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This thesis discusses the use of perturbation theory in the context of financial mathematics, in particular on the use of matched asymptotic expansions in option pricing. Our methods are applied to the ordinary Black-Scholes model for illustration. In this simple example of the Black-Scholes model an exact solution is available, so it is in fact not neccessary to apply the method of asymptotic expansions on this model. However, in case we do apply the method, two artificial layers have to be constructed. Making smart choices for the local variables leads to a transformation of the equations into a heat equation, which can easily be solved. Finally, the results are compared to a Taylor expansion of the exact solution to see that this method is very accurate. After this first instructive model, the method of matched asymptotic expansions is applied to two more advanced models based on papers by Sam Howison and Patrick Hagan et al.. Here, different choices for the scalings are made. The former discusses a fast mean-reverting stochastic volatility model that turns out to have many open ends. In Howison's paper quite a lot of assumptions and simplifications are made. Unfortunately, often the motivation for them is not explicitly given in the paper, and in some cases we even think these assumptions and simplifications are incorrect. The latter examines a new three-parameter stochastic volatility model that successfully prices back the volatility smile as observed in the market nowadays, and that is commonly used. The derivation of this model is the main focus of this thesis. The resulting expression for the implied volatility under the SABR model is obtained by considering the forward and backward Kol-mogorov equations per order in epsilon, making some smart choices for local variables and functions in order to transform them into an equation that looks like a heat equation, which is easier to solve. Recommendations for further investigation on these models would be to consider several different choices for the scalings and see which one works best.

This 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.