# Residue Sum Formula for Pricing Options under the Variance Gamma Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminary Theory

#### 2.1. Multidimensional Residue Calculus

**Definition**

**1**

**.**Let h and ${f}_{i}$, for any index $i\in \{1,\dots ,n\}$, be functions in ${\mathbb{C}}^{n}$, where h is holomorphic. Consider the meromorphic differential n-form

**Definition**

**2**

**.**Consider a proper (the inverse images of a compact set are compact) holomorphic map $g:{\mathbb{C}}^{n}\to G$ where $G={G}_{1}\times \dots \times {G}_{n}$ is a domain (connected open subset of a finite-dimensional vector space) where, for each $j=1,\dots ,n$, ${G}_{j}\subset \mathbb{C}$ is a domain with piecewise smooth boundary. We define a polyhedron Π as the inverse image

**Definition**

**3**

**.**Consider the polyhedron Π and the family of divisors ${\left\{{D}_{i}\right\}}_{i\in \{1,\dots ,n\}}$, they are said to be compatible if for any $i\in \{1,\dots ,n\}$ we get

**Definition**

**4**

**.**Consider the sphere ${S}_{R}=\{z\in \sigma :\u2225z\u2225=R\}$, where $\sigma ={\sigma}_{12\dots n}$ is the boundary of the polyhedron Π. A differential form ${\xi}_{J}$ satisfies the Jordan condition on face ${\sigma}_{{J}^{o}}$, with ${J}^{o}=\{1,\dots ,n\}\backslash J$, if exists a sequence of positive real numbers ${R}_{k}$ that goes to infinity, such that

**Theorem**

**1**

**.**Let ω be a meromorphic form with the polar divisors ${\left\{{D}_{i}\right\}}_{i\in \{1,\dots ,n\}}$ compatible with polyhedron Π. If, for every multi-index J, the differential form ${\xi}_{J}$ satisfies the Jordan condition on ${\sigma}_{{J}^{o}}$, then we get

#### 2.2. One-Dimensional Mellin–Barnes Integral

**Definition**

**5**

**.**The Mellin–Barnes Integral is given by a ratio of products of Gamma functions of linear arguments

#### 2.3. Three-Dimensional Mellin–Barnes Integral

**Theorem**

**2**

**.**Let ω be the three-form integrand of (15) with the characteristic vector $\Delta \ne 0$ and divisors ${D}_{1}$, ${D}_{2}$, and ${D}_{3}$, as defined in (21), being compatible with the admissible polyhedron $\mathrm{\Pi}\subset {\mathrm{\Pi}}_{\Delta}$. Subsequently, the sum formula holds:

## 3. Option Pricing Driven by a Variance Gamma Process

#### 3.1. Mellin–Barnes Representation for a Call Option

**Proposition**

**1**

**.**Let us denote $[log]:=log\frac{S}{K}+(r-q)\tau -\mu \tau $, where $\mu =log\left({\varphi}_{VG}(-i,1;C,G,M)\right)$. Consider the polyhedra ${P}_{1},\phantom{\rule{0.166667em}{0ex}}{P}_{2}\subset {\mathbb{C}}^{3}$, which is defined by

**Proof.**

#### 3.2. Residue Summation Formula for a Call Option

**Theorem**

**3**

**.**The price for the European Call-Option under the Variance Gamma process ${X}_{VG}(\tau ;C,G,M)$ is given by the formula

**Proof.**

#### 3.3. The Greeks

**Theorem**

**4**

**.**The delta, gamma, rho, and theta functions for a European option under the Variance Gamma process ${X}_{VG}(\tau ;C,G,M)$ are given by:

**Delta**is defined as ${\Delta}_{C}:=\frac{\partial C}{\partial S}$, hence:$$\begin{array}{c}\hfill {\Delta}_{C}(S,K,r,q,\tau )=\frac{K{\left(GM\right)}^{C\tau}{e}^{-r\tau}}{S\mathrm{\Gamma}{\left(C\tau \right)}^{2}}\sum _{\begin{array}{c}k=0\\ n=0\\ m=0\end{array}}^{\infty}{\Delta}_{1}+{\Delta}_{2}+{\mathbb{l}}_{[log]>0}{\Delta}_{3},\end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\Delta}_{1}={(-1)}^{n+m}\frac{\mathrm{\Gamma}(C\tau +m)\mathrm{\Gamma}(-2C\tau -k-n-m)}{n!m!\mathrm{\Gamma}(1-C\tau -n)}{M}^{n}{G}^{m}{(-[log])}^{2C\tau +k+n+m},\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\Delta}_{2}={(-1)}^{m}\frac{\mathrm{\Gamma}(C\tau +m)\mathrm{\Gamma}(2C\tau +k-n+m)}{n!m!\mathrm{\Gamma}(1+C\tau +k-n+m)}{M}^{-2C\tau -k+n-m}{G}^{m}{[log]}^{n},\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\Delta}_{3}={(-1)}^{C\tau +m}\frac{\mathrm{\Gamma}(C\tau +n)\mathrm{\Gamma}(C\tau +m)}{n!m!\mathrm{\Gamma}(1+2C\tau +k+n+m)}{M}^{n}{G}^{m}{[log]}^{2C\tau +k+n+m}.\hfill \end{array}$$**Gamma**is defined as ${\mathrm{\Gamma}}_{C}:=\frac{{\partial}^{2}C}{\partial {S}^{2}}$, hence:$$\begin{array}{cc}\hfill {\mathrm{\Gamma}}_{C}(S,K,r,q,\tau )& =\frac{K{\left(GM\right)}^{C\tau}{e}^{-r\tau}}{{S}^{2}\mathrm{\Gamma}{\left(C\tau \right)}^{2}}\sum _{\begin{array}{c}k=0\\ n=0\\ m=0\end{array}}^{\infty}({\mathrm{\Gamma}}_{1}-{\Delta}_{1})+({\mathrm{\Gamma}}_{2}-{\Delta}_{2})+{\mathbb{l}}_{[log]>0}({\mathrm{\Gamma}}_{3}-{\Delta}_{3}),\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathrm{\Gamma}}_{1}& ={(-1)}^{n+m}\frac{\mathrm{\Gamma}(C\tau +m)\mathrm{\Gamma}(1-2C\tau -k-n-m)}{n!m!\mathrm{\Gamma}(1-C\tau -n)}{M}^{n}{G}^{m}{(-[log])}^{-1+2C\tau +k+n+m},\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathrm{\Gamma}}_{2}& ={(-1)}^{m}\frac{\mathrm{\Gamma}(C\tau +m)\mathrm{\Gamma}(-1+2C\tau +k-n+m)}{n!m!\mathrm{\Gamma}(C\tau +k-n+m)}{M}^{1-2C\tau -k+n-m}{G}^{m}{[log]}^{n},\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathrm{\Gamma}}_{3}& ={(-1)}^{C\tau +m}\frac{\mathrm{\Gamma}(C\tau +n)\mathrm{\Gamma}(C\tau +m)}{n!m!\mathrm{\Gamma}(2C\tau +k+n+m)}{M}^{n}{G}^{m}{[log]}^{-1+2C\tau +k+n+m}.\hfill \end{array}$$**Rho**is defined as ${\rho}_{C}:=\frac{\partial C}{\partial r}$, hence:$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\rho}_{C}(S,K,r,q,\tau )=\tau S{\Delta}_{C}(S,K,r,q,\tau )-\tau C(S,K,r,q,\tau )\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{K\tau {\left(GM\right)}^{C\tau}{e}^{-r\tau}}{\mathrm{\Gamma}{\left(C\tau \right)}^{2}}\sum _{\begin{array}{c}k=0\\ n=0\\ m=0\end{array}}^{\infty}({C}_{VG}^{1}-{\Delta}_{1})+({C}_{VG}^{2}-{\Delta}_{2})+{\mathbb{l}}_{[log]>0}({C}_{VG}^{3}-{\Delta}_{3}),\hfill \end{array}$$**Theta**is defined as ${\Theta}_{C}:=\frac{\partial C}{\partial t}=-\frac{\partial C}{\partial \tau}$, hence:$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\Theta}_{C}(S,K,r,q,\tau )=-\frac{K{\left(GM\right)}^{C\tau}{e}^{-r\tau}}{\mathrm{\Gamma}{\left(C\tau \right)}^{2}}\times \hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \sum _{\begin{array}{c}k=0\\ n=0\\ m=0\end{array}}^{\infty}({\theta}_{1}{C}_{VG}^{1}+(r-q-\mu ){\Delta}_{1})+({\theta}_{2}{C}_{VG}^{2}+(r-q-\mu ){\Delta}_{2}+{\mathbb{l}}_{[log]>0}({\theta}_{3}{C}_{VG}^{3}+(r-q-\mu ){\Delta}_{3})),\hfill \end{array}$$$$\begin{array}{cc}\hfill {\theta}_{1}=& Clog\left(GM\right)-r-2C\psi \left(C\tau \right)+C\psi (C\tau +m)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& -2C\psi (-1-2C\tau -k-n-m)+C\psi (1-C\tau -n)+2Clog(-[log\left]\right),\hfill \end{array}$$$$\begin{array}{cc}\hfill {\theta}_{2}=& Clog\left(GM\right)-r-2C\psi \left(C\tau \right)+C\psi (C\tau +m)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& +2C\psi (1+2C\tau +k-n+m)-C\psi (2+C\tau +k+n+m)-2Clog\left(M\right),\hfill \end{array}$$$$\begin{array}{cc}\hfill {\theta}_{3}=& Clog\left(GM\right)-r-2C\psi \left(C\tau \right)+C\psi (C\tau +n)+C\psi (C\tau +m)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& -2C\psi (2+2C\tau +k-n+m)+C\pi i+2Clog\left(\right[log\left]\right),\hfill \end{array}$$

## 4. Numerical Results

#### 4.1. Variance Gamma Formula Values and Behavior

#### 4.2. Convergence of the Variance Gamma Formula

#### 4.3. The Greek Formulas Behavior

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Proof of Theorem 2

**Proof.**

**Lemma**

**A1.**

**Proof.**

**Lemma**

**A2.**

**Lemma**

**A3.**

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**Figure 1.**European Options prices under the Black–Scholes and Variance Gamma models and their differences, for $K=1100$, $\tau =35\times 7/365$ and different values of ${S}_{0}$.

**Figure 2.**Implied Volatility for the empirical data and Black–Scholes and Variance Gamma models, where ${S}_{0}=1124.47$, $K=1100$, and $\tau =35\times 7/365$.

**Figure 3.**Greek functions under the Black–Scholes and Variance Gamma models, for $K=1100$ and $\tau =35\times 7/365$.

**Table 1.**Table containing the values for the Variance Gamma Formula (49) for $n=22$, $m=27$, and $k=7$, denoted by “F”, the Monte Carlo method with 10000 simulated trajectories, denoted by “MC”, and the observed market values are denoted by “Real”.

Time of Maturity | |||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Strike | May | June | September | December | March | June | December | ||||||||||||||

Price | 2002 | 2002 | 2002 | 2002 | 2003 | 2003 | 2003 | ||||||||||||||

F | MC | Real | F | MC | Real | F | MC | Real | F | MC | Real | F | MC | Real | F | MC | Real | F | MC | Real | |

975 | 152.76 | 151.72 | - | 156.80 | 156.19 | - | 166.83 | 168.05 | 161.60 | 176.13 | 176.13 | 173.30 | 184.76 | 183.79 | - | 192.81 | 194.67 | - | 207.41 | 206.05 | - |

995 | 133.42 | 132.40 | - | 138.17 | 137.60 | - | 149.64 | 150.82 | 144.80 | 159.98 | 160.09 | 157.00 | 169.40 | 168.42 | - | 178.07 | 179.85 | 182.10 | 193.65 | 192.31 | - |

1025 | 104.69 | 103.73 | - | 110.74 | 110.27 | - | 124.65 | 125.79 | 120.10 | 136.64 | 136.93 | 133.10 | 147.27 | 146.28 | 146.50 | 156.88 | 158.52 | - | 173.84 | 172.57 | - |

1050 | 81.09 | 80.20 | - | 88.49 | 88.16 | 84.50 | 104.71 | 105.81 | 100.70 | 118.12 | 118.59 | 114.80 | 129.75 | 128.65 | - | 140.10 | 141.69 | 143.00 | 158.15 | 156.92 | 171.40 |

1075 | 57.95 | 57.12 | - | 67.00 | 66.78 | 64.30 | 85.75 | 86.78 | 82.50 | 100.59 | 101.29 | 97.60 | 113.15 | 111.92 | - | 124.20 | 125.73 | - | 143.24 | 142.10 | - |

1090 | 44.39 | 43.58 | 43.10 | 54.58 | 54.41 | - | 74.92 | 75.90 | - | 90.59 | 91.42 | - | 103.68 | 102.36 | - | 115.11 | 116.57 | - | 134.67 | 133.62 | - |

1100 | 35.53 | 34.74 | 35.60 | 46.56 | 46.42 | - | 67.97 | 68.91 | 65.50 | 84.15 | 85.08 | 81.20 | 97.57 | 96.19 | 96.20 | 109.23 | 110.66 | 111.30 | 129.13 | 128.15 | 140.40 |

1110 | 26.88 | 26.12 | - | 38.79 | 38.69 | 39.50 | 61.25 | 62.16 | - | 77.92 | 78.92 | - | 91.65 | 90.18 | - | 103.52 | 104.91 | - | 123.71 | 122.81 | - |

1120 | 18.50 | 17.81 | 22.90 | 31.33 | 31.25 | 33.50 | 54.79 | 55.68 | - | 71.91 | 72.96 | - | 85.90 | 84.37 | - | 97.97 | 99.33 | - | 118.43 | 117.61 | - |

1125 | 14.47 | 13.82 | 20.20 | 27.74 | 27.69 | 30.70 | 51.66 | 52.54 | 51.00 | 68.98 | 70.07 | 66.90 | 83.10 | 81.54 | 81.70 | 95.25 | 96.61 | 97.00 | 115.84 | 115.07 | - |

1130 | 10.60 | 10.00 | - | 24.26 | 24.25 | 28.00 | 48.60 | 49.50 | - | 66.11 | 67.23 | - | 80.34 | 78.76 | - | 92.58 | 93.91 | - | 113.29 | 112.55 | - |

1135 | 7.09 | 6.53 | - | 20.93 | 20.93 | 25.60 | 45.63 | 46.45 | 45.50 | 63.30 | 64.45 | - | 77.63 | 76.04 | - | 89.94 | 91.26 | - | 110.76 | 110.07 | - |

1140 | 5.99 | 5.49 | 13.30 | 17.77 | 17.79 | 23.20 | 42.73 | 43.52 | - | 60.55 | 61.73 | 58.90 | 74.97 | 73.36 | - | 87.35 | 88.66 | - | 108.28 | 107.62 | - |

1150 | 4.66 | 4.26 | - | 12.55 | 12.60 | 19.10 | 37.20 | 37.92 | 38.10 | 55.23 | 56.44 | 53.90 | 69.81 | 68.17 | 68.30 | 82.30 | 83.58 | 83.30 | 103.40 | 102.81 | 112.80 |

1160 | 3.75 | 3.44 | - | 10.02 | 10.09 | 15.30 | 32.06 | 32.72 | - | 50.17 | 51.39 | - | 64.84 | 63.20 | - | 77.43 | 78.65 | - | 98.67 | 98.13 | - |

1170 | 3.08 | 2.82 | - | 8.24 | 8.32 | 12.10 | 27.37 | 27.96 | - | 45.38 | 46.59 | - | 60.09 | 58.46 | - | 72.72 | 73.88 | - | 94.07 | 93.58 | - |

1175 | 2.81 | 2.57 | - | 7.52 | 7.61 | 10.90 | 25.23 | 25.78 | 27.70 | 43.09 | 44.28 | 42.50 | 57.79 | 56.18 | 56.60 | 70.44 | 71.56 | - | 91.82 | 91.36 | 99.80 |

1200 | 1.82 | 1.68 | - | 4.93 | 5.06 | - | 17.15 | 17.54 | 19.60 | 32.79 | 33.78 | 33.00 | 47.14 | 45.62 | 46.10 | 59.70 | 60.65 | 60.90 | 81.11 | 80.74 | - |

1225 | 1.22 | 1.15 | - | 3.35 | 3.55 | - | 12.16 | 12.50 | 13.20 | 24.61 | 25.46 | 24.90 | 37.91 | 36.50 | 36.90 | 50.10 | 50.90 | 49.80 | 71.28 | 70.97 | - |

1250 | 0.84 | 0.80 | - | 2.34 | 2.53 | - | 8.79 | 9.12 | - | 18.59 | 19.29 | 18.30 | 30.17 | 28.88 | 29.30 | 41.67 | 42.34 | 41.20 | 62.30 | 62.03 | 66.90 |

1275 | 0.59 | 0.57 | - | 1.67 | 1.84 | - | 6.45 | 6.76 | - | 14.14 | 14.76 | 13.20 | 23.92 | 22.71 | 22.50 | 34.40 | 34.92 | - | 54.19 | 53.94 | - |

1300 | 0.42 | 0.41 | - | 1.20 | 1.34 | - | 4.79 | 5.08 | - | 10.84 | 11.40 | - | 18.96 | 17.90 | 17.20 | 28.24 | 28.59 | 27.10 | 46.90 | 46.63 | 49.50 |

1325 | 0.31 | 0.30 | - | 0.88 | 0.98 | - | 3.59 | 3.82 | - | 8.35 | 8.85 | - | 15.06 | 14.15 | 12.80 | 23.12 | 23.33 | - | 40.42 | 40.09 | - |

1350 | 0.23 | 0.22 | - | 0.65 | 0.71 | - | 2.71 | 2.92 | - | 6.47 | 6.88 | - | 11.99 | 11.23 | - | 18.92 | 19.01 | 17.10 | 34.71 | 34.31 | 35.70 |

1400 | 0.12 | 0.13 | - | 0.36 | 0.41 | - | 1.58 | 1.72 | - | 3.95 | 4.16 | - | 7.66 | 7.16 | - | 12.67 | 12.67 | 10.10 | 25.37 | 24.87 | 25.20 |

1450 | 0.07 | 0.09 | - | 0.21 | 0.24 | - | 0.95 | 1.03 | - | 2.45 | 2.55 | - | 4.95 | 4.62 | - | 8.51 | 8.46 | - | 18.42 | 17.69 | 17.00 |

1500 | 0.04 | 0.06 | - | 0.13 | 0.14 | - | 0.58 | 0.67 | - | 1.55 | 1.61 | - | 3.23 | 3.03 | - | 5.76 | 5.66 | - | 13.33 | 12.48 | 12.20 |

RMSE | |
---|---|

Black–Scholes | 6.6692 |

Variance Gamma Monte Carlo | 3.9979 |

Variance Gamma Formula | 3.7373 |

Double Series Sum | Double Series Sum | ||||||
---|---|---|---|---|---|---|---|

${\mathit{C}}_{\mathit{VG}}^{\mathit{n}\mathbf{const}}$ (n) | ${\mathit{C}}_{\mathit{VG}}^{\mathit{m}\mathbf{const}}$ (m) | ${\mathit{C}}_{\mathit{VG}}^{\mathit{k}\mathbf{const}}$ (k) | ${\mathit{C}}_{\mathit{VG}}^{\mathit{n}\mathbf{const}}$ (n) | ${\mathit{C}}_{\mathit{VG}}^{\mathit{m}\mathbf{const}}$ (m) | ${\mathit{C}}_{\mathit{VG}}^{\mathit{k}\mathbf{const}}$ (k) | ||

0 | 1573.462 | 1573.462 | 1573.462 | 14 | 3.451 | 5.456 | 0.000 |

1 | 942.085 | 1624.942 | 93.249 | 15 | 1.263 | 2.653 | 0.000 |

2 | 1473.360 | 2138.478 | 14.602 | 16 | 0.430 | 1.277 | 0.000 |

3 | 396.878 | 2185.591 | 0.656 | 17 | 0.137 | 0.609 | 0.000 |

4 | 530.501 | 1834.503 | 0.121 | 18 | 0.041 | 0.288 | 0.000 |

5 | 350.999 | 1328.181 | 0.008 | 19 | 0.011 | 0.135 | 0.000 |

6 | 358.913 | 862.780 | 0.001 | 20 | 0.003 | 0.063 | 0.000 |

7 | 287.068 | 518.316 | 0.000 | 21 | 0.001 | 0.029 | 0.000 |

8 | 214.182 | 294.406 | 0.000 | 22 | 0.000 | 0.013 | 0.000 |

9 | 139.772 | 160.554 | 0.000 | 23 | 0.000 | 0.006 | 0.000 |

10 | 81.751 | 84.932 | 0.000 | 24 | 0.000 | 0.003 | 0.000 |

11 | 42.733 | 43.875 | 0.000 | 25 | 0.000 | 0.001 | 0.000 |

12 | 20.221 | 22.236 | 0.000 | 26 | 0.000 | 0.001 | 0.000 |

13 | 8.716 | 11.090 | 0.000 | 27 | 0.000 | 0.000 | 0.000 |

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Febrer, P.; Guerra, J.
Residue Sum Formula for Pricing Options under the Variance Gamma Model. *Mathematics* **2021**, *9*, 1143.
https://doi.org/10.3390/math9101143

**AMA Style**

Febrer P, Guerra J.
Residue Sum Formula for Pricing Options under the Variance Gamma Model. *Mathematics*. 2021; 9(10):1143.
https://doi.org/10.3390/math9101143

**Chicago/Turabian Style**

Febrer, Pedro, and João Guerra.
2021. "Residue Sum Formula for Pricing Options under the Variance Gamma Model" *Mathematics* 9, no. 10: 1143.
https://doi.org/10.3390/math9101143