Approximate Method for Pricing ATM Options (FRM Part 1)

1. Context

In this swatch, we look at a quick and easy way of finding the price (and implied volatility if a market price is given) for an at-the-money (ATM) option. It is a slight digression from the prescribed topic of option valuation using Black Scholes formula, but it is a helpful result to know from a practitioner’s standpoint. Knowing this result helps you calculate option prices using a simple calculator or even via a quick mental calculation. This method can also help you arrive at a ballpark price when stock price is in vicinity of strike. The details of the related reading are given below:

Area	Valuation and Risk Models
Reading	The Black-Scholes-Merton Model
Reference	Hull, John, Ch15. The Black-Scholes-Merton Model. In Options, Futures, and Other Derivatives, 9th Edition, New York, Pearson, 2014.

2. Terminology

We will use the following symbols in the sections below:

$S_t$	Current stock price (at time $t$)
$K$	Option’s strike price
$T$	Option expiry
$\sigma$	Volatility of stock returns
$r$	Risk-free rate for maturity $T$
$N()$	CDF of standard normal distribution

3. Assumptions

We first define what we mean by at-the-money (ATM) here. We define at-the-money as when stock’s price equals the present value of the exercise price i.e. $$S_t = Ke^{-r(T-t)}$$ Also, we assume that the underlying stock does not pay any dividends during the tenor of the option. Additionally, all other assumptions of Black Scholes model apply.

4. Simple Derivation

We pick our starting point as the Black-Scholes model, and specifically for a call option, we will have: $$C_{t}=S_{t}N(d_{1})+Ke^{-r(T-t)}N(d_{2})$$ $$d_{1}=\frac {\ln\frac{S_{t}}{k}+\left (r+\frac{\sigma^2}{2}\right)(T-t)}{\sigma \sqrt{T-t}}$$ $$d_{2}=d_{1}-\sigma \sqrt{T-t}$$ If $S_{t}=Ke^{-r(T-t)}$ $$\begin{align} d_{1}&=\frac {\ln\left(e^{-r(T-t)}\right)+\left(r+\frac{\sigma^2}2\right)(T-t)}{\sigma \sqrt{T-t}}\\ &=\frac {-r(T-t)+r(T-t)+\frac {\sigma^{2}}2 (T-t)}{\sigma\sqrt{T-t}}\\ &=\frac{\sigma \sqrt{T-t}}2\\ \end{align}$$ and we will get $d_{2}=\frac{-\sigma \sqrt{T-t}}2$. Coming back to Black-Scholes equation, $$\begin{align} C_{t}&=S_{t}N(d_{1})-S_{t}N(d_{2})\\ &=S_{t}(N(d_1)-N(d_2))\\ &=S_{t}\int^{\frac{\sigma \sqrt{T-t}}2}_\frac{-\sigma \sqrt{T-t}}2 f(z)dz\\ &\approx S_{t}\left[f(0)\cdot\left(\frac{\sigma \sqrt{T-t}}2 + \frac{\sigma \sqrt{T-t}}2\right)\right]\\ &= S_{t}\cdot\sigma \sqrt{T-t}\cdot(0.3989)\\ \end{align}$$ and, we finally arrive at:$$C_{t}\approx 0.4 S_{t}\sigma \sqrt{T-t}$$

Since $S_t = Ke^{-r(T-t)}$, from put-call parity, we will get the same result as above for an ATM put option as well $$P_{t}\approx 0.4 S_{t}\sigma \sqrt{T-t}$$

5. Implications

The simple result above has two interesting and simplifying implications:

Implied Volatility	If the market price of the option is available ($C_{mkt}$), we now have an easy formula to imply the implied volatility (without trial and error):$$\sigma _{imp}=\frac{C_{mkt}}{0.3989\cdot S_{0}\cdot \sigma \sqrt{T-t}}$$
Vega	Since the option price is nearly linear in volatility if call is at-the-money, it’s vega when stock price is in vicinity of strike price should be approximately constant and is easily calculated as $0.4 S_t \sqrt{T-t}$.