The delta of the option is the amount of equity market exposure an option has. As a stock price falls by the dividend amount on its ex-date, delta is equal to the exposure to dividends that go ex before expiry. The dividend risk is equal to the negative of the delta.

For example, if you have a call of positive delta, if (expected or actual) dividends rise, the call is worth less(as the stock falls by the dividend amount).If a dividend is substantial, it could be in an investor’s interest to exercise early.

A digital call option is an option that pays 100% if spot expires above the strike price (a digital put pays 100% if spot is below the strike price). The probability of such an option expiring ITM is equal to its delta, as the payoff only depends on it being ITM or not (the size of the payment does not change with how much ITM spot is).

For a vanilla option this is not the case; hence, there is a difference between the delta and the probability of being ITM. This difference is typically small unless the maturity of the option is very long.

While a call can have an infinite payoff, a put’s maximum value is the strike (as spot cannot go below zero). The delta hedge for the option has to take this into account, so a call delta must be greater than the probability of being ITM. Similarly, the absolute value (as put deltas are negative) of the put delta must be less than the probability of expiring ITM. A more mathematical explanation (for European options) is given below:

The difference between delta and probability being ITM at expiry is greatest for long-dated options with high volatility (as the difference between d1 and d2 is greatest for them).

For example, if you have a call of positive delta, if (expected or actual) dividends rise, the call is worth less(as the stock falls by the dividend amount).If a dividend is substantial, it could be in an investor’s interest to exercise early.

**Delta is not the probability option expires ITM**A digital call option is an option that pays 100% if spot expires above the strike price (a digital put pays 100% if spot is below the strike price). The probability of such an option expiring ITM is equal to its delta, as the payoff only depends on it being ITM or not (the size of the payment does not change with how much ITM spot is).

For a vanilla option this is not the case; hence, there is a difference between the delta and the probability of being ITM. This difference is typically small unless the maturity of the option is very long.

**Delta takes into account the amount an option can be ITM**While a call can have an infinite payoff, a put’s maximum value is the strike (as spot cannot go below zero). The delta hedge for the option has to take this into account, so a call delta must be greater than the probability of being ITM. Similarly, the absolute value (as put deltas are negative) of the put delta must be less than the probability of expiring ITM. A more mathematical explanation (for European options) is given below:

**Call delta > Probability call ends up ITM**

Abs (Put delta < Probability put ends up ITMAbs (Put delta < Probability put ends up ITM

- Mathematical proof option delta is different from probability of being ITM at expiry:
- Call delta = N(d1) Put delta = N(d1) - 1

Call probability ITM = N(d2) Put probability ITM = 1 - N(d2)

where:

Definition of d1 is the standard Black-Scholes formula for d1. For more details, see the

section A.7 Black-Scholes Formula.

d2 = d1 - σ T

σ = implied volatility

T = time to expiry

N(z) = cumulative normal distribution

The difference between delta and probability being ITM at expiry is greatest for long-dated options with high volatility (as the difference between d1 and d2 is greatest for them).

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