Delta }_1+C_1{D}_{11}+C_2{D}_{12}+C_3{D}_{13}+C_4{D}_{14}\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathsf{\partial }\widehat{c}(F_1,F_2,T,K)}{\mathsf{\partial }{F}_2}}& =& {\Delta }_2+C_1{D}_{21}+C_2{D}_{22}+C_3{D}_{23}+C_4{D}_{24}\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathsf{\partial }\widehat{c}(F_1,F_2,T,K)}{\mathsf{\partial }\rho }}& =& Z+C_1{D}_{31}+C_2{D}_{32}+C_3{D}_{33}+C_4{D}_{44}\hfill \end{array}$$

(24)

In these expressions, we identify the terms that make the main contributions. For example, for Delta with respect to $F_1$, it is the term ${\Delta }_1$ that is multiplied by $F_1$ (similar to Black Delta $e^{-rT}N\left(d_1\right)$); for Delta with respect to $F_2$, it is the term ${\Delta }_2$ that is multiplied by $F_2$. Rhoza’s decomposition is less obvious. Specifically,

$${\Delta }_1=e^{-rT}M(d_{11},d_{12},\widehat{\rho }),{\Delta }_2=-e^{-rT}M(d_{21},d_{22},\widehat{\rho }),$$

$$Z=-e^{-rT}(F_1{M}_x(d_{11},d_{12},\widehat{\rho }){\displaystyle \frac{{\sigma }_1\sqrt{T}(b\sqrt{1-{\rho }^2}+\rho )}{\sqrt{(1+b^2)(1-{\rho }^2)}}}+F_1{M}_y(d_{11},d_{12},\widehat{\rho }){\displaystyle \frac{{\sigma }_1\sqrt{T}(d\sqrt{1-{\rho }^2}+\rho )}{\sqrt{(1+d^2)(1-{\rho }^2)}}})$$

(25)

The derivatives of the binomial normal CDF $M_x(d_{11},d_{12},\rho )$,$M_y(d_{11},d_{12},\rho )$ can be expressed through the univariate pdf and CDF of the normal distribution:

$$M_x(x,y,\rho )=n\left(y\right)N\left(x,\rho y,\sqrt{1-{\rho }^2}\right)$$

$$M_y(x,y,\rho )=n\left(y\right)N\left(y,\rho x,\sqrt{1-{\rho }^2}\right)$$

where $n\left(x\right)$ is the pdf of the standard normal distribution, and $N(a,\mu ,\sigma )$ is the normal CDF with mean $\mu$ and standard deviation $\sigma$ evaluated at a.

See Appendix E for further details on the given formulas, such as $C_i$ and $D_{ij}$.

Unsurprisingly, the above formulas for Greeks are pretty complex. The identified main terms ${\Delta }_1$, ${\Delta }_2$, and Z are simple but still pretty accurate approximations of the Greeks ${\displaystyle \frac{\mathsf{\partial }\widehat{c}(K,T)}{\mathsf{\partial }{F}_1}}$, ${\displaystyle \frac{\mathsf{\partial }\widehat{c}(K,T)}{\mathsf{\partial }{F}_2}}$, and ${\displaystyle \frac{\mathsf{\partial }\widehat{c}(K,T)}{\mathsf{\partial }\rho }}$ in most situations. This will be verified by our numerical tests.

2.5. Two Examples

We illustrate the details of the sector and line formulas on two examples and compare them with the MC and numerical integration benchmarks.

Table 1. Two examples of calculations of spread options by using the line and sector formulas.

Case	${\mathit{F}}_1$	${\mathit{F}}_2$	K	T	${\mathit{\sigma }}_1$	${\mathit{\sigma }}_2$	$\mathit{\rho }$	r	MC	Num Int	Line	Sector
1	100	90	10	1	0.2	0.36	0.7	0	9.2676	9.2677	9.2661	9.2675
2	100	70	30	1	0.3	0.5	0.9	0	6.2108	6.211	6.0072	6.1860

demonstrates the comparison of the values. The two benchmarks are very close; the sector formula is more accurate, especially for the second case of higher correlations.

Table 2. Line parameters.

Case	Line	${\mathit{d}}_1$	${\mathit{d}}_2$	${\mathit{d}}_3$	a	b
1	9.2661	0.1430	−0.1182	0.1649	−0.2669	1.2732
2	6.0072	0.2724	0.0515	0.2783	−0.3123	0.5090

gives the ingredients of the line formula (19) for the two examples.

Table 3. Sector parameters, part I.

Case	Sector	${\mathit{d}}_{11}$	${\mathit{d}}_{12}$	${\mathit{d}}_{21}$	${\mathit{d}}_{22}$	${\mathit{d}}_{31}$	${\mathit{d}}_{32}$	$\tilde{\mathit{\rho }}$
1	9.2675	0.1465	0.1568	−0.1151	−0.1039	0.1726	0.1753	0.9993
2	6.1860	0.2860	0.5045	0.0358	0.3341	0.3667	0.4251	0.8576

and

Table 4. Sector parameters, part II.

Case	a	b	c	d	${\mathit{x}}_1$	${\mathit{y}}_1$	${\mathit{x}}_2$	${\mathit{y}}_2$	u	v
1	−0.287	1.330	−0.278	1.229	0.686	0.739	−0.867	−0.477	1.273	−0.267
2	−0.489	0.883	−0.432	0.185	0.378	0.982	−0.567	−0.730	0.509	−0.312

provide the details of the sector formula (20) for the same examples.

In

Table 5. Sector Greeks vs. Greeks from numerical integration.

Case	Sector Delta ${\mathit{D}}_1$	Sector Delta ${\mathit{D}}_2$	Sector Rhoza Z	Delta ${\mathit{F}}_1$ num. int	Delta ${\mathit{F}}_2$ num. int	Rhoza num. int
1	0.5540	−0.4501	−10.9953	0.5525	−0.4416	−11.001
2	0.5672	−4758	−26.6393	0.5470	−0.4572	−26.6744

, we list the approximate sector Greeks, Deltas, and Rhoza for our examples: $D_1$, (22), $D_2$, (23), and Z, (24). We compare them with the Greeks obtained by the finite difference of the numerical integration formula (we used a small bump size of $0.001$).

In the first case, the sector Deltas and true Deltas are very close, within a $0.3\%$ difference. In the second case, the difference in Deltas is bigger but still reasonable, within $4\%$. Rhozas are remarkably close in both cases.

3. Numerical Results

3.1. Monte Carlo Simulation

To verify the accuracy of our closed-form pricing and Greek formulas using the sector formula, we selected the Monte Carlo simulation method as a benchmark. To ensure the efficiency and accuracy of the MC results, we applied antithetic variables and control variate techniques. The methodology is described in Appendix F.

We chose ATM options, as these are the most sensitive to inputs.

To verify the accuracy of the MC results, we performed the following steps:

• Randomly generate inputs for five ATM spread call option samples from the following range: $F_1=100$, ${\displaystyle \frac{K}{F_2}}\in (0.05,0.2)$, $F_1-F_2-K=0$, $T=1$, $r=0$, ${\sigma }_1=2$, ${\sigma }_2\in (0.2,0.6)$, $\rho \in (0.7,0999)$.
• For each ATM sample, repeat the simulation 100 times, and for each simulation, use 10,000,000 paths.
• For a given ATM sample and each sample of 100 simulation sets, calculate the percentage differences between the MC results and the approximation results using the line formula (19) and the sector formula (20). Calculate the percentage errors in the Deltas as well. (We chose percentage errors to remove the dependence of price levels.) Calculate the benchmark Delta from MC using the central finite difference and the bump size $ϵ=0.0001$.
• Calculate the standard deviation of 100 MC results for option values and Deltas and compare them with the average errors over the same 100 simulation sets for option prices and Deltas.

From the results in

Table 6. Monte Carlo simulation accuracy, values.

	Price
No.	MC Std	Line	Sector
1	0.0001	−0.0209	−0.0024
2	0.0000	−0.0079	−0.0006
3	0.0000	−0.0170	−0.0014
4	0.0000	−0.0068	−0.0005
5	0.0000	−0.0004	0.0000

and

Table 7. Monte Carlo simulation accuracy, Deltas.

		Delta
No.	MC Std	Line	Sector	MC Std	Line	Sector
1	0.0018	0.0193	0.0061	0.0018	−0.0471	−0.0328
2	0.0022	0.0090	0.0024	0.0008	−0.0188	−0.0117
3	0.0037	0.0158	0.0050	0.0018	−0.0449	−0.0330
4	0.0021	0.0086	0.0024	0.0008	−0.0174	−0.0108
5	0.0005	0.0020	0.0006	0.0004	−0.0026	−0.0012

(the left three columns in Table 7 correspond to ${\Delta }_{F_1}$ and the right three to ${\Delta }_{F_2}$), we can observe that the simulation errors are immaterial in comparison with the formula errors. This implies that our simulation method can serve as a reasonable benchmark for our tests.

3.2. Pricing Accuracy under Different Scenarios

To test the performance of our sector formula and line formula and compare them with the previous formulas from [9,11,14], we implemented our numerical tests in three different scenarios. In each of the scenarios, we randomly chose the inputs and obtained a distribution of percentage errors vs. the MC benchmark prices. We list the statistics of this distribution below.

3.2.1. Scenario 1: Low Strikes and Correlations

In this scenario, we randomly generated 500 ATM samples with $F_1=100$, $T=1$, $r=0$, ${\sigma }_1\in (0.2,0.5)$, ${\sigma }_2\in (0.2,0.5)$, ${\displaystyle \frac{K}{F_2}}\in (0.05,0.2)$, and $\rho \in (0.6,0.8)$. The results are given in

Table 8. Pricing accuracy of Scenario 1.

	Kirk	Bjerksund	Li	Tangent Line	Sector
Mean	0.03%	0.04%	0.00%	0.07%	0.01%
Std	0.04%	0.05%	0.00%	0.08%	0.01%
Max	0.22%	0.26%	0.02%	0.45%	0.05%
Min	0.00%	0.00%	0.00%	0.00%	0.00%
Median	0.02%	0.03%	0.00%	0.04%	0.01%

.

3.2.2. Scenario 2: Higher Strikes and Correlations and ${\sigma }_2-\rho {\sigma }_1\le 0$

In this scenario, we randomly generated 500 ATM samples with $F_1=100$, $T=1$, $r=0$, ${\sigma }_1\in (0.3,0.4)$, ${\displaystyle \frac{{\sigma }_2}{\rho {\sigma }_1}}\in (0.5,1)$, ${\displaystyle \frac{K}{F_2}}\in (0.2,0.5)$, and $\rho \in (0.8,0.99)$. See

Table 9. Pricing accuracy of Scenario 2.

	Kirk	Bjerksund	Li	Tangent Line	Sector
Mean	0.13%	0.05%	0.00%	0.05%	0.01%
Std	0.11%	0.07%	0.00%	0.07%	0.01%
Max	0.57%	0.64%	0.04%	0.36%	0.05%
Min	0.00%	0.00%	0.00%	0.00%	0.00%
Median	0.09%	0.03%	0.00%	0.02%	0.00%

.

3.2.3. Scenario 3: Higher Strikes and Correlations and ${\sigma }_2-\rho {\sigma }_1>0$

In this scenario, we randomly generated 500 ATM samples with $F_1=100$, $T=1$, $r=0$, ${\sigma }_1\in (0.3,0.4)$, ${\displaystyle \frac{{\sigma }_2}{\rho {\sigma }_1}}\in (1.5,2)$, ${\displaystyle \frac{K}{F_2}}\in (0.2,0.5)$, and $\rho \in (0.8,0.99)$. See

Table 10. Pricing accuracy of Scenario 3.

	Kirk	Bjerksund	Li	Tangent Line	Sector
Mean	1.83%	1.93%	1.39%	3.68%	0.44%
Std	1.79%	1.77%	4.58%	3.82%	0.47%
Max	18.52%	18.55%	57.53%	37.20%	3.76%
Min	0.08%	0.28%	0.00%	0.36%	0.04%
Median	1.29%	1.38%	0.20%	2.49%	0.29%

.

From the above numerical results, we can observe that in Scenario 1 and Scenario 2, all five methods are accurate. Among them, the formula in [14] and our sector formula are the most accurate. In Scenario 3, for higher strikes and higher correlations, as expected, all of the errors become larger. Some of the errors are significant. In contrast, the sector formula performs much better, and even in extreme cases, the errors are tolerable.

3.2.4. Pricing Speed

From the 1500 samples in the previous three scenarios, we summarize the calculation speed in

Table 11. Pricing speed comparison.

	Kirk	Bjerksund	Li	Tangent Line	Sector
Mean	0.0002	0.0003	0.0003	0.0003	0.0008
Std	0.0000	0.0001	0.0000	0.0001	0.0001
Max	0.0003	0.0004	0.0004	0.0005	0.0013
Min	0.0001	0.0002	0.0002	0.0002	0.0006
Median	0.0002	0.0003	0.0003	0.0004	0.0008

.

From the above statistics, we see that, for the sector formula, it takes twice as long as other methods to obtain the price of a spread call on average. The reason for this is that, in the sector formula, we use the bivariate normal CDF, which is more complex than the standard normal CDF.

3.3. Greek Accuracy and Speed

3.3.1. Three Methods in Comparison with MC Simulation

In this part, we verify the accuracy of the formulas for Greeks given in Proposition 3. We also check that the main terms give a reasonable approximation in most cases. We use the Monte Carlo Greeks as a benchmark. (Another benchmark for Greeks can be obtained through the numerical integration method by differentiating the integrand with respect to price or correlation and calculating the integral.)

We employ the following three methods:

• The first method is to use the numerical Greeks calculated by the central finite difference using the formulas in Proposition 3.

  $$\begin{array}{ccc}\hfill {{\displaystyle \frac{\mathsf{\partial }\widehat{c}(F_1,F_2,T,K)}{\mathsf{\partial }{F}_1}}}^{\left(ϵ\right)}& =& {\displaystyle \frac{\widehat{c}(F_1+ϵ)-\widehat{c}(F_1-ϵ)}{2ϵ}}\hfill \end{array}$$

  (26)

  $$\begin{array}{ccc}\hfill {{\displaystyle \frac{\mathsf{\partial }\widehat{c}(F_1,F_2,T,K)}{\mathsf{\partial }{F}_2}}}^{\left(ϵ\right)}& =& {\displaystyle \frac{\widehat{c}(F_2+ϵ)-\widehat{c}(F_2-ϵ)}{2ϵ}}\hfill \end{array}$$

  (27)

  $$\begin{array}{ccc}\hfill {{\displaystyle \frac{\mathsf{\partial }\widehat{c}(F_1,F_2,T,K)}{\mathsf{\partial }\rho }}}^{\left(ϵ\right)}& =& {\displaystyle \frac{\widehat{c}(\rho +ϵ)-\widehat{c}(\rho -ϵ)}{2ϵ}}\hfill \end{array}$$

  (28)
• The second method is to use the formulas given in Proposition 3.
• The third method is to use only the main terms, i.e., ${\Delta }_1$, ${\Delta }_2$, and Z.

3.3.2. Delta Accuracy and Speed

We generated 500 ATM samples with $F_1=100$, ${\sigma }_1=0.2$, $T=1$, and $r=0$ and randomly chose $K/F_2\in (0.05,0.5)$, ${\sigma }_2\in (0.2,0.6)$, and $\rho \in (0.7,0.99)$. For the numerical Greeks, the bump size is $ϵ=0.0001$. As before, the benchmark Delta from MC is calculated using a central finite difference and $ϵ=0.0001$. The statistics of the distribution of percentage errors in Greeks for the three methods in comparison with the MC benchmark are listed in

Table 12. Delta accuracy.

	${\mathbf{\Delta }}_{{\mathit{F}}_1}$			${\mathbf{\Delta }}_{{\mathit{F}}_2}$
	Numerical	Formula	Main Term	Numerical	Formula	Main Term
Mean	0.42%	0.42%	1.01%	4.45%	4.45%	5.73%
Std	0.45%	0.46%	1.04%	4.98%	4.99%	6.18%
Max	2.39%	2.38%	5.05%	25.01%	25.08%	33.40%
Min	0.00%	0.00%	0.00%	0.11%	0.11%	0.18%
Median	0.25%	0.26%	0.66%	2.57%	2.57%	3.72%

. The errors in ${\Delta }_{F_2}$ are perceptibly bigger than the errors in ${\Delta }_{F_1}$. The reason for this discrepancy is the errors in our sector formula, mainly caused by $F_2$-related parts. This leads to a bigger error in the main term, ${\Delta }_2$, and consequently to a bigger overall error.

To compare the speed of Delta calculations, we randomly chose 10 samples and compare the statistics in

Table 13. Delta speed.

	Numerical	Formula	Main Term
Mean	0.0037	0.0085	0.0006
Std	0.0004	0.0008	0.0001
Max	0.0040	0.0097	0.0007
Min	0.0031	0.0074	0.0005
Median	0.0037	0.0086	0.0006

.

The above results affirm the validity of our Delta formulas in Proposition 3 and also show that the main terms ${\Delta }_1$ and ${\Delta }_2$ are reasonable approximations in most situations. It is much faster than the other methods, and its errors are tolerable.

3.3.3. Rhoza Accuracy and Speed

We generated 500 ATM samples with fixed $F_1=100$, ${\sigma }_1=0.2$, $T=1$, and $r=0$, randomly choosing $K/F_2\in (0.05,0.5)$, ${\sigma }_2\in (0.2,0.6)$, and $\rho \in (0.7,0.99)$. For the numerical formula, we took $ϵ=0.0001$. The errors in Rhoza for the three methods are given in comparison with the MC benchmark. The statistics of the distribution of percentage errors are listed in

Table 14. Rhoza accuracy.

	Numerical	Formula	Main Term
Mean	0.10%	0.11%	0.23%
Std	0.13%	0.18%	0.26%
Max	1.60%	2.70%	1.78%
Min	0.00%	0.00%	0.00%
Median	0.05%	0.05%	0.14%

.

To compare the Delta speed, we chose 10 samples and compare the statistics in

Table 15. Rhoza speed.

	Numerical	Formula	Main Term
Mean	0.0014	0.0048	0.0000
Std	0.0002	0.0002	0.0000
Max	0.0017	0.0050	0.0000
Min	0.0011	0.0045	0.0000
Median	0.0014	0.0048	0.0000

.

The above results validate our Rhoza formula in Proposition 3 and demonstrate that the main term Z gives reasonable accuracy in most practical situations. It is much faster than the other methods, and its errors are tolerable.

4. Applications of the Sector Formula to the Market Data of Commodity Spread Options

Now, we turn our attention to the real commodity data. We apply our sector formula to the market data of commodity spread options taken from the CME site. Specifically, we analyze calendar spread options (“CSOs”).

As stated in the CME white paper “Trading Energy Spread Option on Energy Futures”, “Calendar Spread Options (CSOs) are options on the spread between two different futures expirations. The Energy futures term structure represents the time value of Energy market variables, such as storage costs, seasonality, and supply/demand conditions. Calendar Spread Options provide a leveraged means of hedging against, or speculating on, a change in the shape of the futures term structure”.

Employing our sector formula, market future prices, and volatilities, we calculated the implied correlations, matching the given quotes. We also applied the numerical integration formula with the same inputs to back up implied correlations for verification purposes. The calendar correlations are vital in the evaluation of commodity assets, such as storage, as well as widespread commodity options, such as swaptions.

We chose to use settlement prices, which are determined by trading activity during a given settlement period and, therefore, represent the fair market value of a commodity or financial derivative. We analyzed the market data of CSOs for three important commodities: WTI, Natural Gas, and Corn. These data change daily. We used website materials and white papers on CSOs published by the CME group.

4.1. WTI CSOs

The Crude Oil futures and options markets are global and are the most liquid and actively traded commodity contracts in the world. The forward term structure of Crude Oil is guided mainly by supply/demand, storage, and production. The shape of the forward curve, backwardation vs. contango, has important implications for inventory management. CSOs provide efficient tools for hedging to minimize the impact of the changes in the shape or trade to express views on oil price movements.

We started with WTI CSOs and used settlement prices from the CME site, https://www.cmegroup.com/markets/energy/crude-oil/light-sweet-crude.settlements.options.html#optionProductId=2952 (CME WTI CSOs, URL accessed on 30 April 2024 and 7 May 2024). The CME Chapter 390 states, “WTI calendar spread option is an option to assume a long position in the first expiring Light Sweet Crude Oil futures contract in the spread and a short position in the second expiring Light Sweet Crude Oil futures contract in the spread”. Two futures entering the spread could be consecutive (1-month calendar spread option) or 2, 3, 6, or 12 months apart.

Currently, the only actively traded CSOs that we can find on the CME site are 1-month CSOs, starting with the prompt month and going 12 months forward. The biggest total open interest is observed for the first three months. At the time of writing this section, the prompt month was July 2024, and the total open daily interest for July, August, and September 2024 was around ∼95,000–99,000 contracts. It gradually decreased to ∼66,000 for December and practically disappeared thereafter. The strikes of the options are listed in increments, typically of USD $0.25$, starting with $-0.75$ up to USD 3.

We use the settlement prices (at the end of the day) of one-month calendar spread options and record the contemporaneous settlement future prices of the contracts on the same day. The implied volatilities are derived (“implied”) from the settlements of options on WTI futures (as of the same day); we use ATM options. We chose to employ only actively traded options with a sizable amount of open interest. These are one-month CSOs and strikes close to ATM strikes: $K_{ATM}=F_1-F_2$. We use Treasury rates for discounts.

In

Table 16. Implied correlations for different strikes K, 24 December/25 January CSO, call quotes as of 30 April 2024, $F_1=77.95$, $F_2=77.34$, ${\sigma }_1=0.2534$, ${\sigma }_2=0.2537$, and $K_{ATM}=0.61$.

Case	Strike K	Quote	Implied $\mathit{\rho }$ Sector	Implied $\mathit{\rho }$ num. Integration
1	0.25	0.5	0.9987	0.9987
2	0.75	0.32	0.9977	0.9977
3	1	0.25	0.9974	0.9974
4	1.25	0.2	0.997	0.997
5	1.5	0.15	0.9968	0.9968
6	2	0.1	0.9960	0.9961

and

Table 17. Implied correlations for different strikes K, 24 July/24 August CSO, call quotes as of 7 May 2024, $F_1=78.06$, $F_2=77.68$, ${\sigma }_1=0.252$, ${\sigma }_2=0.2456$, and $K_{ATM}=0.38$.

Case	Strike K	Quote	Implied $\mathit{\rho }$ Sector	Implied $\mathit{\rho }$ num. Integration
1	0.5	0.17	0.9963	0.9963
2	0.75	0.13	0.9943	0.9923
3	1	0.1	0.9923	0.9923
4	1.25	0.08	0.9902	0.9902
5	1.5	0.06	0.9896	0.9896
6	2	0.04	0.9842	0.9842

, we list strikes, settlement call CSO prices, and future prices of the underlying contracts and report the calculated implied correlations (“correlation smile”) employing our sector formula, and we compare them to the implied correlations obtained with the use of the numerical integration. The results are practically the same.

The ATM correlations from the one-month 24 July/24 August CSO are slightly lower than the correlations from the one-month 24 December/25 January CSO, as they should be, since the first contract is closer for the first CSO; see [16].

4.2. Natural Gas CSOs

The Natural Gas term structure is defined by seasonality. Natural Gas storage is a vital part of the Natural Gas delivery system and helps to balance supply and demand. During the winter season, from November to March, gas consumption peaks as a result of increased heating demand. During the summer season, from April to October, gas demand decreases while production continues, and excess Natural Gas can be stored. Natural Gas storage can be replicated by the option to inject excessive gas during the injection season and sell it during the withdrawal period, factoring in financing and storage costs. CSOs provide physical storage operators and professional traders with opportunities to hedge or benefit from moves in Natural Gas futures spreads.

As with WTI, we use the settlement prices of Natural Gas CSOs, https://www.cmegroup.com/markets/energy/natural-gas/natural-gas.settlements.options.html#optionProductId=2946 (CME NG CSOs, URL accessed on 6 June 2024). We record the contemporaneous settlement future prices of the contracts on the same day and the implied ATM volatilities.

As of 6 June 2024, CME listed the settlement prices for one-month CSOs, with the first contract going from July 2024 through 25 March (missing December), two-month CSOs, and three-month CSOs, with the first contract being in July 2024 or October 2024. Unsurprisingly, the biggest total open daily interest of ∼369,000 contracts is observed for the three-month spread for October/January; then, the three-month spread for July/October involved ∼45,000 contracts. The pattern of daily interest for one-month CSOs is seasonal, with the biggest total open daily interest for September/October, ∼53,000, and then for the prompt month CSOs in July/August. Another actively traded one-month CSO involved spreads from October to November and March to April. Strikes for a one-month CSO are typically given in increments of USD $0.05$. The biggest range of strikes can be found for October/January, with strikes starting at $-2.5$ and going to $-0.3$.

We transform CSOs with negative strikes (calls) into CSOs with positive strikes (calls) using the relationship in (5). We choose liquid CSOs with sizable open interest, involving important spreads in Natural Gas futures such as October/November, October/January, March/April, and July/October. We list the resulting implied correlations in

Table 18. Implied correlations for different strikes K as of 6 June 2024, Natural Gas CSOs.

Months	${\mathit{F}}_1$	${\mathit{F}}_2$	${\mathit{\sigma }}_1$	${\mathit{\sigma }}_2$	Strike K	Quote	Implied $\mathit{\rho }$ Sector	Implied $\mathit{\rho }$ num. Integration
July/October	2.757	2.888	0.6844	0.5719	−0.2	0.087	0.9943	0.9943
July/October	2.757	2.888	0.6844	0.5719	−0.15	0.054	0.9914	0.9914
July/October	2.757	2.888	0.6844	0.5719	−0.1	0.031	0.9862	0.9862
July/October	2.757	2.888	0.6844	0.5719	−0.05	0.016	0.9815	0.9815
October/January	2.888	3.901	0.5719	0.5372	−1.25	0.353	0.9428	0.9429
October/January	2.888	3.901	0.5719	0.5372	−1	0.353	0.9450	0.9451
October/January	2.888	3.901	0.5719	0.5372	−0.75	0.192	0.9481	0.9482
October/January	2.888	3.901	0.5719	0.5372	−0.7	0.078	0.9500	0.9500
October/January	2.888	3.901	0.5719	0.5372	−0.6	0.061	0.9555	0.9556
October/November	2.88	3.22	0.5719	0.537	−0.3	0.065	0.9897	0.9897
March/April	3.34	3.107	0.5411	0.4360	1	0.08	0.9350	0.9351

.

We observe weaker correlations between months in different seasons, such as October and January (see [16]). The resulting correlations based on the two methods are close; for example, for the last CSO, the difference is within $0.02$ correlation points ($0.0002$). We also checked the implied correlation based on the Kirk formula (7) using the last March/April CSO. (The spread for Mar/Apr is called the “widowmaker” because of its potential devastating losses.) The Kirk formula’s implied correlation was $0.9445$, one correlation point higher than the implied correlation from the sector formula.

4.3. Agricultural CSOs

The standardized trading months for agricultural products, such as Corn, Soybean, and Wheat futures, reflect the seasonal patterns for planting and harvesting the underlying crop. During the planting months, the source of grain that is available for sale or purchase by end users is from crops that were harvested during the previous harvest season, which is considered the “old crop”. On the other hand, during the harvest months, the newly harvested crop, called the “new crop”, comes to market, and supply is higher. Specifically, Corn is considered an old crop in March, May, July, August, and early September. November, December, and January represent the new crop. As with WTI and Natural Gas, agricultural CSOs represent effective hedging and trading tools.

As of 6 June 2024, CME listed three nearby successive futures calendar spreads: 1-month CSOs, with the first month being July, September, and December; a 12-month December/December CSO; a December/July CSO; and a July/December CSO. The most actively traded CSO for the old/new crop spread is Jul/Dec, with a total open daily interest of ∼15,000 contracts. The next active spread involves December/December, with ∼10,000 contracts, and then consecutive spread in July/August, with a total open daily interest of ∼9000 contracts. The strikes are typically listed in increments of USD $0.01$ for consecutive spreads and in increments of USD $0.05$ for other spreads.

We considered Corn CSOs that included three nearby successive futures calendar spreads and one old crop/new crop calendar spread, https://www.cmegroup.com/markets/agriculture/grains/corn.settlements.options.html#optionProductId=2702 (CME Corn SCO, URL accessed on 6 May 2024). We chose liquid CSOs with sizable open interest: July/September, a consecutive spread within the same season (old/old crop), and July/December, an old/new crop spread. We list the resulting implied correlations in

Table 19. Implied correlations for different strikes K as of 5 June and 10 June 2024, Corn CSOs.

Months	${\mathit{F}}_1$	${\mathit{F}}_2$	${\mathit{\sigma }}_1$	${\mathit{\sigma }}_2$	Strike K	Quote	Implied $\mathit{\rho }$ Sector	Implied $\mathit{\rho }$ num. Integration
July/September	4.5175	4.5625	0.2447	0.293	−0.05	0.02375	0.9795	0.9795
July/December	4.3925	4.59	0.2212	0.2387	−0.2	0.03875	0.9094	0.9094
July/December	4.3925	4.59	0.2212	0.2387	−0.15	0.02	0.9014	0.9014
July/December	4.3925	4.59	0.2212	0.2387	−0.1	0.01	0.8884	0.8884
July/December	4.3925	4.59	0.2212	0.2387	−0.05	0.005	0.8725	0.8725
July/December	4.3925	4.59	0.2212	0.2387	0	0.0925	0.8554	0.8554

.

We spot stronger correlations for contracts in the same crop season, old/old (July/September), and weaker correlations for contracts in different crop seasons, old/new (July/December).

The strength of the correlation skew can be measured by the slope of correlation vs. strike normalized by the spread, $k_s={\displaystyle \frac{K}{F_1-F_2}}$. Choosing an important CSO with the maximum number of strikes (December/January for WTI, October/January for Natural Gas, and July/December for Corn), we detect a strong positive slope for Corn, ∼0.05, and negative slopes for Natural Gas and WTI. The WTI slope is much weaker than the Natural Gas slope: ∼−0.001 vs. ∼−0.017. The positive slope for the Corn correlation skew is the result of the current unusual negative spread between July and December (normally positive). The results are illustrated in Figure 6.

4.4. Calculation of Model Valuation Adjustment of CSOs Using the Sector Formula for Greeks

So far, we have demonstrated the use of the sector formula to calculate implied correlations, employing market data for different commodities. Now, we would like to show how sector Greeks, in particular, sensitivity to correlation, i.e., Rhoza, can be utilized to calculate a model valuation adjustment to the value of the spread option due to uncertainty in the correlation input.

If the correlation is assessed through historical data, its estimate $\widehat{\rho }$ is a random variable, and the uncertainty in its value leads to uncertainty in the spread option value. The uncertainty in the correlation estimate can be measured either by a standard error or using, for example, a $95\%$ confidence interval, in which the true correlation resides. Since the spread option is a decreasing function of correlation, for a long position to arrive at a fair value of the option, one has to use the higher value of the correlation $\widehat{\rho }+\Delta \rho$. For small $\Delta \rho$, the fair value can be calculated as follows:

$$V=\widehat{V}-MVA$$

where $\widehat{V}$ is the value of the spread option calculated for $\widehat{\rho }$, and $MVA$ is the model valuation adjustment:

$$MVA\approx {\displaystyle \frac{\mathsf{\partial }V}{\mathsf{\partial }\rho }}\Delta \rho$$

The statistical error of the estimator $\widehat{\rho }$ of a Pearson correlation coefficient $\rho$ calculated on N bivariate sample pairs $(x_i,y_i)$, $i=1,...N$ (log-returns), can be found by using the Fisher z transform; see [17]:

$$\widehat{z}={\displaystyle \frac{1}{2}}ln\left({\displaystyle \frac{1+\widehat{\rho }}{1-\widehat{\rho }}}\right)$$

If $(x,y)$ has a bivariate normal distribution with correlation $\rho$, then $\widehat{z}$ is approximately normally distributed with the mean

$$\mu ={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1+\rho }{1-\rho }}\right)$$

and standard deviation $\sigma ={\displaystyle \frac{1}{\sqrt{N-3}}}$. Consequently, $\Delta \rho$ can be calculated as

$$\Delta \rho ={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{e^{2z_U}-1}{e^{2z_U}+1}}-{\displaystyle \frac{e^{2z_L}-1}{e^{2z_L}+1}}\right)$$

(29)

where

$$z_U=\mu +\beta \sigma ,z_U=\mu -\beta \sigma$$

are the left and right points of the confidence interval, with $\beta =1$ for the standard error and $\beta =1.96$ for the $95\%$ confidence interval.

We illustrate the procedure with an example. Suppose today is 7 June 2024, and we need to price a Natural Gas CSO with an unusual term that is not traded in the market, for example, a 24 September/25 February CSO. All of the inputs are observable in the market, such as future prices, implied volatilities, and discount rates, but not correlation. We choose to use historical data to estimate the correlation using the log-returns of September 2023 and February 2024 NG future prices. Since calendar correlations depend on the time to the first contract (see [16]), we selected a similar time interval for the calculation, 1 June 2023 to 29 August 2023, the expiration date of the September contract (similar to the current situation). The calculation of the correlation between log-returns using the data gives an estimate of the correlation of $\widehat{\rho }=90\%$ with a statistical error $\Delta \rho =0.0249$ (the relatively large statistical error in this case is caused by the small number of observation points (62)).

Table 20. Calculation of model valuation adjustment for NG CSO 24 September/25 February, $F_1=2.971$, $F_2=3.83$, ${\sigma }_1=0.6048$, ${\sigma }_2=0.5730$, and $K=-0.9$.

$\widehat{\mathit{\rho }}$	$\mathbf{\Delta }\mathit{\rho }$	Value Sector	Value num. int	Rhoza num. int.	Sector Rhoza Z	MVA
0.9	0.0249	0.2224	0.2225	−0.6678	−0.6661	0.0167

gives the details of the evaluation and Greeks for the CSO. We list the statistical error of the correlation estimate, the sensitivity Rhoza obtained by the finite difference of the numerical integration formula, and the approximate value of Rhoza given by the main term Z (25).

The model valuation adjustment is about $7.5\%$ of the option value. The MVA calculated by using the approximate value Z of Rhoza from the sector formula is practically the same. The advantage is that a sector Rhoza Z does not require revaluations of the CSO by numerical integration or Monte Carlo and can be calculated analytically.

5. Conclusions

The problem of the accurate valuation and calculation of the Greeks of a spread option is very important for practitioners, especially in the field of commodities. Even though there are numerous approximations, including the famous Kirk formula, it is crucial to understand the limitations of approximations and reveal situations where those approximations do not work properly.

In this paper, we present yet another closed-form approximation of the spread option value and Greeks by using double integration and carefully analyzing the exercise boundary. Depending on the sign of ${\sigma }_2-\rho {\sigma }_1$, we identified two major cases with different types of boundaries: in the first case, the asymptotic slopes of the boundary have the same sign, and in the second case, the boundary has a U-shape, with the asymptotic slopes having different signs. The previous research on pricing spread options by integration was based on the assumption of a small curvature of the boundary close to the origin. In such situations, the boundary can be approximated reasonably well by a straight line, and the pricing formulas are expressed in terms of the univariate normal CDF. We identified situations in which the curvature of the boundary close to the origin is not small, and the previous integration formulas produced bigger errors. We derived two general integration formulas for spread option payoff over straight-line- and sector-type boundaries. We found the optimal way to choose a straight line or a particular sector to find the best approximation of the spread option value. The sector formula is expressed in terms of the bivariate normal CDF.

We worked out formulas for Greeks, in particular, Deltas and sensitivity to correlation. We identified the main terms that give reasonable approximations of Greeks in most situations. We compared our results in terms of accuracy and speed with those obtained by three previous methods, including Kirk and two integration formulas in different types of scenarios, using Monte Carlo simulations as a benchmark. Our method shows good accuracy and superiority to the previous methods in some cases.

We demonstrated applications of our sector formula to the market data of calendar spread options for three major commodities: WTI, Natural Gas, and Corn. The data were listed on the CME site as of 30 April, 7 May, 5 June, and 6 June 2024. Specifically, we showed the particular effectiveness of the formula for the calculation of implied correlations and Greeks. We illustrated how the sensitivity to correlation, Rhoza, can be utilized to calculate a model valuation adjustment to a CSO value due to the uncertainty in the correlation input to arrive at a fair value of the derivative.