The Adjoint method in Financial Sensitivity Analysis

This thesis investigated the adjoint method in estimating   financial sensitivities within a Monte Carlo framework. We gave a detailed   overview of the main Monte Carlo methods for the estimation of sensitivities,   and showed how the pathwise method in combination with the likelihood ratio   estimator allows to estimate sensitivities of derivatives with discontinuous   payoff functions. In part I the aforementioned estimation techniques were   applied and implemented numerically at the example of the Libor market model.   This model featured a multi-dimensional process and was therefore   particularly suited for the use of adjoints. The numerical implementation   showed that the use of adjoints significantly sped the simulation time up. In   part II we took an in-depth look at the adjoint pathwise method at the   example of stochastic volatility models. The advantages of the adjoint method   as well as the pitfalls were pointed out and verified numerically. In   addition, we showed how these estimated sensitivities can be used to   calibrate a financial model to market data. To summarize the results in a   general manner, the adjoint method is particularly useful when the objective   is to estimate multiple sensitivities of a small number of derivatives or   portfolios, multi-dimensional sensitivities of a small number of derivatives   or portfolios, sensitivities of derivatives that have a high likelihood of   zero payout, or sensitivities of financial models with a high number of input   parameters, such as time-dependent parameters whose evolution is unknown. On   the other hand, the adjoint method should not be used when the objective is   to estimate few sensitivities or sensitivities of a large number of   derivatives that differ only in their payoff function.

Sensitivities of financial securities play a fundamental   role in hedging risk and calibrating models. In cases where no closed-form   solution for a sensitivity is known, Monte Carlo methods such as the pathwise   estimator form a convenient alternative. One drawback of the standard   pathwise estimator is that the computational cost scales linearly in the   number of input parameters of the security. In this thesis, we investigate   the adjoint pathwise method, which allows the computation of all   sensitivities of a security at a total computational cost that is constant   relative to the computational cost of simulating the price, irrespective of the   number of input parameters. After giving a general introduction to   sensitivity estimation and extending the pathwise estimator by help of the   likelihood ratio method to estimate sensitivities of securities with   discontinuous payoffs, the thesis is split up in two parts. In the first part   the different techniques are applied and numerically implemented at the   example of caplets and swaptions within the Libor market model. In the second   part the adjoint method is applied in the context of stochastic volatility   models (SVM). In particular, sensitivities of European calls within the   Heston model as well as the lognormal SVM with time-dependent parameters are   estimated with the adjoint method and contrasted with the standard pathwise   method. Using adjoint sensitivities, the lognormal SVM is calibrated to   market data in a quadratic programming approach.