What is Algorithmic or Automatic Differentiation?
Artificial Intelligence & Machine Learning
Algorithmic or Automatic Differentiation (AD) is a computational technique used to accurately and efficiently calculate the derivatives of functions expressed as computer programs. Unlike numerical differentiation, which approximates derivatives using finite differences, AD leverages the chain rule to decompose complex functions into a sequence of elementary operations, ensuring precise and efficient gradient computations.
How automatic differentiation relates to symbolic differentiation.
How is it useful for machine learning development?
Enables the efficient and accurate computation of gradients! Which are essential for optimizing model parameters during training. By automating the differentiation process, AD allows for the seamless implementation of gradient-based optimization algorithms, such as gradient descent, thereby speeding up the development and improving the performance of complex neural networks and other machine learning models.
Automatic Adjoint Differentiation (AAD) can improve both sensitivity calculations and pricing by interpreting prices as sensitivities. Through perturbation techniques, AAD efficiently computes the gradients of prices with respect to input parameters, which are essential for both risk management and optimization in financial models. This dual capability enhances the accuracy and efficiency of pricing complex financial derivatives.
Today we are taking a look at the work of Adil Reghai & Othmane Kettani where they demonstrate the pricing improvement.
A Sensitivity Based Approach for CVA Computation by Adil Reghai, Othmane Kettani :: SSRN
Since the sub-prime crisis, addressing counterparty credit risk and Wrong Way Risk has become essential in the valuation and risk management of credit derivatives. Calculating Credit Valuation Adjustment (CVA) for portfolios containing complex products can be particularly challenging. This paper introduces a thorough approach to model CVA and Wrong Way Risk for equity portfolios. The method employs a first-order approximation of exposures, enabling the derivation of closed-form formulas for CVA. Wrong Way Risk becomes modeled using a Gaussian copula, linking the counterparty’s default to the underlying portfolio.
Gaussian copula is a statistical tool used to describe the dependency structure between multiple variables. As a result of transforming their joint distribution into a standard normal distribution. It allows for the modeling of complex dependencies by capturing the correlation between variables, making it particularly useful in finance for risk management and pricing derivatives.
The model became evaluated on two portfolios under different credit risk scenarios, demonstrating a computationally efficient retrieval of theoretical CVA.
Kettani & Reghai also discuss notable properties of the Gaussian copula in the context of Wrong Way Risk modeling.
With a comprehensive method to price CVA begins with deriving a first-order approximation of the contract’s market value based on its sensitivities to spot variations. This approximation allows for a closed-form expression for expected exposures.
Kettani & Reghai tested the model on two equity portfolios: one containing an Equity Return Swap (ERS) and another where a call option became added to the ERS.
Model results were compared to theoretical values, showing that the theoretical CVA could become retrieved using their approach, with necessary adjustments to model parameters as maturity increases. Additionally, modified the CVA formula to account for Wrong Way Risk through a Gaussian copula, which models the dependence structure between the counterparty‘s default and exposure. The impact of the Gaussian copula correlation parameter on CVA became analyzed under two different credit risk scenarios—high and low credit spreads—thus, revealing interesting properties of the Gaussian copula in CVA modeling.
A Sensitivity Based Approach for CVA Computation by Adil Reghai, Othmane Kettani :: SSRN
