Understanding the Implied Volatility Surface

Implied volatity is a $3$-dimensional representation of the implied volatilities of options on any given underlying asset, plotted as a function, $\mathcal{F}(\cdot) = \sigma_{\text{imp}}$ of time to maturity $\tau$ and moneyness $M$ where,

$$M_{\text{call}} = \frac{S}{K} \hspace{3mm} \text{ if call} \\[3mm] M_{\text{put}} = \frac{K}{S} \hspace{3mm} \text{ if put} \\[3mm]$$

where $S$ is the current price of the asset.

Assume we have a call at $K = X_c$, meaning we have the right to buy at any point before the time at which the contract expires and the option has a value of $S > (K = X)$, then $M > 1$.

Now assume we have the same scenario but $S < (K = X)$, then $M < 1$.

Now assume we have a put at $K = X_p$, meaning we have the right to sell at any point prior to the expiration of a contract.

If $S > K$, then $M < 1$, but if $K > S$ then $M > 1$.

Then ultimately, the formulation for moneyness $M$ becomes the delimiter for knowing the value of the option (whether call or put) at a given time $t < T$, where if $M > 1$, we're in the money, else our option has no profitable value.

Aside from $M$, the $\text{IVS}$ is a function of a second variable $\tau$ the time to maturity. If $t$ is current time, $T$ is time at maturity, $\tau = T - t$ is time to maturity.

To compute the implied volatility at $\tau$, we need a couple of inputs to our function $\mathcal{F}(\cdot)$.

Market option price, $C_{\text{intrinsic}} = \max(S - K, 0)$ if call or $P_{\text{intrinsic}} = \max(K - S, 0)$ if put, the underlying price $S$, the strike price $K$, time to maturity $\tau$, risk free rate $r$.

The Black-Scholes model provides the theoretical price for an option (European, as it only accounts for the price at $T$) as,

$$C_{\text{BS}}(S, K, \tau, r, \sigma) = SN(d_1) - Ke^{-r\tau}N(d_2) \hspace{3mm} | \hspace{3mm} \text{Theoretical Call Price} \\[3mm] P_{\text{BS}}(S, K, \tau, r, \sigma) = Ke^{(-r(\tau))}N(- d_2) - SN(-d_1) \hspace{3mm} | \hspace{3mm} \text{Theoretical Put Price}$$

where $N(d_i)$ is the CDF of the normal distribution function of the standard normal distribution at $d_i$ (or equivanlently the PDF integrated up to $d_i$) $\int_{-\infty}^{d_i} \frac{1}{\sqrt{2\pi}}e^{\frac{-z^2}{2}}$.

$d_1$ and $d_2$ are defined as:

$$d_1 = \frac{\ln(M_{\text{call}}) + (\frac{r - q + \sigma^2}{2})(\tau)}{\sigma \sqrt{\tau}} \\[3mm] d_2 = d_1 - \sigma \sqrt{\tau}$$

The goal is to find $\sigma_{\text{imp}}$ by solving for $C_{\text{BS}} = C_{\text{market}}$ or $P_{\text{BS}} = P_{\text{market}}$.

You can find $\sigma_{\text{imp}}$ via the Newton-Raphson method, which essentially tries to find the roots of a function such that $f(\cdot) = 0$

Given that we're trying to find a $C_{\text{BS}} = C_{\text{market}}$, we can construct a function as,

$$f(\sigma) = C_{\text{BS}}(S, K, \tau, r, \sigma) - C_{\text{market}}$$

with the goal of finding $\sigma$ such that $f(\sigma) = 0$

Newton-Raphson method iteratively adjusts $\sigma$ until $f(\sigma_n) = 0$ as,

$$\sigma_{n + 1} = \sigma - \frac{f(\sigma_n)}{f'(\sigma_n)}$$

where $f'(\sigma_n)$ is simply the partial derivative of $f(\sigma_n)$ with respect to $\sigma_n$,

$$f'(\sigma_n) = \frac{\partial C_{\text{BS}}}{\partial \sigma_n} = S\sqrt{\tau} \cdot \frac{∂N(d_1)}{∂\sigma}$$

defining the infinitesimal rate of change in the price of the call with respect to the volatility.

Once you computed $\sigma_n$, you simply construct the $3$-dimensional plot of the function by plotting $\sigma_n$, $M$, and $\tau$,