Real-Time Risk Management: An AAD-PDE Approach

We apply adjoint algorithmic differentiation (AAD) to the risk management of securities when their price dynamics are given by partial differential equations (PDE). We show how AAD can be applied to forward and backward PDEs in a straightforward manner. In the context of one-factor models for interest rates or default intensities, we show how price sensitivities are computed reliably and orders of magnitude faster than with a standard finite-difference approach. This significantly increased efficiency is obtained by combining (i) the adjoint forward PDE for calibrating model parameters, (ii) the adjoint backward PDE for derivatives pricing, and (iii) the implicit function theorem to avoid iterating the calibration procedure

Joint Distribution of Passage Times of an Ornstein-Uhlenbeck Diffusion and Real-Time Computational Methods for Financial Sensitivities

This thesis analyses two broad problems: the computation of financial sensitivities, which is a computationally expensive exercise, and the evaluation of barriercrossing probabilities which cannot be approximated to reach a certain precision in certain circumstances. In the former case, we consider the computation of the parameter sensitivities of large portfolios and also valuation adjustments. The traditional approach to compute sensitivities is by the finite-difference approximation method, which requires an iterated implementation of the original valuation function. This leads to substantial computational costs, no matter whether the valuation was implemented via numerical partial differential equation methods or Monte Carlo simulations. However, we show that the adjoint algorithmic differentiation algorithm can be utilised to calculate these price sensitivities reliably and orders of magnitude faster compared to standard finite-difference approaches. In the latter case, we consider barrier-crossing problems of Ornstein-Uhlenbeck diffusions. Especially in the case where the barrier is difficult to reach, the problem turns into a rare event occurrence approximation problem. We prove that it cannot be estimated accurately and robustly with direct Monte Carlo methods because of the irremovable bias and Monte Carlo error. Instead, we adopt a partial differential equation method alongside the eigenfunction expansion, from which we are able to calculate the distribution and the survival functions for the maxima of a homogeneous Ornstein-Uhlenbeck process in a single interval. By the conditional independence property of Markov processes, the results can be further extended to inhomogeneous cases and multiple period barrier-crossing problems, both of which can be efficiently implemented by quadrature and Monte Carlo integration methods

Option Pricing with Orthogonal Polynomial Expansions

We derive analytic series representations for European option prices in polynomial stochastic volatility models. This includes the Jacobi, Heston, Stein-Stein, and Hull-White models, for which we provide numerical case studies. We find that our polynomial option price series expansion performs as efficiently and accurately as the Fourier transform based method in the nested affine cases. We also derive and numerically validate series representations for option Greeks. We depict an extension of our approach to exotic options whose payoffs depend on a finite number of prices.Comment: forthcoming in Mathematical Finance, 38 pages, 3 tables, 7 figure

On the calibration of stochastic volatility models: A comparison study

We studied the application of gradient based optimization methods for calibrating stochastic volatility models. In this study, the algorithmic differentiation is proposed as a novel approach for Greeks computation. The “payoff function independent” feature of algorithmic differentiation offers a unique solution cross distinct models. To this end, we derived, analysed and compared Monte Carlo estimators for computing the gradient of a certain payoff function using four different methods: algorithmic differentiation, Pathwise delta, likelihood ratio and finite differencing. We assessed the accuracy and efficiency of the four methods and their impacts into the optimisation algorithm. Numerical results are presented and discussed

AAD and least-square Monte Carlo: fast Bermudan-style options and XVA Greeks

We show how Adjoint Algorithmic Differentiation (AAD) can be used to calculate price sensitivities in regression-based Monte Carlo methods reliably and orders of magnitude faster than with standard finite-difference approaches. We present the AAD version of the celebrated least-square algorithms of Tsitsiklis and Van Roy (2001) and Longstaff and Schwartz (2001). By discussing in detail examples of practical relevance, we demonstrate how accounting for the contributions associated with the regression functions is crucial to obtain accurate estimates of the Greeks, especially in XVA applications

Adjoint Venture: Fast Greeks with Adjoint Algorithmic Differentiation

This dissertation seeks to discuss the adjoint approach to solving affine recursion problems (ARPs) in the context of computing sensitivities of financial instruments. It is shown how, by moving from an intuitive 'forward' approach to solving a recursion to an 'adjoint' approach, one might dramatically increase the computational efficiency of algorithms employed to compute sensitivities via the pathwise derivatives approach in a Monte Carlo setting. Examples are illustrated within the context of the Libor Market Model. Furthermore, these ideas are extended to the paradigm of Adjoint Algorithmic Differentiation, and it is illustrated how the use of sophisticated techniques within this space can further improve the ease of use and efficiency of sensitivity calculations

Mathematics of Quantitative Finance

The workshop on Mathematics of Quantitative Finance, organised at the Mathematisches Forschungsinstitut Oberwolfach from 26 February to 4 March 2017, focused on cutting edge areas of mathematical finance, with an emphasis on the applicability of the new techniques and models presented by the participants