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

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## Abstract

**:**

## 1. Introduction

## 2. Analytical Results of Spread Option Evaluation

#### 2.1. Setup and Assumptions

#### 2.1.1. Call Spread Option

#### 2.1.2. Exercise Boundary

- 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.

#### 2.1.3. Boundary Approximation

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

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

#### 2.2. Domain of Integration in Prices and Returns

**Corollary**

**1.**

**Proof.**

#### 2.3. Closed-Form Spread Option Formulas

**Proposition**

**1.**

**Proof.**

**sector formula**.

**Proposition**

**2.**

**Proof.**

#### 2.4. Closed-Form Greeks

**Proposition**

**3.**

#### 2.5. Two Examples

## 3. Numerical Results

#### 3.1. Monte Carlo Simulation

- Randomly generate inputs for five ATM spread call option samples from the following range: ${F}_{1}=100$, $\frac{K}{{F}_{2}}}\in (0.05,0.2)$, ${F}_{1}-{F}_{2}-K=0$, $T=1$, $r=0$, ${\sigma}_{1}=2$, ${\sigma}_{2}\in (0.2,0.6)$, $\rho \in (0.7,0999)$.
- 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 $\u03f5=0.0001$.
- 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.

#### 3.2. Pricing Accuracy under Different Scenarios

#### 3.2.1. Scenario 1: Low Strikes and Correlations

#### 3.2.2. Scenario 2: Higher Strikes and Correlations and ${\sigma}_{2}-\rho {\sigma}_{1}\le 0$

#### 3.2.3. Scenario 3: Higher Strikes and Correlations and ${\sigma}_{2}-\rho {\sigma}_{1}>0$

#### 3.2.4. Pricing Speed

#### 3.3. Greek Accuracy and Speed

#### 3.3.1. Three Methods in Comparison with MC Simulation

- The first method is to use the numerical Greeks calculated by the central finite difference using the formulas in Proposition 3.$$\begin{array}{ccc}\hfill {{\displaystyle \frac{\mathsf{\partial}\widehat{c}({F}_{1},{F}_{2},T,K)}{\mathsf{\partial}{F}_{1}}}}^{\left(\u03f5\right)}& =& {\displaystyle \frac{\widehat{c}({F}_{1}+\u03f5)-\widehat{c}({F}_{1}-\u03f5)}{2\u03f5}}\hfill \end{array}$$$$\begin{array}{ccc}\hfill {{\displaystyle \frac{\mathsf{\partial}\widehat{c}({F}_{1},{F}_{2},T,K)}{\mathsf{\partial}{F}_{2}}}}^{\left(\u03f5\right)}& =& {\displaystyle \frac{\widehat{c}({F}_{2}+\u03f5)-\widehat{c}({F}_{2}-\u03f5)}{2\u03f5}}\hfill \end{array}$$$$\begin{array}{ccc}\hfill {{\displaystyle \frac{\mathsf{\partial}\widehat{c}({F}_{1},{F}_{2},T,K)}{\mathsf{\partial}\rho}}}^{\left(\u03f5\right)}& =& {\displaystyle \frac{\widehat{c}(\rho +\u03f5)-\widehat{c}(\rho -\u03f5)}{2\u03f5}}\hfill \end{array}$$
- 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

#### 3.3.3. Rhoza Accuracy and Speed

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

#### 4.1. WTI CSOs

#### 4.2. Natural Gas CSOs

#### 4.3. Agricultural CSOs

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

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Lemma**

**1.**

## Appendix B

**Proof**

**of**

**Lemma**

**2.**

## Appendix C

#### Conditions for “Big” Curvature

**Proof**

**of**

**Proposition**

**1.**

## Appendix D

**Proof**

**of**

**Proposition**

**2.**

## Appendix E

#### Delta and Rhoza Formula Coefficients of Proposition 3

## Appendix F

#### Appendix F.1. Monte Carlo Simulation

#### Appendix F.2. Sample Cleaning

#### Appendix F.3. Antithetic Variables

#### Appendix F.4. Control Variates

#### Appendix F.5. Greek Simulation

## References

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**Figure 2.**Boundaries in the case of ${\sigma}_{2}-\rho {\sigma}_{1}\le 0$: ${\sigma}_{1}=0.4$, ${\sigma}_{2}=0.2$, $T=1$.

**Figure 3.**Boundaries in the case of ${\sigma}_{2}-\rho {\sigma}_{1}>0$: ${\sigma}_{2}=0.5$, ${\sigma}_{1}=0.36$, $T=1$.

**Figure 4.**Approximations of the exercise boundary in the case ${\sigma}_{2}-\rho {\sigma}_{1}\le 0$.

**Figure 6.**Correlation skew for WTI (December/January), NG (October/January), and Corn (July/December).

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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% |

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% |

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% |

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 |

${\mathbf{\Delta}}_{{\mathit{F}}_{1}}$ | ${\mathbf{\Delta}}_{{\mathit{F}}_{2}}$ | |||||
---|---|---|---|---|---|---|

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% |

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 |

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% |

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 |

**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 |

**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 |

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 |

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 |

**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 |

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## Share and Cite

**MDPI and ACS Style**

Galeeva, R.; Wang, Z.
Sector Formula for Approximation of Spread Option Value & Greeks and Its Applications. *Commodities* **2024**, *3*, 281-313.
https://doi.org/10.3390/commodities3030017

**AMA Style**

Galeeva R, Wang Z.
Sector Formula for Approximation of Spread Option Value & Greeks and Its Applications. *Commodities*. 2024; 3(3):281-313.
https://doi.org/10.3390/commodities3030017

**Chicago/Turabian Style**

Galeeva, Roza, and Zi Wang.
2024. "Sector Formula for Approximation of Spread Option Value & Greeks and Its Applications" *Commodities* 3, no. 3: 281-313.
https://doi.org/10.3390/commodities3030017