Marek Musiela gave a very interesting talk at Columbia the other evening about trying to use a derivatives framework to handle optimal asset allocation. A discussion arose about what makes a model useful for trading derivatives. I always come back to the same answer: practitioners use models in finance to take you from the prices of liquid market-traded instruments and use them to estimate the prices of illiquid instruments. Black-Scholes takes you from stock and bond prices to options prices or convertible bond prices. In a real sense, the models are interpolators from known boundaries to the middle ground.
Ambitious theoretical models that start bottom up often have elegant theoretical variables (a stochastic pde that describes the evolution of volatility, for example) that have to be made up and don’t directly relate to observed or traded quantities. You then have to use heavy machinery to calibrate the model to the value of the liquid instruments and use it to calculate the values of the illiquid ones. The link between liquid and illiquid goes through a substrate of hedden variable that is invisible. A practitioner model tries to eliminate as much of that substrate as possible. There should be a short path from calibration to value.
Therefore, it’s easier for users of the models to have the inputs and their stochastic description be quantities or parameters they can have an intuitive handle on. You can work with stochastic variables that are directly observed. This is more or less what market models do.
Barrier options valuation is another example. In order to value a barrier option, you can build an arbitrage-free stochastic volatility model, calibrate it as best you can, make assumptions for correlations that you are uncertain about, and then value the exotic. Another way to go at it, that I’ve seen practitioners do, is to approximately replicate the exotic option out of a portfolio of vanilla options, and then let the vanilla options themselves become stochastic, and compute the effect of that stochasticity on the value of the exotic option. It’s less theoretically defensible, and may violate some axioms of valuation, but it may also give a better handle on the dynamics of the exotic, since there’s a shorter path from liquid to illiquid.