Calculate uncertainty of option expiring ITM

Question

I know it is pretty straightforward to determine the probability that an option will expire OTM -- basically a 0.10 delta call will have a 10% probability of being ITM at expiration (see this question). I'm having a bit of trouble formulating my question, but I think it comes down to how much confidence can I have in that (in this case 10%) probability?

How could I calculate the uncertainty of that probability? For example, an option with a very large gamma may have a 10% probability that it will expire OTM, yet have large uncertainty in that 10% estimate since delta will change quickly with small changes in the underlying.

Edit trying to add some clarification:

As Canardini says, I think I may be forgetting the assumptions of the underlying model or not even really understanding what model I'm using. And as Alex C states, what I'm probably interested in is how sensitive the 10% estimate is to changes in either time or the underlying.

Some thoughts I'm trying to wrap my head around:

• $\Delta$ is approximately the CDF for Black-Scholes
• I think I'm interested in the CDF's sensitivity to price changes or time changes
• We can infer the market assigned PDF from $\frac{\partial^2 C}{\partial K^2}$
• The slope of the CDF equals the PDF

Is the curvature $\frac{\partial^2 C}{\partial K^2}$ telling me the sensitivity of the CDF to price changes? This is sort of along the lines of my original intuition that a higher gamma increases the sensitivity of the 10% probability estimate.

Answer 1

From the moment you are talking about probability, you define a measure, and a model. Usually , when you hear Delta 10, it means that under the risk-neutral measure, and very likely, the Black model, you have a probability of 10 % to exercise, everything is incorporated in that 10%, whether gamma or vega is high or not. This number is exact, it is not a random variable.

Now , if you believe in this model, you could create a process $X(t,S_t)$ $$X(t,x)=\mathbb{P}(S_T>K|S_t=x)$$, this variable may help you.

Answer 2

I agree with Canardini that the probability of 0.10 is exact, it does not have an uncertainty around it.

If you are interested in how this probability will evolve in the next day, you could calculate two stock prices for tomorrow $S^+$ assuming a rise by 2 standard deviations of daily return from today's price and $S^-$ assuming a drop in price by 2 standard deviations. Then you can calculate tomorrows probability in these two cases and you will have a kind of confidence intervals for tomorrow's probability estimate.

Answer 3

I think the question you are asking how certain you can be that 10% is the right probability is really how accurate is your model, and how can you attest to its accuracy.

For example: A man will toss a coin. What is the probability that it comes up heads?

You will say 50%, and this probability is not a random variable as @canardini states. But how do you know 50% is an accurate model. You can test it against 10,000 of his throws and see if close to 5000 came up heads.

As an example of when your model might be wrong

A man will roll a die. What is the probability of rolling a one?

You say 1/6th, but what if after 60,000 rolls you have 20,000 ones? Then you conclude your model is not very accurate and the die must be loaded. You adapt your model and hope for better accuracy.

This is why the volatility smile was introduced to option pricing to account for an underlying probability distribution of pricing moves that was not (log) normal.

The only way to my mind to test your model is to use historical data, although I may be wrong.