As a physicist, I come in peace! I was wondering if anyone had some interesting thoughts on the "physical" nature of martingales with respect to the concept of observers. Namely, take for example the notion of GBM with respect to an asset price, so you have the SDE:
$dS = \mu S dt + \sigma S dW$,
where $W$ is the standard Wiener process. Now, we know we can turn this into a martingale by considering the transformation $S^{*} = e^{-rt} S$, where $r$ is the risk-free rate. This more broadly comes from a "change of measure" via the Radon-Nikodym derivative, which I will not go into detail here.
I was just curious if one can say that there are two "universes", one in which observers "measure"/"observe" $S(t)$, and one in which observers "measure" / "observe" $S^{*}$. Is this a correct statement to say?
Thanks.
I suppose there can't be a universe where both are true, and in fact neither are probably true. In any event there's only one "physical" measure, i.e. the probability distribution we imagine actually represents the distribution of outcomes for the stock price.
(This doesn't mean that it isn't possible for two different observers to believe in two different measures, or two different models entirely. Nor is it impossible to imagine different worlds in which there are different price processes for the underlying thanks to different investor beliefs and preferences.)
The physical measure cannot really be observed. Both the form of the model and the parameters might be chosen to fit some stylized facts from history and theory. We can of course statistically validate/reject/fit it historically, but we only get one historical path, which is a far cry from an observation of the whole distribution. Also the usual caveats apply that just cause the model was reasonable in the past doesn't mean it will be in the future.
The alternative measure in which $S$ grows at the risk-free rate is not the physical measure, but a pricing measure. It is the measure in which the market price is equal to the expected (discounted) value. This is not necessarily true in the physical measure because market participants attach a risk premium and don't price according to expected value.
Now say we have some time horizon and some discrete set of outcomes for the stock price at that time (this discrete set can limit to a continuum but let's keep it discrete for illustration). In a "complete market" where one can synthesize any possible payoff by a self-financing replicating strategy, one can replicate a security that pays off $\$1$ in a particular outcome. These are sometimes called Arrow securities. Because these (and any payoffs) can be replicated risklessly, they have a unique price in terms of the underlying... whatever you need to buy the initial replicating portfolio. And since if you buy all of them, you get $\$1$ no matter what, their values must sum to one (assuming zero interest rates).
They also must be non-negative value, since they never lose money, or it would be an arbitrage opportunity. So for each state you have a positive price, and they add up to one. This can be viewed as a probability distribution on the set of outcomes! Furthermore, the price will be zero if and only if the possibility of the outcome is zero (again by no arbitrage). This means it will be an equivalent probability distribution / measure to the true one (equivalent means they have the same set of certain and impossible outcomes). This new probability distribution, constructed from the prices of the arrow securities, is the pricing measure. The price of any payoff is straightforwardly equal to its expectation value under the pricing measure as one must simply buy the requisite arrow securities to construct the payoff. Note that the pricing measure is unconstrained, other than needing to be equivalent to the physical measure, so it doesn't represent the distribution of outcomes in the "real world". Also, it is not necessarily believed by anyone.
Now, if there were some observers who happened to be risk-neutral with respect to risk in the underlying, they would price according to this measure (since they price according to EV). So the measure is often called the risk-neutral measure.
Another way to look at it is that the fact that there is a complete market forces us to all price as if we're risk-neutral and the stock grows at the risk-free rate, even if it doesn't. This is because the price of derivatives doesn't have anything to do with the market's risk preferences (since the prices are determined entirely by replication), so we can pick any risk preferences we want for the market. The choice that it is risk free simultaneously makes the stock grow at the risk-free rate and makes the price the expected discounted value of the payoff. So in this way we can imagine the risk-neutral measure as a physical measure of an imaginary market (a different universe, if you will).
This is an oversimplification since complete markets are only required for uniqueness, not existence of the pricing measure, but hopefully it gets the idea across.