We are delighted to welcome Professor Uwe Naumann from the University of Aachen to Singapore. Professor Naumann has very kindly offered to share his knowledge with us at a roundtable event that will be hosted after work on the 18th February.

Professor Naumann will deliver a presentation and host a Q&A on Adjoint Parameter Estimation in Computational Finance.

“How sensitive are the values of the outputs of my computer program with respect to changes in the values of the inputs? How sensitive are these first-order sensitivities with respect to changes in the values of the inputs? How sensitive are the second-order sensitivities with respect to changes in the values of the inputs? . . .”

Computational scientists, engineers, and economists as well as quantitative analysts in computational finance tend to ask these questions on a regular basis. They write computer programs in order to simulate diverse real-world phenomena. The underlying mathematical models often depend on a possibly large number of (typically unknown or uncertain) parameters. Values for the corresponding inputs of the numerical simulation programs can, for example, be the result of (typically error-prone) observations and measurements. If very small perturbations in these uncertain values yield large changes in the values of the outputs, then the feasibility of the entire simulation becomes questionable. Nobody should make decisions based on such highly uncertain data.

Quantitative information about the extent of this uncertainty is crucial. First- and higher-order I sensitivities of outputs of numerical simulation programs with respect to their inputs (also first and higher derivatives) form the basis for various approximations of uncertainty. They are also crucial ingredients of a large number of numerical algorithms ranging from the solution of (systems of) nonlinear equations to optimization under constraints given as (systems of) partial differential equations. This talk describes a set of techniques for modifying the semantics of numerical simulation programs such that the desired first and higher derivatives can be computed accurately and efficiently. Computer programs implement algorithms. Consequently, the subject is known as Algorithmic (also Automatic) Differentiation (AD).

The calibration of unknown or uncertain parameters in financial models is a prime application for adjoint AD. For example, discrepancies between simulated and observed payoffs can be minimzed by optimizing corresponding least-squares objectives with respect to the N free parameters using first- or second-order numerical methods. Approximation of the required gradients / Hessians by finite difference quotients yields a computational cost that is linear / quadratic in N. Adjoint AD allows for gradients to be computed with a computational cost that is independent of N (typically at a constant factor of the cost of the underlying simulation that ranges between 4 and 20). Hessians can be obtained at linear (in N) cost. Suppose that a single payoff simulation as a function of 40 uncertain free parameters takes 5 seconds. A single sequential gradient approximation by finite differences would take at least 205 seconds. The speedup obtained by using adjoint AD would range between 2 and 10. The savings become even more substantial if large parameter spaces are considered. Last but not least, the numerical values obtained are accurate up to machine precision (no truncation).

Professor Naumann also works with the Numerical Algorithms Group (NAG) who specialise in delivering trusted, high quality numerical computing software and high performance computing services.

7 City, the home of the CQF have offered to host the event at their offices on Robinson Road. Click here for a map.

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