Fixed Income Derivatives Pricing In Practice

In this lesson, we wrapped up our discussion of model calibration.

The pricing philosophy is the same for all types of models:

1. Specify a model under the Q(theta) dynamics where theta is a vector of parameters, such as a(i) and b(i).
2. Price all securities using the formula:

      3. Choose theta parameters such that the market prices of the liquid securities agree with the model             prices.

These three steps are formally known as the calibration procedure.

The calibration problem usually requires minimizing a sum of squares equation:

where:

P(i)(model) is the model price of the i-th calibration security

P(i)(market) is the model price of the i-th calibration security

w(i) is a positive weight reflecting the importance of the i-th security or the confidence we have in its market price

theta(prev) = previously calibrated model parameters

and lambda is a parameter reflecting relative importance of remaining close to the previous calibration.

Once we have minimized this equation we can use the model to hedge or price more illiquid securities.

One problem, however, is that this equation is very difficult to solve.  It is a non-convex optimization problem with many local minima and therefore many solutions.  As market conditions change from minute-to-minute and hour-to-hour, we may need to recalibrate the model frequently.  If the model was in fact, "right," then we would only need to calibrate once.  So in practice our model is not right, and markets are too complex for there to be a "right model." However, through risk-neutral pricing at the model level, we can extrapolate/interpolate in an arbitrage free manner.