Martingale Measures, Numéraires, and Market Completeness

# The martingale property - why only for Black Scholes model?


# The martingale property - why only for Black Scholes model?












In the Black Scholes model, the normalized price process is a martingale under the martingale measure $Q$.

What I don't understand is, why is this limited to the Black Scholes model only? Does this not hold for the general model where $$dS(t) = S(t)\alpha(t,S(t))dt + S(t)\sigma(s,S(t))dW(t)?$$

At least I was asked to prove this property, and I did not need to assume that $\alpha$ and $\sigma$ were constants. So was my proof wrong?

The way I proved it was that I first found out $d\Pi(t)$ using the neutral risk evaluation formula, and then I used the multi-dimensional Ito formula on the ratio of the price process over the riskfree asset. The dt part just cancelled out, and so I was left with only dW(t), and then it's a martingale.

EDIT:

In fact, here in Bjork's text. He too states the martingale property in the generalized section with the model as stated above, and then shortly after he has a subsection where he focuses strictly on the standard Black Scholes model. This merely adds to my confusion: if the result holds generally, why does he claim in the theorem that it only holds in the Black Scholes model? And why does he not put this theorem inside subsection 7.5 where he limits himself to precisely the Black Scholes model??

## Answer by Quantuple (score 3)

https://quant.stackexchange.com/a/31536

It is not a property per se.

Rather, it is the absence of arbitrage opportunities which requires that "normalised" prices of self-financing investment strategies emerge as martingales under some probability measure equivalent to the physical measure, where by "normalised" one should understand, "prices expressed with respect to a certain numéraire", a numéraire being any, positive-valued, tradable asset which will serve as a relative basis to express the value of some other assets/self-financing strategies.

This is obviously not limited to the BS model. Any valuation model working under the "no free lunch" assumption should verify that.

Yet, for pure diffusion models like BS, such equivalent martingale measures are unique (read: there exists a unique measure per type of numéraire, the risk-neutral measure being associated to the risk-free money market account numéraire). This is not the case for stochastic volatility models or models that include random jumps. We talk about (market) model (in)completeness.