# Duration of option

can anyone please confirm the formula for the duration of an option

thanks.

Yup - it’s the duration of the underlying*delta of the option*(price of underlying/price of option)

the last term represents the leverage of using the lower prices option to gain exposure to the higher priced underlying security. Note puts will have negative duration and calls will have positive…tho this is accounted for in the option’s delta

Is this really in the 2014 curriculum? Don’t remember it being referred to as such

Delta x duration of underlying x price of underlying / price of option

There’s also the hedge ratio = (Price of hedge x duration of hedge / price of ctd x duration of ctd ) x beta x conversion factor.

This is standard formula for changing duration of portfolio. In this case target duration = 0.

so for a ATM call on 2 year t-bond the duration would be approx (assuming price of 1 bond is 100,000USD and price of 1 option is 20dollars):

0.5 * (2*0.75) * 100,000 / 20 = 3750 (doesn’t that seem a lot to you guys ? or my numbers are too irrealistic ? or is there a quantity factor somewhere missing here ?)

isn’t the price of 1 bond always going to be really larger than the price of 1 option ?

thanks guys

If the bond expired up 1% 101,000. You return on investment would be 1000/20 = 5000%. Seem reasonable. In the money at expiration the delta of the call is 1. So it’s 1 * 10000/20 = 5000 MD, which is correct.

thanks a lot.