Option deltas across strikes and time

Quantitative Finance Level 2

Which of the following options have the largest delta value when the stock is trading at 100?

Assume that interest rate is 0, and that there are no dividends

Call on the 120 strike with 1 month to expiry Call on the 120 strike with 3 months to expiry Call on the 80 strike with 3 months to expiry Call on the 80 strike with 1 month to expiry

1 solution

Chew-Seong Cheong
Mar 19, 2015

Delta of a call ranges from $0$ to $1$ , while delta of a put ranges from $-1$ to $0$ . Delta (absolute value) is larger when the option is further in the money and decreases as the option moves to at the money and out of the money. ITM delta increases while OTM delta decreases as the expiration date is getting nearer. The largest delta is when the option is $\boxed{\text{furthest in the money and nearest to expiration date.}}$ The following graph from www.TheOptionsGuide.com shows how delta varies from ITM to OTM and with time to expiration.

Is there a intuitive explanation for this?

Hint: Consider an alternative definition of delta.

Calvin Lin Staff - 6 years, 2 months ago

Okay.

Delta $\dfrac {\partial C}{\partial S} = N(d_1)$ , where $N(x)$ is the standard normal cumulative distribution function and $d_1 = \dfrac {1}{\sigma \sqrt{t}} \left[ \ln {\left(\frac{S}{K}\right)} + \left( r + \frac {\sigma^2}{2} \right) \right]$ . Therefore, delta is close to $0$ when deep OTM and far from expiry and $1$ when deep ITM and near expiry.

Chew-Seong Cheong - 6 years, 2 months ago

The alternative definition that I'm referring to is the dual delta, which is the "probability that the call will end up in the money". So, for a lower strike will have a higher delta. And if the call is in the money, then with a shorter time to expiry it would be more likely to end up in the money.

Calvin Lin Staff - 6 years, 2 months ago

Option deltas across strikes and time

1 solution

0 pending reports