Using Binary Event Odds in Risk-Neutral Option Pricing
Summary
The document examines whether bookmaker odds can inform option prices when an underlying asset moves up or down in response to a binary event. The proposed setup combines a real-world probability for the positive event outcome with an assumed expected absolute move, while requiring the underlying price to have zero expected change in a one-step model.
A symmetric binomial example illustrates the constraint: if the only possible moves are equal-sized gains and losses, a martingale condition fixes their pricing probabilities at equal weights, regardless of the bookmaker's estimate. The discussion suggests that event odds cannot simply be inserted as risk-neutral probabilities while retaining the stated move sizes and martingale condition. It offers a conceptual example, not a full pricing framework; it does not resolve how richer payoff distributions, risk premia, or alternative market assumptions might incorporate event information.
Key ideas
- A proposed event-based model combines bookmaker odds, expected absolute move size, and a zero expected price change.
- In a symmetric two-outcome model with equal-sized moves, the martingale condition implies equal pricing probabilities.
- Real-world event odds cannot automatically be used as risk-neutral probabilities when the move sizes and martingale constraint are fixed.
- The example does not address richer distributions or risk premia.
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# Use of real-world probabilities in options pricing: binary event with continuous effect # 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) - $\Bbb P(\Delta X \geq 0) = 0.75$ - $\Bbb E|\Delta X| = 0.02$ - $\Bbb E(\Delta X) = 0$ 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?
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