In a recent training session we were confronted with an apparently tricky issue regarding the delta of an option. We were demonstrating to a group of graduate trainees in a bank how to use a simple Cox-Ross-Rubenstein tree for pricing vanilla options. However, before we could talk about CRR tree we wanted to demonstrate a simple equi-probable tree (50% probability of asset going up and down) to price a call option and explain the concepts of option sensitivity. We shouldn't have taken this approach but we did and right away we ran into a conceptual problem. The issue was with equating delta of an option with that of the probability of exercise, i.e. the probability that the stock will finish in the money.

Many traders, especially the old school types, who learnt the business on the floor and not through a Ph.D. degree, had to go through a steep learning curve to figure out what delta is. On the first day of the job there was a tendency of the traders to equate the delta of an option to the probability of the exercise. And it usually stuck for a long time until he got some experience in exotic options like binaries or if the volatility caused a blow up.

These days we learn about options through Financial Engineering degrees and text books (though some great textbooks were there in those days as well, such as the book by Rubenstein). And to tell you the truth, the present day approach of structured learning by first understanding of the math behind option pricing is very helpful even if you are contemplating spending the rest of your life as a trader.

So let's see how we landed up in trouble. Call it a slip of tongue, call it stupidity, call it lack of experience or sufficient knowledge, one of us in a grand gesture declared that the delta of a vanilla option, though not equal but is very close to the probability of exercise. And perhaps in his enthusiasm for all the math that lies beneath he went to tell to the starry eyed (from previous nights party), bored and disinterested graduates that delta is equal to the integral of the payoff between strike price and infinity for a call under a risk neutral measure and all that stuff. So there was this five minute interlude - an unnecessary digression, since we were actually demonstrating a simple binomial tree - of "math stuff".

This was pure disaster. One graduate trainee raised his hand and said: "you are totally wrong, delta is never the probability of exercise or even close to it and you don't need integral calculus to show it. I can show it using a simple decision tree". Check and mate! The graduate trainee deserves to get a huge signing bonus and we deserve to go back to farming, or writing stupid stories on the web which is what we do best.

Here was his analysis (which should have been our analysis). Let's say that a stock is trading at $10 and there is a 50% probability that in a year's time it will go up to $15 and 50% probability that it will go down to $5. Now let us assume that the strike price of a call option is $10 (at the money). A 50% up move means a 50% probability of exercise. If this is the case then the first derivative of the call option with respect to the underlying, the delta, will be given by:

So far so good! The delta of the option is equal to the probability of exercise which is also 50%. Therefore, can we say that for at the money options the delta of an option is always equal to the probability of exercise? The answer is NO. First, note that the probability of exercise which is 50% is not the risk neutral probability but a real probability measure (which is not the correct valuation measure for derivatives). Secondly, the volatility impact is being neglected in the above simplistic model (which we will show in the Part II of this article).

Now let's see what happens if the strike price is different from today's spot price. Say the strike price is $12. Then the value of delta will be:

This shows that for the above out of the money option the delta of the option is different from the probability of exercise. Though the probability of exercise is 50% the delta of the option is 30%. Actually, this is a very fundamental and moot point, something we often fail to capture in our learning process. The delta of an option, not matter the strike, should always be greater than the probability of exercise according to the Black-Scholes formula. That is being contradicted above. This goes to show that an equi-probable tree, with 50% probability of up and down move, is not the right approach to model derivatives and rightly so, all pricing is done under risk neutral probability measure (and not real probability measure).

If we further change the strike price to say $14 which is deep out of the money we will see using the above formula from the tree that the delta of the call will be 15% far away from the probability of exercise of 50%.

In the second part of this article we will talk a bit more about this issue and tackle it using the Black-Scholes formulation. However we conclude the following:

- The delta of an option is not equal to and may never be even close to the probability of exercise of the option; in fact, as we will show in part II, delta of an option is always greater than the probability of exercise;

- Equi-probable tree approach is a tricky one and one can run into problems, which in the case of interest rates modelling become a total disaster; therefore, it is always better to understand binomial trees by starting off with the Cox-Ross-Rubenstein tree using risk neutral probabilities.

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