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Calculating Probability of occurrence in options

Aesop

Joined
4/18/13
Messages
4
Points
11
As the title says I am requesting some help some guidance on how to go about calculating the probability of occurrence in option strikes. I am not as mathematically incline as most on this site but I understand there are different ways to go about this and for this reason I am stuck. Any advice or direction would be greatly appreciated. Thanks
 
Did I phrase this question wrong? Why am I not getting any responses?
 
From an intuitive and non rigorous way, you could think in this fashion:

Get the evolution of stock price as standard normal variable (N), ie. ln(St) is normally distributed, (St is a geometric brownian motion), reduce it to standard normal form: z= (ln(St)-E[ln(t)])/StdDev[St]. Now say for a digital call option, it would be exercised only if the stock price St >K (strike) or ln(St)>ln(K). Hence the probability of its exercise is the probability that z corresponding to a price St is > z corresponding to K ie. (say: x=lnK-E[ln(t)])/StdDev[St]. ) Since z is standard normally distributed, hence the probability is 1-N(x). Similary for digital put, the probability is z corresponding to St<x, hence N(x).

It is bit roughly presented but to have a better understanding, you might want to have a look at the derivation of black-scholes from the first principle.
 
You did phrase it wrong. I assume you mean getting implied probabilities from options.

A vertical spread is approximately proportional to the probability that the underlying will finish "above" the average of the strikes (sign adjusted). The closer the strikes the better the approximation. This is represents a CDF.

A butterfly is approximately proportional to the probability that the spot will finish "at" the middle strike. Again, the closer the strikes the better. This is represents a PDF.
 
Yike Lu said:
You did phrase it wrong. I assume you mean getting implied probabilities from options.

A vertical spread is approximately proportional to the probability that the underlying will finish "above" the average of the strikes (sign adjusted). The closer the strikes the better the approximation. This is represents a CDF.

A butterfly is approximately proportional to the probability that the spot will finish "at" the middle strike. Again, the closer the strikes the better. This is represents a PDF.
Click to expand...


What is the difference between implied probabilities and probability of occurrence? You are still trying to figure out the chance of an event occurring, right?
 
Faisal said:
From an intuitive and non rigorous way, you could think in this fashion:

Get the evolution of stock price as standard normal variable (N), ie. ln(St) is normally distributed, (St is a geometric brownian motion), reduce it to standard normal form: z= (ln(St)-E[ln(t)])/StdDev[St]. Now say for a digital call option, it would be exercised only if the stock price St >K (strike) or ln(St)>ln(K). Hence the probability of its exercise is the probability that z corresponding to a price St is > z corresponding to K ie. (say: x=lnK-E[ln(t)])/StdDev[St]. ) Since z is standard normally distributed, hence the probability is 1-N(x). Similary for digital put, the probability is z corresponding to St<x, hence N(x).

It is bit roughly presented but to have a better understanding, you might want to have a look at the derivation of black-scholes from the first principle.
Click to expand...


So I would have to look into a BSM?
 
The delta of an option with a given strike is (approximately) the probability of the underlying finishing beyond that strike level at expiration of the option. What that delta is, intuitively, depends on the shape and level of the implied volatility smile to that expiration date which itself implies a particular distribution function for the underlying's level at expiration.
 
if you are thinking about barrier options, you can take a look at "Brownian Bridge" which can be used to calculate the probability of the stock price cross a barrier(price level) in between two time points, given the value of the two time points and associated price levels.
 
Aesop said:
As the title says I am requesting some help some guidance on how to go about calculating the probability of occurrence in option strikes. I am not as mathematically incline as most on this site but I understand there are different ways to go about this and for this reason I am stuck. Any advice or direction would be greatly appreciated. Thanks
Click to expand...

everything you can extract from BS (i mean implied distrib) is not "real" distrib, but risk-neutral. work with log-normal process directly (see faisal post)
 
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