The current Thesis is devoted to comprehensive studies of comparison, or stochastic domination, theorems. It presents a combination of theoretical research and practical ideas formulated in several specific examples. Previously known results and their place it the theory of stochastic processes and stochastic differential equations is reviewed. This part of the work yielded three new theoretical results, formulated as theorems. Two of them are extensions of commonly used methods to more sophisticated processes and conditions. The third theorem is proven using previously not exploited technique. The place of all three results in the global theory is demonstrated by examining interconnections and possible distinctions between old and new theorems. Second and equally important part of the work focuses on more practical issues. Its main goal is to demonstrate where and how various theoretical findings can be applied to typical financial problems, such as option pricing, hedging, risk management and others. The example chapter summarizes the best of the obtained results in this direction. / Mathematical Finance

Value-at-Risk (VaR ) is an industrial standard for monitoring market risk in an investment portfolio. It measures potential osses within a given confidence level. VaR was first used by major financial institutions in the early 1990’s, and widely developed after the release of J.P. Morgan’s Riskmetrics Technical Document in 1996. The efficient calculation, implementation, interpretation and optimization of VaR are a challenge in the practice of risk management when the number of market factors in the portfolio is high. In this thesis, we are concerned with the quadratic analytical estimation of VaR and we present a methodology for an approximation to VaR that is based on the principal components of a sensitivity-adjusted covariance matrix. The result is an explicit expression in terms of portfolio deltas, gammas, and the mean and covariance matrix. It can be viewed as a non-linear extension of the linear model given by the delta-normal-VaR of RiskMetrics, a standard calculation for the risk in the financial sector. We obtain an asymptotic expansion for VaR in the limit when the confidence level approaches 1 and precise estimates of the reminder. We then optimize the approximated VaR with respect to the gradient or delta of the portfolio, a quantity which can be changed by trading the underlying assets (stocks), without entering into any derivative transactions. This analysis provides an optimal trading strategy of the portfolio that minimizes the risk.

Value-at-Risk (VaR ) is an industrial standard for monitoring market risk in an investment portfolio. It measures potential osses within a given confidence level. VaR was first used by major financial institutions in the early 1990’s, and widely developed after the release of J.P. Morgan’s Riskmetrics Technical Document in 1996. The efficient calculation, implementation, interpretation and optimization of VaR are a challenge in the practice of risk management when the number of market factors in the portfolio is high. In this thesis, we are concerned with the quadratic analytical estimation of VaR and we present a methodology for an approximation to VaR that is based on the principal components of a sensitivity-adjusted covariance matrix. The result is an explicit expression in terms of portfolio deltas, gammas, and the mean and covariance matrix. It can be viewed as a non-linear extension of the linear model given by the delta-normal-VaR of RiskMetrics, a standard calculation for the risk in the financial sector. We obtain an asymptotic expansion for VaR in the limit when the confidence level approaches 1 and precise estimates of the reminder. We then optimize the approximated VaR with respect to the gradient or delta of the portfolio, a quantity which can be changed by trading the underlying assets (stocks), without entering into any derivative transactions. This analysis provides an optimal trading strategy of the portfolio that minimizes the risk.

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This thesis investigates implied volatility in general classes of stock price models. To begin with, we take a very general view. We find that implied volatility is always, everywhere, and for every expiry well-defined only if the stock price is a non-negative martingale. We also derive sufficient and close to necessary conditions for an implied volatility surface to be free from static arbitrage. In this context, free from static arbitrage means that the call price surface generated by the implied volatility surface is free from static arbitrage. We also investigate the small time to expiry behaviour of implied volatility. We do this in almost complete generality, assuming only that the call price surface is non-decreasing and right continuous in time to expiry and that the call surface satisfies the no-arbitrage bounds (S-K)+≤ C(K, τ)≤ S. We used S to denote the current stock price, K to be a option strike price, τ denotes time to expiry, and C(K, τ) the price of the K strike option expiring in τ time units. Under these weak assumptions, we obtain exact asymptotic formulae relating the call price surface and the implied volatility surface close to expiry. We apply our general asymptotic formulae to determining the small time to expiry behaviour of implied volatility in a variety of models. We consider exponential L??vy models, obtaining new and somewhat surprising results. We then investigate the behaviour close to expiry of stochastic volatility models in the at-the-money case. Our results generalise what is already known and by a novel method of proof. In the not at-the-money case, we consider local volatility models using classical results of Varadhan. In obtaining the asymptotics for local volatility models, we use a representation of the European call as an integral over time to expiry. We devote an entire chapter to representations of the European call option; a key role is played by local time and the argument of Klebaner. A novel alternative that is especially useful in the local volatility case is also presented.

This thesis investigates implied volatility in general classes of stock price models. To begin with, we take a very general view. We find that implied volatility is always, everywhere, and for every expiry well-defined only if the stock price is a non-negative martingale. We also derive sufficient and close to necessary conditions for an implied volatility surface to be free from static arbitrage. In this context, free from static arbitrage means that the call price surface generated by the implied volatility surface is free from static arbitrage. We also investigate the small time to expiry behaviour of implied volatility. We do this in almost complete generality, assuming only that the call price surface is non-decreasing and right continuous in time to expiry and that the call surface satisfies the no-arbitrage bounds (S-K)+≤ C(K, τ)≤ S. We used S to denote the current stock price, K to be a option strike price, τ denotes time to expiry, and C(K, τ) the price of the K strike option expiring in τ time units. Under these weak assumptions, we obtain exact asymptotic formulae relating the call price surface and the implied volatility surface close to expiry. We apply our general asymptotic formulae to determining the small time to expiry behaviour of implied volatility in a variety of models. We consider exponential L??vy models, obtaining new and somewhat surprising results. We then investigate the behaviour close to expiry of stochastic volatility models in the at-the-money case. Our results generalise what is already known and by a novel method of proof. In the not at-the-money case, we consider local volatility models using classical results of Varadhan. In obtaining the asymptotics for local volatility models, we use a representation of the European call as an integral over time to expiry. We devote an entire chapter to representations of the European call option; a key role is played by local time and the argument of Klebaner. A novel alternative that is especially useful in the local volatility case is also presented.

In this thesis, we study three topics in optimal portfolio selection that are relevant to the theory of forward investment performance processes. In Chapter 1, we develop a connection between the classical mean-variance optimisation and time-monotone forward performance processes for infinitesimal trading times. Namely, we consider consecutive mean-variance problems and we show that, for an appropriate choice of the corresponding mean-variance trade-off coefficients, the wealth process that is generated converges (as the trading interval goes to zero) to the optimal wealth process generated by a time-monotone forward performance process. The choice of the trade-off coefficients is made in accordance to the evolution of the risk tolerance process of the forward performance process. This result allows us to provide a fresh view on the issue of time-consistency of mean-variance analysis, for we propose a method to update mean-variance risk preferences forward in time. As a by-product, our convergence theorem generalises a result by Gyöngy (1998) on the convergence of the Euler scheme for SDEs. We also provide novel results on the Lipschitz regularity of the local risk tolerance function of forward investment performance processes. The material in this chapter is joint work with Marek Musiela and Thaleia Zariphopoulou. Chapter 2 combines forward investment theory and partial information. Specifically, we construct forward investment performance processes in models where the drift is a random variable distributed according to a known distribution. The forward performance processes we consider are of the type U(t,x) = u(t,x, R_t), where R. denotes the process of cumulative excess returns, and u(t,x,z):[0,∞) × ℝ imes ℝ^N ⟶ ℝ is such that u(t,.,z) is a utility function satisfying Inada's conditions. We derive the Hamilton-Jacobi-Bellman (HJB) equation for u(.). The HJB equation is linearised into the ill-posed heat equation; then, using the multidimensional version of Widder's theorem, we fully characterise the solutions to this equation in terms of a collection of positive measures; the result is an integral representation of the convex conjugate function of u(t,.,z). We construct several examples, and we show how these can be combined, in the dual domain, to generate mixtures of forward investment performance processes. We also show that the volatility of these processes is intrinsic, in that it is not generated by changes of numéraire/measure. In Chapter 3, we provide an extension of the Black-Litterman model to the continuous time setting. Our extension is different from, and complements that of, Frey, Gabih, and Wunderlich (2012) and Davis and Lleo (2013). Specifically, we develop a novel robust estimator of instantaneous expected returns which is continuously shrunk towards the predictions of an asset pricing theory, such as the CAPM. We derive this estimator fairly explicitly and study some of its properties. As in the Black-Litterman model, such an estimator can be used to make optimal asset allocation problems in continuous time more robust with respect to estimation errors. We provide explicit solutions to the problem of maximising expected power utility of terminal wealth, when our estimator is used to estimate the drift. As an example, we illustrate our results explicitly in the case of a multifactor model, where Arbitrage Pricing Theory predicts that alphas should be approximately zero.

In the field of uncertainty quantification, the effects of parameter uncertainties on scientific simulations may be studied by integrating or approximating a quantity of interest as a function over the parameter space. If this is done numerically, using regular grids with a fixed resolution, the required computational work increases exponentially with respect to the number of uncertain parameters – a phenomenon known as the curse of dimensionality. We study two methods that can help break this curse: discrete least squares polynomial approximation and kernel-based approximation. For the former, we adaptively determine sparse polynomial bases and use evaluations in random, quasi-optimally distributed evaluation nodes; for the latter, we use evaluations in sparse grids, as introduced by Smolyak. To mitigate the additional cost of solving differential equations at each evaluation node, we extend multilevel methods to the approximation of response surfaces. For this purpose, we provide a general analysis that exhibits multilevel algorithms as special cases of an abstract version of Smolyak’s algorithm. In financial mathematics, high-dimensional approximation problems occur in the pricing of derivatives with multiple underlying assets. The value function of American options can theoretically be determined backwards in time using the dynamic programming principle. Numerical implementations, however, face the curse of dimensionality because each asset corresponds to a dimension in the domain of the value function. Lack of regularity of the value function at the optimal exercise boundary further increases the computational complexity. As an alternative, we propose a novel method that determines an optimal exercise strategy as the solution of a stochastic optimization problem and subsequently computes the option value by simple Monte Carlo simulation. For this purpose, we represent the American option price as the supremum of the expected payoff over a set of randomized exercise strategies. Unlike the corresponding classical representation over subsets of Euclidean space, this relax- ation gives rise to a well-behaved objective function that can be globally optimized using standard optimization routines.

The study of expected utility maximization in continuous-time stochastic market models dates back to the seminal work of Merton 1969 and has since been central to the area of Mathematical Finance. The associated stochastic optimization problems have been extensively studied. The problem formulation relies on two strong underlying assumptions: the ability to specify the underpinning market model and the knowledge of the investor's risk preferences. However, neither of these inputs is easily available, if at all. Resulting issues have attracted continuous attention and prompted very active and diverse lines of research. This thesis seeks to contribute towards this literature and questions related to both of the above issues are studied. Specifically, we study the implications of certain qualitative properties of the utility function; we introduce, and study various aspects of, the notion of robust forward investment criteria; and we study the investment problem associated with risk- and ambiguity-averse preference criteria defined in terms of quasiconcave utility functionals.

In real markets, to some degree, every trade will incur a non-zero cost and will influence the price of the asset traded. In situations where a dynamic trading strategy is implemented these liquidity effects can play a significant role. In this thesis we examine two situations in which such trading strategies are inherent to the problem; that of pricing a derivative contingent on the asset and that of executing a large portfolio transaction in the asset. The asset's finite liquidity has been incorporated explicitly into its price dynamics using the Bakstein-Howison model [4]. Using this model we have derived the no-arbitrage price of a derivative on the asset and have found a true continuous-time equation when the bid-ask spread in the asset is neglected. Focussing on this pure liquidity case we then employ an asymptotic analysis to examine the price of a European call option near strike and expiry where the liquidity effects are shown to be most significant and closed-form expressions for the price are derived in this region. The asset price model is then extended to incorporate the empirical fact that an asset's liquidity mean reverts stochastically. In this situation the pricing equation is analyzed using the multiscale asymptotic technique developed by Fouque, Papanicolaou, and Sircar [22] and a simplified pricing and calibration framework is developed for an asset possessing liquidity risk. Finally, the derivative pricing framework (both with and without liquidity risk) is applied to a new contract termed the American forward which we present as a possible hedge against an asset's liquidity risk. In the second part of the thesis we investigate how to optimally execute a large transaction of a finitely liquid asset. Using stochastic dynamic programming and attempting only to minimize the transaction's cost, we first find that the optimal strategy is static and contains the naive strategy found in previous studies, but with an extra term to account for interest rates neglected by those studies. Including time risk into the optimization procedure we find expressions for the optimal strategy in the extreme cases when the trader's aversion to this risk is very small and very large. In the former case the optimal strategy is simply the cost-minimization strategy perturbed by a small correction proportional to the trader's level of risk aversion. In the latter case the problem is shown to be much more difficult; we analyze and derive implicit closed-form solutions to the much-simplified perfect liquidity case and show numerical results to demonstrate the agreement of the solution with our intuition.