Calibration problems – An inverse problems view

Heinz W Engel presents techniques that have been applied to various types of inverse problems in science and engineering

When pricing structured or derivative financial instruments, the typical steps a quant has to do are the following:

1. Choose a model for the movement of the underlying(s)

2. Identify (“calibrate”) the model parameters from market prices of liquid instruments

3. Calculate the fair value of the structured instrument by appropriate numerical techniques. We will concentrate on step 2 in this article.

The techniques we will present have been applied to various types of inverse problems in science and engineering and should be relevant also in computational finance. At my research
groups, we applied advanced inverse problems techniques successfully e.g. to the following problems

– Reconstruct reinforcement bars in concrete from measurements of a scattered magnetic field [Engl, Neubauer]
– Determine optimal cooling strategies in continuous casting and hot rolling of steel (inverse heat conduction problems), e.g. [Binder, Engl, Vessella], [Binder et al.], [Chen et al.]
– Inferring the structure of ion channels from measurements (an inverse problem in a coupled system of partial differential equations similar to the semiconductor equations): [Burger, Eisenberg, Engl], [Burger et al.]

Let us continue with an example from finance:

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