This thesis consists of ve research papers written during the period March 2014 - April 2016. The papers can be read independently and their abstracts are: 1. European Option Pricing with Stochastic Volatility Models under Pa- rameter Uncertainty. We consider stochastic volatility models under parameter uncertainty and investigate how model derived prices of European options are affected. We let the pricing parameters evolve dynamically in time within a specied region, and formalise the problem as a control problem where the control acts on the parameters to maximise/minimise the option value. Through a dual representation with backward stochastic dierential equations, we obtain explicit equations for Heston's model and investigate several numerical solutions thereof. In an empirical study, we apply our results to market data from the S&P 500 index where the model is estimated to historical asset prices. We nd that the conservative model-prices cover 98% of the considered market-prices for a set of European call options. 2. The Fundamental Theorem of Derivative Trading. When estimated volatilities are not in perfect agreement with reality, delta hedged option portfolios will incur a non-zero prot-and-loss over time. However, there is a surprisingly simple formula for the resulting hedge error, which has been known since the late '90s. We call this The Fundamental Theorem of Derivative Trading. This paper is a survey with twists of that result. We prove a more general version of it and discuss various extensions and applications, from incorporating a multi-dimensional jump framework to deriving the Dupire-Gyongy-Derman-Kani formula. We also consider its practical consequences both in simulation experiments and on empirical data thus demonstrating the benets of hedging with implied volatility. 3. Risk Minimization in Electricity Markets. This paper analyses risk management of xed price, unspecied consumption contracts in energy markets. We model the joint dynamics of spot-price and consumption of electricity, study expected loss minimisation for dierent loss functions and derive optimal static hedge strategies based on forward contracts. These strategies are tested empirically on 2012-2014 Nordic market data and compared to a simpler hedge strategy which is widely employed by the industry. Results show that our suggested hedge outperforms the commonly used with a higher reward-to-risk ratio, which can be exploited to release a premium from the contract. The realised cumulative prot-and-loss from our suggested hedge is greater for almost every single one-month period of the considered data, whilst the hourly realised payout results in a 66% out-performance probability. 4. Stochastic Volatility for Utility Maximisers. Using martingale methods we derive bequest optimising portfolio weights for a rational investor who trades in a bondstock- derivative economy characterised by a generic stochastic volatility model. For illustrative purposes we then proceed to analyse the specic case of the Heston economy, which admits explicit expressions for plain vanilla Europeans options. By calibrating the model to market data, we nd that the demand for derivatives is primarily driven by the myopic hedge component. Furthermore, upon deploying our optimal strategy on real market prices, we nd only a very modest improvement in portfolio wealth over the corresponding strategy which only trades in bonds and stocks. 5. Volatility is Log-Normal, But not for the Reason you Think. Stochastic volatility models have increased enormously in popularity since their introduction in the late eighties. Not the least for hedging and option pricing purposes since they do well in tting the implied volatility surface. In fact, their pricing ability is often the reason for advocating such a model, whilst their ability to capture the underlying dynamics is loosely motivated. In this paper we test for what is a good model of volatility based on the latter perspective: we brie y review three wellknown stochastic volatility models, and concentrate on the instantaneous variance in Heston's model, a log-normal model and in the 3-over-2 model. Since volatility is a non-observable process, we employ the technique of realized volatility to obtain variance measurements and from these we form a goodness-of-t analysis based on the concept of uniform residuals. To assess the model-classication ability of our analysis, we perform a Monte Carlo study. We then apply the methodology in an empirical study, where our results show that the log-normal model