Calculating Option Price in Your Head
07 Dec 2018

TL;DR

$C = P = 0.4 \times S \times \sigma\sqrt{t}$

Derivation

We know that in Black Scholes a Call option has price

$C = SN(d_1) - K e^{-rt} N(d_2)$

where

$d_{1,2} = \frac{\log\frac{S}{K} + (r \pm \frac{1}{2}\sigma^2)t }{\sigma\sqrt{t}}$

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

$d_{1,2} = \pm \frac{1}{2} \sigma\sqrt{t} \\ C = S\big[ N(\frac{1}{2} \sigma\sqrt{t} ) - N(- \frac{1}{2} \sigma\sqrt{t} ) \big]$

Now comes the approximation

$N(x) \approx N(0) + N^{(1)}(0)x + \frac{1}{2}N^{(2)}(0)x^2 \\ N(x) - N(-x) \approx 2xN^{(1)}(0)$

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

$C = 0.4 \times S \times \sigma\sqrt{t}$

From Put-Call Parity, we have

$P = C - (F-K) = C = 0.4 \times S \times \sigma\sqrt{t}$

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

$C = P = 0.4 \times \sigma \sqrt{t}$

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

$C_{\%} = P_{\%} = 0.4 \times \sigma_{(\%)} \sqrt{t}$

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

Exercise

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

$\begin{align} C &\approx 0.4 \times \sigma_{(\%)} \sqrt(t) \\ &= 0.4 \times 7 \times 0.5 \\ &= 1.4\% \end{align}$

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

Calculating Option Price in Your Head 07 Dec 2018

TL;DR

Derivation

In Practice

Exercise

Calculating Option Price in Your Head
07 Dec 2018