Occasionally I despair of the way people teach quantitative finance. They put so much effort into teaching students to do the heavy lifting of equivalent martingale measures. Having learned options theory before this stuff became de rigueur, I think it’s overkill. I’m reluctantly in favor of teaching it to financial engineers because they have to live in a world when that’s how so many other people talk. But I wish people didn’t talk that way, didn’t talk about the fundamental theorem of finance, didn’t teach it as a branch of rigorous mathematics.
I like to argue that you can get most of the results you need in derivatives via the following common sense:
- Write down a stochastic process for the underlying assets. These are the scenarios you will allow.
- Calibrate the model, which means you must make its scenario parameters consistent with the current prices of the simpler liquid risky tradeable securities. If you calibrate to forwards and futures, this will force you into a risk-neutral world. Anytime you use your model to calculate the value of a simple risky security whose risk you have somehow specified (even incorrectly) and whose market price you know, your model must match that price—if it doesn’t, you’re starting off from the wrong place.
I believe this insight captures most of the meaning behind the use of martingale measures. But it’s based on the common sense that a model is not too useful if it produces the wrong values for things you already know the price of. This is true even if you have discontinuous and unhedgeable moves in your assets.