In the commodity and energy markets, there are two kinds of risk that traders and analysts are concerned a lot about: multiple underlying risk and average price risk. Spread options, swaps and swaptions are widely used to hedge multiple underlying risks and Asian (average price) options can deal with average price risk. But when those two risks are combined together, then we need to consider Asian spread options and Asian-European spread options for hedging purposes. For an Asian or Asian-European spread call option, its payoff depends on the difference of two underlyings' average price or of one average price and one final (at expiration) price. Asian and Asian-European spread option pricing is challenging work. Even under the basic assumption that each underlying price follows a log-normal distribution, the average price does not have a distribution with a simple form. In this dissertation, for the first time, a systematic analysis of Asian spread option and Asian-European spread option pricing is proposed, several original approaches for the Black-Scholes-Merton model and a special stochastic volatility model are developed and some numerical computation tests are conducted as well.

This thesis includes three chapters on electricity and natural gas prices. In the first chapter, we give a brief introduction to the characteristics of power prices and propose a mean reversion jump diffusion model, in which jump intensity depends on temperature data and overall system load, to model electricity prices. Compared to the models used in the literature, we find the model proposed in this chapter is better to capture the tail behavior in the electricity prices. In the second chapter, we use the model proposed in the first chapter to simulate the spark spread option and value the power generations. In order to simulate power generation, we first propose and estimate mean reversion jump diffusion model for natural gas prices, in which jump intensity is defined as a function of temperature and storage. Combing the model with the electricity models in chapter 1, we find that the value of power generation is closer to the real value of the power plants as reflected in the recent market transaction than one obtains from many other models used in literature. The third chapter investigates extremal dependence among the energy market. We find a tail dependence that exceeds the Pearson correlation ρ, which means the traditional Pearson correlation is not appropriate to model tail behavior of oil, natural gas and electricity prices. However, asymptotic dependence is rejected in all pairs except Henry Hub gas return and Houston Ship Channel gas return. We also find that extreme value dependence in energy market is stronger in bull market than that in bear market due to the special characteristics in energy market, which conflicts the accepted wisdom in equity market that tail correlation is much higher in periods of volatile markets from previous literature.

In this dissertation, we investigate the spread option pricing problem with stochastic interest rate. First, we will review the basic concept and theories of stochastic calculus, give an introduction of spread options and provide some examples of spread options in different markets. We will also review the market efficiency theory, arbitrage and assumptions that are commonly used in mathematical finance. In Chapter 3, we will review existing spread pricing models and term-structure models such as Vasicek Mode, and the Heath-Jarrow-Morton framework. In Chapter 4, we will use the martingale approach to derive a partial differential equation for the price of the spread option with stochastic interest rate. In Chapter 5, we will study the spread option numerically. We will conclude this dissertation with ideas for future research.

In financial markets, spread option is a derivative security with two underlying assets and the payoff of the spread option depends on the difference of these assets. We consider American style spread option which allows the owners to exercise it at any time before the maturity. The complexity of pricing American spread option is that the boundary of the corresponding partial differential equation which determines the option price is unknown and the model for the underlying assets is two-dimensional.In this dissertation, we incorporate the stochasticity to the interest rate and assume that it satisfies the Vasicek model or the CIR model. We derive the partial differential equations with terminal and boundary conditions which determine the American spread option with stochastic interest rate and formulate the associated free boundary problem. We convert the free boundary problem to the linear complimentarity conditions for the American spread option, so that we can go around the free boundary and compute the option price numerically. Alternatively, we approximate the option price using methods based on the Monte Carlo simulation, including the regression-based method, the Lonstaff and Schwartz method and the dual method. We make the comparisons among the option prices derived by the partial differential equation method and Monte Carlo methods to show the accuracy of the result.

Spread options are derivative securities, which are written on the difference between the values of two underlying market variables. They are very important tools to hedge the correlation risk. American style spread options allow the holder to exercise the option at any time up to and including maturity. Although they are widely used to hedge and speculate in financial market, the valuation of the American spread option is very challenging. Because even under the classic assumptions that the underlying assets follow the log-normal distribution, the resulting spread doesn't have a distribution with a simple closed formula. In this dissertation, we investigate the American spread option pricing problem. Several approaches for the geometric Brownian motion model and the stochastic volatility model are developed. We also implement the above models and the numerical results are compared among different approaches.

傳統上在 LIBOR市場模型架構下,評價固定交換商品一般是透過模地卡羅模擬。在本文中,吾人在此模型架構下推導出一個遠期交換利率的近似動態,並在一因子的架構下提供一個固定交換利差選擇權與固定交換輪棘選擇權的近似評價公式。數值結果顯示這兩者之相對誤差甚小。此外對於這兩個產品,吾人亦提供一個有效率的避險方法。 / The derivatives of the constant maturity swap (CMS) are evaluated by the LIBOR market model (LMM) implemented by Monte Carlo methods in the previous researches. In this paper, we derive an approximated dynamic process of the forward-swap rate (FSR) under LMM. Based on the approximated dynamics for the FSR under one factor model, CMS spread options and CMS ratchet options are valued by the no-arbitrage method in approximated analytic formulas. In the numerical analysis, the relative errors between the Monte Carlo simulations and the approximated closed form formulas are very small for CMS spread options and CMS ratchet options and we also provide an efficient hedging method for these products under one factor LMM.

This master thesis should provide reader with an overview of the European spread options evaluation using the fast Fourier transform numerical method. The first and second part of the thesis deal with the theoretical foundations of Fourier analysis and existing approaches of spread option valuation under two and three-factors frameworks (namely GBM - geometric Brown motion and SV - stochastic volatility). The third part describes extention of Hurd-Zhou (2010) valuation method by tool for call and put spread options pricing in case of negative or zero strikes. Extension will be compared with Monte Carlo simulation results from a variety of perspectives, including computing complexity and implementation requirements. Dempster-Hong model, Hurd-Zhou model and Monte Carlo simulation are implemented and tested in R (programming language).

Depuis la succession des récentes crises financières, les « dérivés de crédit » connaissent une notoriété médiatique très intense qui dépasse la seule sphère des spécialistes. Créés au début des années 1990, ils sont des instruments financiers de transfert du risque de crédit. Ils autorisent tant la protection que la spéculation. Ils sont juridiquement documentés par des conventions-Cadres proposées par l’International Swaps and Derivatives Association (ISDA), et dans une très petite mesure, par la Fédération Bancaire Française en France. Ils regroupent cinq grandes catégories de contrat : « credit default swap » ou « contrat d’échange sur le risque de crédit », « credit linked notes » ou « dérivé de crédit titrisé », « credit spread option » ou « option sur écart de taux », « credit spread forward » ou « dérivé sur écart de taux » et « total rate of return swap » ou « dérivé de transfert total de rendement ». La nature et la diversité des « dérivés de crédit » posent depuis toujours de sérieuses difficultés de qualification dans de nombreux pays. En droit français, si une qualification commune semble émerger, celle d’instrument financier, elle est hélas insuffisante à apporter un régime juridique complet. Un travail complémentaire de qualification est indispensable pour chaque contrat membre des « dérivés de crédit ». / Nowadays, since financial crisis, « credit derivatives » are famous. Born in 1990’s, they transfer the credit risk. They are speculation’s instrument or margin’s instrument. International Swaps and Derivatives Association (ISDA), and the Fédération Bancaire Française (in France), point to pattern juridical agreement. Credit derivatives include five big sort of agreement : « credit default swap » (« contrat d’échange sur le risque de crédit »), « credit linked notes » (« dérivé de crédit titrisé »), « credit spread option » (« option sur écart de taux »), « credit spread forward » (« dérivé sur écart de taux ») and « total rate of return swap » (« dérivé de transfert total de rendement »). Their variety and essence ask difficult question of juridical appreciation in many countries. In French law, credit derivatives are « instrument financier ». But this juridical appreciation is incomplete. Every sort of agreement must being individually studies.