Options have a Greek story. Of particular importance is the option delta. This week, we discuss the argument that the delta of an option is approximately the probability of the option ending in the money (ITM).

Model world

As with other financial models, the Black-Scholes-Merton (BSM) model makes assumptions that do not necessarily hold in the real-world markets. These assumptions are necessary to build a model that can provide a solution to determine an option value (the model is said to be tractable). The delta of an option is an output of the model and is, therefore, affected by these assumptions.

The BSM model has a term called N(d2) that denotes the probability of an option ending ITM in a risk-neutral world. Suffice it to understand that this is a world where traders are not anxious about an adverse movement in the underlying. That is, it does not matter if the underlying declines when you are long on a call option or if the underlying moves up when you are long on a put option because you can create a perfectly-hedged portfolio.

Then, there is another term called N(d1). This term is more like an adjustment factor to ensure that an option does not take a negative value. Options give you the right to buy or sell an underlying; a right cannot have a negative value. So, now we have N(d2) that is the probability of an option ending ITM using model parameters and N(d1) that is an adjustment factor. The delta of an option is N(d1). Hence, the argument that delta is only an approximation of an option ending ITM. That said, delta can be a useful statistic.

BSM model basics
Assumptions do not necessarily hold in the real-world markets
N(d2) is the probability of option ending ITM using model parameters
N(d1) is adjustment factor to ensure an option does not take negative value

Suppose the delta of an option is 0.60. It means that the probability of the option ending ITM is approximately 0.60. So, if you hold the option till expiry, there is a 60 per cent chance that the option will end ITM. This does not mean that you will generate profits at expiry; that is a function of the stock price, the strike price, and the premium you pay to buy the option.

Also read
Getty Images/iStockphoto

Mastering Derivatives: Call vs. Put set-ups for price reversals

Suppose you buy a near-month out-of-the-money (OTM) 500 strike call for 45 points. The option would be ITM at expiry if the stock closes at, say, 520. But you would have still lost 25 points on your trade. Why? At expiration, the time value of the option is zero. So, you lose 45 points. The intrinsic value of 20 points (520 less 500) is not enough to offset this loss of time value. You, therefore, need another factor to determine whether to set up the trade - by how much the option is likely to be ITM (moneyness of the option) at expiry.

Some traders use delta to arrive as an approximate probability of the option ending at a particular price. How? Suppose you have a view that the stock is likely to move to 600. You now want to find the probability that the stock will trade at 600 at option expiry. The argument is that you should check the delta of 600 call! If its delta is 0.25, the approximate probability you will make a 55-point profit is 0.25 (600 minus 500 minus 45). As you can see, this is an approximation.

Optional reading

Delta changes as the underlying changes. Even the adjustment of the gamma to the delta does not correct the approximation. True, you can determine the delta implied in the actual price of the option, just as you do for implied volatility. But implied delta is estimated using the BSM model. So, it is only as correct as the assumptions underlying the model. For instance, an assumption is normal distribution of asset returns that does not hold in real-world markets. So, you should be cautious in incorporating delta into your trading decision.

(The author offers training programs for individuals to manage their personal investments)

Related Topics
comment COMMENT NOW