Application of Malliavin Calculus and Wiener chaos to option pricing theory.

This dissertation provides a contribution to the option pricing literature by means of some recent developments in probability theory, namely the Malliavin Calculus and the Wiener chaos theory. It concentrates on the issue of faster convergence of Monte Carlo and Quasi-Monte Carlo simulations for the Greeks, on the topic of the Asian option as well as on the approximation for convexity adjustment for fixed income derivatives. The first part presents a new method to speed up the convergence of Monte- Carlo and Quasi-Monte Carlo simulations of the Greeks by means of Malliavin weighted schemes. We extend the pioneering works of Fournie et al. (1999), (2000) by deriving necessary and sufficient conditions for a function to serve as a weight function and by providing the weight function with minimum variance. To do so, we introduce its generator defined as its Skorohod integrand. On a numerical example, we find evidence of spectacular efficiency of this method for corridor options, especially for the gamma calculation. The second part brings new insights on the Asian option. We first show how to price discrete Asian options consistent with different types of underlying densities, especially non-normal returns, by means of the Fast Fourier Transform algorithm. We then extends Malliavin weighted schemes to continuous time Asian options. In the last part, we first prove that the Black Scholes convexity adjustment (Brotherton-Ratcliffe and Iben (1993)) can be consistently derived in a martingale framework. As an application, we examine the convexity bias between CMS and forward swap rates. However, for more complicated term structures assumptions, this approach does not hold any more. We offer a solution to this, thanks to an approximation formula, in the case of multi-factor lognormal zero coupon models, using Wiener chaos theory

Generalized Bent-Cable Methodology for Changepoint Data: A Bayesian Approach

The choice of the model framework in a regression setting depends on the nature of the data. The focus of this study is on changepoint data, exhibiting three phases: incoming and outgoing, both of which are linear, joined by a curved transition. These types of data can arise in many applications, including medical, health and environmental sciences. Piecewise linear models have been extensively utilized to characterize such changepoint trajectories in di erent scientific fields. However, although appealing due to its simple structure, a piecewise linear model is not realistic in many applications where data exhibit a gradual change over time. The most important aspect of characterizing a changepoint trajectory involves identifying the transition zone accurately. It is not only because the location of the transition zone is of particular interest in many areas of study, but also because it plays an important role in adequately describing the incoming and the outgoing phases of a changepoint trajectory. Note that once the transition is detected, the incoming and the outgoing phases can be modeled using linear functions. Overall, it is desirable to formulate a model in such a way that it can capture all the three phases satisfactorily, while being parsimonious with greatly interpretable regression coe cients. Since data may exhibit an either gradual or abrupt transition, it is also important for the transition model to be flexible. Bent-cable regression is an appealing statistical tool to characterize such trajectories, quantifying the nature of the transition between the two linear phases by modeling the transition as a quadratic phase with unknown width. We demonstrate that a quadratic function may not be appropriate to adequately describe many changepoint data. In practice, the quadratic function of the bent-cable model may lead to a wider or narrower interval than what could possibly be necessary to adequately describe a transition phase. We propose a generalization of the bent-cable model by relaxing the assumption of the quadratic bend. Specifically, an additional transition parameter is included in the bent-cable model to provide su cient flexibility so that inference about the transition zone (i.e., shape and width of the bend) can be data driven, rather than pre-assumed as a specific type. We discuss the properties of the generalized model, and then propose a Bayesian approach for statistical inference. The generalized model is then demonstrated with applications to three data sets taken from environmental science and economics. We also consider a comparison among the quadratic bent-cable, generalized bent-cable and piecewise linear models in terms of goodness of fit in analyzing both real-world and simulated data. Moreover, we supplement the motivation for our generalized bent-cable methodology via extensive simulations – we simulate changepoint data under some realistic assumptions, and then fit the quadratic bent-cable, generalized bent-cable and piecewise linear models to each of the simulated data sets to compare the performance of these models with respect to the overall quality of fit. A sensitivity analysis is also performed to investigate the sensitivity of Bayesian inference to prior specifications. This study suggests that the proposed generalization of the bent-cable model can be valuable in adequately describing changepoint data that exhibit either an abrupt or gradual transition over time

Essays in financial asset pricing

Three essays in financial asset pricing are given; one concerning the partial differential equation (PDE) pricing and hedging of a class of continuous/generalized power mean Asian options, via their (optimal) Lie point symmetry groups, leading to practical pricing formulas. The second presents high-frequency predictions of S&P 500 returns via several machine learning models, statistically significantly demonstrating short-horizon market predictability and economically significantly profitable (beyond transaction costs) trading strategies. The third compares profitability between these [(mean) ensemble] strategies and Asian option Δ-hedging, using results of the first. Interpreting bounds on arithmetic Asian option prices as ask and bid values, hedging profitability depends largely on securing prices closer to the bid, and settling midway between the bid and ask, significant profits are consistently accumulated during the years 2004-2016. Ensemble predictive trading the S&P 500 yields comparatively very small returns, despite trading much more frequently. The pricing and hedging of (arithmetic) Asian options are difficult and have spurred several solution approaches, differing in theoretical insight and practicality. Multiple families of exact solutions to relaxed power mean Asian option pricing boundary-value problems are explicitly established, which approximately satisfy the full pricing problem, and in one case, converge to exact solutions under certain parametric restrictions. Corresponding hedging parameters/ Greeks are derived. This family consists of (optimal) invariant solutions, constructed for the corresponding pricing PDEs. Numerical experiments explore this family behaviorally, achieving reliably accurate pricing. The second chapter studies intraday market return predictability. Regularized linear and nonlinear tree-based models enjoy significant predictability. Ensemble models perform best across time and their return predictability realizes economically significant profits with Sharpe ratios after transaction costs of 0.98. These results strongly evidence that intraday market returns are predictable during short time horizons, beyond that explainable by transaction costs. The lagged constituent returns are shown to hold significant predictive information not contained in lagged market returns or price trend and liquidity characteristics. Consistent with the hypothesis that predictability is driven by slow-moving trader capital, predictability decreased post-decimalization, and market returns are more predictable midday, on days with high volatility or illiquidity, and during financial crises