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50-Delta Options

A

Adam

Member
In section 20.4 of Hull (9th global), it defines that "at-the-money option is a call option with a delta of 0.5". Using BS formula, we know that delta of ATM option is not exactly 0.5. So here can we say the definition is only approximate?
 
Xu Shi said:
In section 20.4 of Hull (9th global), it defines that "at-the-money option is a call option with a delta of 0.5". Using BS formula, we know that delta of ATM option is not exactly 0.5. So here can we say the definition is only approximate?
Click to expand...
The delta of a call option is given by: N(d1) where d1 = [log(S/K) + (r - q + 0.5*sigma^2) * (T-t)] / (sigma*(T-t)^0.5).
The notation follows the usual conventions [not reproduced here as I'm too lazy to write it down :)].

It follows that the delta = 0.5 iff d1 = 0. The result is exactly true 1) AT expiration when the (T-t) term is zero; and 2) the stock is trading at-the-money; i.e. S=K.

Otherwise, the said statement is only approximately true (where the stock trades ATM): it becomes an increasingly 'poor' approximation as time to expiration, implied volatility or interest rates increase.

Play around with an online option pricing calculator if you want to 'eye ball' the dynamics.
 
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