# Contributions to the financial mathematics of energy markets

Contributions to the financial mathematics of energy markets

Author Contributor Faculty Date2008-02-01

AbstractThis thesis provides several contributions to quantitative finance for energy markets: electricity price modelling, implying oil price volatilities, pricing and hedging of exotic commodity options. Electricity spot prices are characterized by spikes (jumps) because electricity is non-storable. A widely used model for stochastic component of electricity spot prices, a mean-reversion jump-diffusion model, is only partially successful to capture spikes. We propose the so-called potential Lévy model, incorporating a potential function and a class of Lévy process, i.e. those with Α-stable distributions. In this model, after a jump, the potential function has higher mean-reversion rate than the 'normal' mean-reversion rate. Modelling stochastic price fluctuations using an Α-stable distribution has several advantages: disentangling stochastic price fluctuations as a part of stochastic dynamics and jump dynamics, and assumption that the jump inter-arrival times are exponentially distributed are not necessary. This distribution is also heavy-tailed enough to capture spikes. The implied volatility obtained from liquid option prices by inverting the Black-Scholes formula is often considered as the best volatility forecast. The Black-Scholes model assumes a constant volatility for options on the same underlying asset. In practice, the implied volatilities vary across the strike prices and the times to maturity. We develop the so-called semi-parametric model for fitting the implied volatility surface, incorporating the simplicity of a parametric method and the flexibility of a non-parametric method. Such a model can capture the smile, skew or smirk shape and can deal with limited amount of option price data. A basket option is a convenient market risk management tool for a company whose portfolio consists of several assets. The difficulty in valuing basket options is that the weighted sum of log-normal random variables is not log-normally distributed anymore, which is the key assumption in the famous Black-Scholes model. Moreover, a basket may have negative values (if some basket weights are negative). Hence, the Black-Scholes model cannot be applied. To solve this problem, we introduce a so-called GLN (Generalized log-normal) distribution which we can use to approximate a general basket distribution. We propose the so-called GLN approach to valuation and hedging a basket option. The main attractions of this approach are: it is easy to implement since it provides closed form expressions for the basket options price and the greeks, can deal with basket of several assets with negative weights. The GLN approach also allows to obtain the implied correlation between assets in the basket by inverting the closed formula of the basket option's price. We extend the GLN approach for pricing and hedging of Asian basket options.

Subjectelectricity spot prices

mean-reversion jump-diffusion model

potential L¿é

vy model

&alpha

-stable distribution

implied volatility surface

semi-parametric model

Black-Scholes (or Black) model

Asian option

basket option

Asian basket option

GLN (generalized log-normal) distribution

GLN approach

http://resolver.tudelft.nl/uuid:01bfcad0-9a56-454b-a30e-dc2fd3394f12

ISBN978-90-9022713-9

Part of collectionInstitutional Repository

Document typedoctoral thesis

Rights(c) 2008 Permana, F.J.