A Workout in Computational Finance

15 Parameter Calibration and Inverse Problems

Up to this point, we have considered various models for the movement of the underlying(s), and have assumed their parameters to be provided by an unspecified external source. The typical workflow in practical quantitative finance simulations is as follows:

1. Choose an appropriate model for the movement of the underlying.
2. Determine the parameters of this model such that model prices fit given market data, or, alternatively, manually choose the parameters.
3. Use one of the numerical methods described in the previous chapters to calculate a fair value or a Greek of a derivative or structured financial instrument.
4. (optional) Modify the market data to run scenario analyses or stress tests for risk management purposes.

In the present chapter, we will focus on step 2 above. Note that there is a huge difference between applications based on, say, thermodynamics and quantitative finance problems: The heat conductivity of a specific material at room temperature is, within measurement errors, the same when measured by different engineers at different times. In finance, on the other hand, the market prices of bonds are possibly made by the same market participants who want to determine model parameters. As a result, self-fulfilling prophecies or overreaction of markets may occur. Here, we do not take into account such important behavioral aspects of price formation, but accept the market prices of the instruments used for calibration at face value. ...