# 💎

## Derivation

We know that in Black Scholes a Call option has price

where

For an ATMF, at-the-money forward, option, $K=F=Se^{rt}$, thus

Now comes the approximation

We know $N^1(0)=\frac{1}{\sqrt{2\pi}}\approx0.4$, it’s time to calculate the below formula in your head

From Put-Call Parity, we have

## In Practice

To make your calculation even faster, here is the table of $\sqrt{t}$,

1m 2m 3m 6m 9m 1y
0.29 0.41 0.50 0.71 0.87 1.00

Forex options premium are often paid in asset currency, which means the formula can further be simplied to

Forex options prices are often quoted in percentage of asset currency, i.e.

where $\sigma_{(\%)}=\frac{\sigma}{100}$ is volatility in percentage.

## Exercise

Q: USDJPY 3m ATM vol is quoted at 7%, what is the premium?

A:

So the premium is 1.4% per USD, e.g., for a 3m Call option to buy 100 mio USD and sell JPY at ATM strike, the premium is roughly 1.4 mio USD. Nb we ignored the small difference between ATM and ATMF strikes.

Til next time,
Jianfeng at 22:34