Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Pricing energy spread options with variance gamma-driven Ornstein-Uhlenbeck dynamics

Tim Leung, Kevin W. Lu

Abstract

We consider the pricing of energy spread options for spot prices following an exponential Ornstein-Uhlenbeck process driven by a sum of independent multivariate variance gamma processes, which gives rise to mean-reverting, infinite activity price dynamics. Within this class of driving processes, the Esscher transform is used to obtain an equivalent martingale measure with a focus on the weak variance alpha-gamma process. By deriving an analytical formula for the cumulant generating function of the innovation term, we obtain a pricing formula for forwards and apply the FFT method of Hurd and Zhou to price spread options. Lastly, we demonstrate how the model should be both estimated on energy prices under the real world measure and calibrated on forward or call prices, and provide numerical results for the pricing of spread options.

Pricing energy spread options with variance gamma-driven Ornstein-Uhlenbeck dynamics

Abstract

We consider the pricing of energy spread options for spot prices following an exponential Ornstein-Uhlenbeck process driven by a sum of independent multivariate variance gamma processes, which gives rise to mean-reverting, infinite activity price dynamics. Within this class of driving processes, the Esscher transform is used to obtain an equivalent martingale measure with a focus on the weak variance alpha-gamma process. By deriving an analytical formula for the cumulant generating function of the innovation term, we obtain a pricing formula for forwards and apply the FFT method of Hurd and Zhou to price spread options. Lastly, we demonstrate how the model should be both estimated on energy prices under the real world measure and calibrated on forward or call prices, and provide numerical results for the pricing of spread options.

Paper Structure

This paper contains 28 sections, 11 theorems, 88 equations, 9 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Variance gamma and related process
  4. Lévy-Driven Ornstein-Uhlenbeck Processes
  5. Analytical formula for cgf of the innovation term
  6. Energy derivatives
  7. Energy market model
  8. Esscher transform
  9. Analytical formula for cgf of the log return
  10. Pricing energy derivatives
  11. Forward prices
  12. Call options
  13. Spread options
  14. Implementation
  15. Simulated data example
  16. ...and 13 more sections

Key Result

Lemma 3.1

Let ${\bf Z}\sim L^n(\boldsymbol{\mu},{\boldsymbol{\Sigma}},{\cal Z})$ and $t>0$. Let ${\cal D}_{{\bf Z}^*(t)} = e^{-\lambda t}{\cal D}_{{\bf Z}}$. If $\boldsymbol{\theta}\in {\cal D}_{{\bf Z}^*(t)}$, then is finite, and also $\kappa_{{\bf Z}}(e^s\boldsymbol{\theta})$ is finite for all $s\in[0,\lambda t]$. Furthermore, cfgstar holds for all $\boldsymbol{\theta}\in\mathbb{C}^n$ such that $\operato

Figures (9)

  • Figure 1: Forward price curve $\tau \mapsto F_1(8,8+\tau)$, where the MPR is ${\bf h} = \gamma(-0.1,-0.03)$, $\gamma=-1,1,10$. All parameters are otherwise taken from Section \ref{['modelparsec']} but $b_{21}=b_{31}=0$ for all curves except the one with cyclical component.
  • Figure 2: Average error using the FFT method for $N=2^k$, $k=6,7,\dots 10$, and $\overline{\theta}=40,80$. The plot is on a log scale.
  • Figure 3: Spread option price with $K=3.6$ as a function of the LDOUP parameters.
  • Figure 4: Sample path of the log price $Y_k(t)$ (black), true seasonality function $\Lambda_k(t)$ (green) and estimated seasonality function $\widehat{\Lambda}_k(t)$ (red) for $k=1,2$.
  • Figure 5: True (black) and estimated (red) pdf of $R_k(T_e,T_o){\,\vert\,} {\cal F}_{T_e}$ under $\mathbb{Q}_{\bf h}$ and the corresponding means $\widetilde{\mu}_k$ (vertical line) for $k=1,2$.
  • ...and 4 more figures

Theorems & Definitions (33)

  • Lemma 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Corollary 3.5
  • Remark 3.6
  • Lemma 4.1
  • proof
  • ...and 23 more