Risk-Neutral Measure

Okay, I admit that I never carefully thought this through - basically, why should a derivative's price process be a martingale under the so called risk neutral measure?

Here is the answer: at the first place: there isn't any probability issue involved in the pricing of a derivative! The risk neutral measure is an artificial measure assigned to different outcomes of all the underlying assets, such that the discounted of them are martingales. Because these individual martingales, in their differential form, can be expressed as X_i(t)*dW_i(t), it follows that any martingale process can be expressed as a linear combination of these individual underlying assets. In other words - this derivative can be perfectly hedged.

So what happened when there are more than one risk neutral measure? It means that there are more "fundamentals" than actual assets available. The conclusion is that a position in the derivative can not be hedged by the existing underlying. However the market still does not admit any arbitrage (proof is easy).

What happened when there is no risk neutral measure? The market admits arbitrage because we can find a linear combination of the underlying to form a portfolio that drives non-negative returns in all scenario while getting positive returns for some. The proof is not included here. Intuitively a lack of risk-neutral measure would mean some assets are having an unfair price for their risks - fully expressible by that of other assets and can be properly hedged, therefore profits can be received from a proper portfolio.

PRACTICE: Solve Hull-White interest rate model by classical PDE ways (Green's function maybe).