Let's say I have to price options on instrument X with a multitude of strikes. For simplicity, assume that X only makes one move during the options' lifetime, and this move is affected by some binary "yes"/"no" event - e.g. an option on the 30 minutes SP500 move after the primaries results for Clinton vs. Sanders are made public (in time when there was still some intrigue there). Assume also, that in case of "yes" X will go up and in case of "no" it will go down.
To improve this pricing w.r.t. market, a colleague of mine suggested the following idea. He goes to a bookies website, and sees their odds. For example, they suggest that yes is 75% probable. He also has his estimate of the standard deviation of the move, e.g. he expects an absolute move of 2%. He then uses the bookies data to scale the moves so that: (the $\Delta X$ here is the move)
- $\Bbb P(\Delta X \geq 0) = 0.75$
- $\Bbb E|\Delta X| = 0.02$
- $\Bbb E(\Delta X) = 0$
He claims that the 3rd condition makes it legit to incorporate the bookies information into pricing, since he makes the X be a martingale in this 1-step model. I agree with the latter fact, but I don't think he can incorporate this bookies information there at all, at least in the classical pricing framework.
For example, let's consider the case when X follows the binomial model, and only makes moves of 2% up or down (so that the 2nd condition is satisfied). No matter what bookies website would suggest, we would still use 50/50 probabilities for the moves here (to satisfy the 3rd condition). Of course, we don't have enough flexibility to change probabilities after we assured 2nd and 3rd conditions, but I think the underlying point here is: this bookies information can (and should) only be used to estimate the mean of the move, not relative probabilities, and in the pricing world the mean is fixed. Am I right?