## Saturday, 5 January 2013

### The possibility effect and the certainty effect redraw the expected utility curve

The expected utility of an outcome or a collection of outcomes is the various utilities (or values, often financial values, that is, cash amounts) multiplied by their probability of happening.  (Assuming that such a probability was stable and estimable in the first place).  So for any binary bet where the payoffs are $P_0$ and $P_1$ respectively you can work out the expected utility for for any valid probability $p$ that $P_1$ happens.  As $p$ ranges from 0 to 1 you ought to see the expected utility trace out a straight line between $P_0$, when $p=0$ up to $P_1$ when the outcome is certain.

With call and put options, the delta tells you the risk neutral probability that the instrument will be exercised.  So following Kahneman's observation that we tend to operate not by expected utility (which would be the line y=x in the above chart)but by over-valuing low probability events and under-valuing nearly certain events, maybe there's an inherent tendency for the very out of the money call to be too valuable in the marketplace, which would mean some strategy like covered calls (where you're selling that over-priced call) would have an edge.  I got the decision weights above from a table in Kahneman's book.

Conversely a low delta put is also a way out of the money put so buying protection against tail risk is expensive not only for local supply and demand reasons.

However, against all that is the thought that this is fundamentally how humans set decision weights, so don't expect them to 'converge' to expected utility.  In other words the market price isn't over-valued as such from a human valuation perspective, only with respect to a theory of expected utility.  Perhaps this effect will get smaller with the rise of algorithms making the trades, on the assumption that they'll be programmed to be more like utility expectation machines.

This raises an interesting question - on average is the implied volatility of a way-OTM call expected to 'converge down' to the implied volatility which would be in play assuming the market valued the call on an expected utility basis?  Or is it the expected utility valuation which needs to move closer towards the possibility-effect inspired market price?

By the way this point is totally separate from the fact that the Black-Scholes model itself introduces biases in calculating deep out of the money options, especially under conditions of high volatility.  That is an issue or weakness in a model when you compare the model's prediction of the fair value of the volatile OTM option compared to where the market is on it at that point in time.   One is a weakness of Black-Scholes, the other is possibly a weakness of the more fundamental expected utility approach.  The Black-Scholes weakness is related to the assumptions of lognormality assumed in the stock process, whereas the expected utility weakness, if it is indeed a weakness, is that it doesn't model a fairly stable human behavioural bias, namely the possibility effect and the certainty effect.