The delta of an option is frequently considered to be the same as the probability that an option will be exercised, i.e., the probability that the option will be in the money at maturity. There is, however, a difference, especially when it comes to long-dated options on volatile stocks.

**A Simple Binomial Tree Example**

Assume the stock price today is USD100 and it will be either USD150 or USD50 when the European call option expires (a one-step binominal tree) with 50% probability respectively, that is:

If option strike price=100, then the probability to exercise =0.5 and

Delta= ((150-100)-0)/ (150-50) = 0.5.

If option strike price=120, then the probability to exercise =0.5 and

Delta= ((150-120)-0)/ (150-50) = 0.3.

If option strike price =149.99999, then the probability to exercise=0.5 and

Delta= ((150-149.99999)-0)/ (150-50) = 0 (approximately).

This simple example shows that delta and probability to exercises are different. More specifically, option delta--the hedging ratio--does not only care about the probability of the option ending up in the money but also how deep the option is in, or out, of the money, as the final payoff of the option depends on where spot is in relation to the strike.

**Black-Scholes Formula**

Option delta and the probability to exercise are also distinguished in the Black-Scholes formula. Recall that the pricing formula for a European call is:

Call option price c = S_{0*} N(d1) PV(K) *N(d2),

Where: S_{0} = Current Stock Price.

PV(K) = Present Value of Strike Price

N(d1) and N(d2) are cumulative probability distribution

functions for a normal distribution (i.e. it is the probability

that such a normal variable will be less than d1 or d2),

and d2= d1- _* (check)T (_ is the stock volatility and T is the

time to maturity).

By definition, we immediately have N(d1) as the option delta, representing the changing rate of the option price as a result of the stock price change. It can be further shown that N(d2) actually is the probability the option will be exercised. Since d1 is always larger than d2, it follows the cumulative probability function N(d1)--the option delta--should always be larger than N(d2)--the probability to exercise.

Further, because the difference between d1 and d2 is _* (check)T, such a difference will be more significant for long dated options (large T) on highly volatile (larger ?) equity stocks, hence the difference between the option delta and probability to exercise.

**Intuitive Explanation**

Instead of digging into mathematical calculations, we are able to derive an intuitive explanation from the Black-Scholes formula, if we just accept that N(d1) is the option delta and N(d2) is the probability to exercise.

Suppose for a moment, N(d1) is the same as N(d2), the exercise probability. Then, the value of a European call option would be:

Value = ( Probability of Exercise ) * ( S - PV(K) )

Now suppose the option is out-of-the-money, but there is still time until maturity. The present value, S-PV(K), of the expected exercise pay-off could well be negative, which would result in a negative option value. This cannot be correct, since an option can never have a negative value (due to limited liability/downside). Therefore N(d1) needs to be larger than N(d2) so the option value will not be negative when the stock price drops below the strike.

The reason for this is that the above formula acts as if it was known what the option holder will get when the option is exercised and the only uncertainty is if the option will be exercised. But this is clearly not the case: there is uncertainty about the stock price at maturity and the higher the volatility, the longer the option life, the more stock price uncertainty. Due to the limited liability/downside of an option higher uncertainty can only be beneficial, so that the value formula of the option needs to be adjusted to reflect this extra value, which is reflected in the excess of N(d1) over N(d2).

Technically, option pricing theory will attribute the difference between N(d1), the option delta and N(d2), the probability to exercise, to probability calculations in different "measures." Mathematics-oriented readers may consult **Tomas Bjork**'s *Arbitrage Theory in Continuous Time* or similar publications.

**Conclusions**

The delta of an option is not necessarily equal to the probability to exercise the option. In general, option delta is larger than the probability to exercise and the difference becomes more significant with respect to long dated options on volatile equity stocks. That is because delta incorporates not only the probability the option will be exercised but also the amount the option is in-the-money. The Black-Scholes formula also shows that delta has to be adjusted by more than the probability to exercise in order to reflect the option's value increase as a result of stock price uncertainties, and such adjustment becomes more significant for long dated options on volatile equity stocks.

*This week's Learning Curve was written by* *Winston Wenyan Ma**, president of the financial derivatives and risk management club at the* *University of Michigan Business School**and a former associate in the derivatives group at* *Davis Polk & Wardwell**in New York.*