Using a Bull Spread to Express a Terminal Probability View
Summary
The document asks how to trade a forecast that the underlying will finish above a chosen strike more often than option prices imply. It distinguishes this terminal probability view from a general volatility view and notes that buying an out-of-the-money call may have a low chance of profit even when the forecast is right. The proposed approach uses the Breeden-Litzenberger relationship between call prices and the risk-neutral distribution. A finite difference of call prices around a strike approximates the cumulative probability of finishing below that strike, and the difference corresponds to a static bull spread position.
This gives a way to obtain direct exposure to a probability measure through option prices rather than relying on a single far out-of-the-money call. The discussion is theoretical and does not provide market data, transaction costs, or a complete trade construction for a physical probability forecast. It also cautions that option prices encode risk-neutral probabilities, which may differ from an investor's real-world probabilities.
Key ideas
- Option prices encode the risk-neutral distribution rather than the physical distribution.
- The Breeden-Litzenberger relationship links strike derivatives of call prices to terminal probabilities.
- A finite difference of call prices around a strike approximates a cumulative probability.
- A bull spread can provide exposure to that cumulative probability estimate.
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Full text
# Best way to trade probability density
# Best way to trade probability density
From the option chain of a security, we can calculate the implied probability density at the maturity $T$ (assume the options are European. Now suppose we have our own view/prediction on the probability density of the underlying price at time $T$. What is the best way to trade this view?
For example, say the implied probability of the underlying will end above $150$ is $5\%$ and our own view of that is $10\%$. We would think that the upside calls are underpriced. However, if we express our view by simply buying an upside call, the chance for us to have profit is still very small even if our view is totally correct. Trading volatility does not seem to be quite relevant also since the view is about the terminal probability distribution.
To be more specific, in the situation described above, what are the good ways trading the view? Buying a 150 call only gives you a $10\%$ chance of winning even if our view is $100\%$ correct. Are there any other ways so that the winning probability is higher but still give us a positive expectation?
## Answer by Daneel Olivaw (score 1)
https://quant.stackexchange.com/a/61622
Note that option prices contain distributional information about the risk-neutral measure $\mathcal{Q}$, not the physical one.
That being said, per the well-known Breeden-Litzenberger formula, the undiscounted price $\widehat{C}$ of a European call option on a security $S$ with strike $K$ and expiry $T$ is: $$\widehat{C}(t,K,T)=\int_K^\infty(S_T-K)\text{d}\mathcal{Q}(S_T)$$ Differentiating with respect to $K$: $$\frac{\partial \widehat{C}}{\partial K}(t,K,T)=-\int_K^\infty \text{d}\mathcal{Q}(S_T)$$ That is: $$\mathcal{Q}(S_T\leq K)=1+ \frac{\partial \widehat{C}}{\partial K}(t,K,T) $$ Approximating by finite central differences: $$\mathcal{Q}(S_T\leq K)\approx 1+ \frac{\widehat{C}(t,K+\delta,T)-\widehat{C}(t,K-\delta,T)}{2\delta} $$ The second term is a static position in a bull spread, which give you direct exposure to the cumulative probability density.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.