This dissertation is comprised of five research papers written during the period January 2013 -December 2015. Their abstracts are: The Fundamental Theorem of Derivative Trading. When estimated volatilities are not inperfect agreement with reality, delta hedged option portfolios will incur a non-zero profitand-loss over time. There is, however, a surprisingly simple formula for the resulting hedgeerror, which has been known since the late 90s. We call this The Fundamental Theoremof Derivative Trading. This paper is a survey with twists of that result. We prove a moregeneral version of it and discuss various extensions (including jumps) and applications (includingderiving the Dupire-Gy¨ongy-Derman-Kani formula). We also consider its practicalconsequences both in simulation experiments and on empirical data thus demonstrating thebenefits of hedging with implied volatility. Numerical Stochastic Control Theory with Applications in Finance. Analytic solutionsto HJB equation in mathematical finance are relatively hard to come by, which stresses theneed for numerical procedures. In this paper we provide a self-contained exposition of thefinite-horizon Markov chain approximation method as championed by Kushner and Dupuis.Furthermore, we provide full details as to how well the algorithm fares when we deploy itin the context of Merton type optimisation problems. Assorted issues relating to implementationand numerical accuracy are thoroughly reviewed, including multidimensionality andthe positive probability requirement, the question of boundary conditions, and the choice ofparametric values. Stochastic Volatility for Utility Maximisers Part I. From an empirical perspective, thestochasticity of volatility is manifest, yet there have been relatively few attempts to reconcilethis fact with Merton’s theory of optimal portfolio selection for wealth maximisingagents. In this paper we present a systematic analysis of the optimal asset allocation in aderivative-free market for the Heston model, the 3/2 model, and a Fong Vasicek type model.Under the assumption that the market price of risk is proportional to volatility, we can deriveclosed form expressions for the optimal portfolio using the formalism of Hamilton-JacobixiiiBellman. We also perform an empirical investigation, which strongly suggests that there inreality are no tangible welfare gains associated with hedging stochastic volatility in a bondstockeconomy. Stochastic Volatility for Utility Maximisers Part II. Using martingale methods we derivebequest optimising portfolio weights for a rational investor who trades in a bond-stockderivativeeconomy characterised by a generic stochastic volatility model. For illustrativepurposes we then proceed to analyse the specific case of the Heston economy, which admitsexplicit expressions for plain vanilla Europeans options. By calibrating the model to marketdata, we find that the demand for derivatives is primarily driven by the myopic hedge component.Furthermore, upon deploying our optimal strategy on real market prices, we findonly a very modest improvement in portfolio wealth over the corresponding strategy whichonly trades in bonds and stocks. Optimal Hedge Tracking Portfolios in a Limit Order Book. In this paper we developa control theoretic solution to the manner in which a portfolio manager optimally shouldtrack a targeted D, given that he wishes to hedge a short position in European call optionsthe underlying of which is traded in a limit order book. Specifically, we are interested in theinterplay between posting limit and market orders respectively: when should the portfoliomanager do what (and at what price)? To this end, we set up an Hamilton-Jacobi-Bellmanquasi variational inequality which we can solve numerically. Our scheme is shown to bemonotone, stable and consistent. Finally, we provide a concrete numerical study, comparingour algorithm with more na¨ıve approaches to delta-hedging.
Further to these papers we provide an extensive appendix which summarises standard resultsfrom martingale pricing, PDE methods (analytic and numerical), and stochastic control theory.The work presented in this section is mostly (though not exclusively) derivative, and canbe read as a friendly reminder with respect to the body of theory underpinning the re