Risk Neutral Paradox: Strange Phenomena Observed by Delbaen-Schachermayer

If markets are not complete then there are several risk neutral measures (equivalent to the “physical” measure).  That means that for a new instrument there might be several prices consistent with the other prices.  In reality markets are not complete, especially if there are default prone bonds  the way the default intensity is modelled can have infinitely many solutions.  2005 I've worked with an extension part of the theorem which Delbaen did long time ago (1989):  if all martingales are continuous (this is a hypothesis that is invariant for an equivalent change of measure, the martingales are of course not invariant but the continuity is), then there are two alternatives:

- either there is a unique EMM (market is then complete)

- or the set of absolutely continuous risk neutral measures is so big that it does not contain extreme points.  The latter means that there are many ways to model the prices.

The fact that there are many risk neutral measures gave rise to the study of special risk neutral measures (if they exist, see Delbaen Schachermayer for some pitfalls)

- variance minimal measure

- Foellmer — Schweizer measure

- entropy minimal measure

- and with some phantasy one can make the list infinitely long (if time permits).

By the way Girsanov showed the theorem but for Brownian Motion Maruyama did it before. It would be better to call it Girsanov-Maruyama.  There is also an extension due to Lenglart (for absolutely continuous changes).

Delbaen-Schachermayer gave counter examples where risk neutral evaluation for the final value of a stock can produce a lower price than todays market price. This does not permit arbitrage since going short on the stock would yield a non-admissible strategy.  In this case the stock price becomes a local martingale, not a martingale. We also see examples where using one risk neutral measure we reproduce the stock price but for another risk neutral measure we get a smaller value.  Meaning:  for one risk neutral measure the stock price is a martingale, for the other it is only a local martingale. We even see that such a phenomena is present in every incomplete market (where all martingales are continuous — to make it easier).  These strange things are sometimes overlooked.  D-S also gave a characterisation of the final values for which such phenomena does not exist (to be honest it is only sufficient as D-S analysis shows).  There are nice Banach spaces behind this analysis.

Does this help (or confuses)?