# Use of real-world probabilities in options pricing: binary event with continuous effect

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)

1. $\Bbb P(\Delta X \geq 0) = 0.75$
2. $\Bbb E|\Delta X| = 0.02$
3. $\Bbb E(\Delta X) = 0$

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?

• Could you clarify the difference between "the expected absolute move" and "the expected move", with respect to points 3 and 4. – oliversm Jun 10 '16 at 8:29
• @oliversm: clarified, also renumerated - now it's 2 and 3. The previous 2nd condition $\Bbb P(\Delta X < 0) = 0.25$ was obsolete. – Ulysses Jun 10 '16 at 8:39
• You seem to refer to a "classical pricing framework". Classically, options are priced in a risk-neutral world, where the whole idea is that we don't care about historical probabilities (here your bookies' odds, or the up/down probabilities in a binomial tree). If you choose to care about real probabilities, then you should also discount depending on your risk aversion. Maybe this discussion can help: quant.stackexchange.com/questions/8274/…? – Quantuple Jun 10 '16 at 8:52
• Also remember that it's not really $X$ which is a martingale but (1) it's discounted price, which assumes the existence of a risk-free asset and (2) this is only true if you assume that investing in $X$ is a self-financing strategy, which is not the case for stocks/indices paying dividends. – Quantuple Jun 10 '16 at 8:57
• @Quantuple: for simplicity I assume the interest rate/cost-of-carry to be $0$ - as I said, it is a very short time horizon, and these things don't matter there even if they are present. Regarding the use of real probabilities: the risk-neutral condition 3. is always in place, but you can also calibrate your model to include other information: market sentiment, probabilities of some events etc. as long as they go orthogonal to risk-neutrality. – Ulysses Jun 10 '16 at 13:21