The issue of jumps puts a different spin on the question. If there is a scheduled event with a jump distribution,then the P and Q distributions have to have the same support. Taking your example, suppose theP distribution will jump up or down but only to two possibilities, say up 5% or down 8%.Then, the Q distribution must also and *only* either jump up 5% or down 8%. The relative probabilities can be different under Q (as long as both are positive).If the Q-distribution allows *anything else*, there will be an arbitrage opportunity.That's because bets under the Q-distribution on *anything else* occurring will command a positive price under Q, but are impossible under P.In reality, there will be a continuum of outcomes. However, suppose we knew that the absolute value of the jump-return would be at least 2% under P.Then, the same property would have to hold under Q and so perhaps this is the way to, more realistically, make a bi-modal connection between the two distributions.
Last edited by Alan
on March 9th, 2016, 11:00 pm, edited 1 time in total.