Term Structure of Volatility under Black-Scholes

In this research, we extracted a term structure from the implied volatility skewness under the Black-Scholes framework and S&P 500 adjusted close prices. With the daily close prices over the period of 1970-2018, we calibrated a local volatility function by fitting the BS-implied density based on the Dupire formula (which is the aggregation of a bunch of log-normal densities at each strike, since volatility is not constant and depended on strike) to the observed S&P 500 log return distribution.

We started by assuming that the local volatility is a function of the log-in-the-moneyness $x$ for a fixed maturity window of $T$ business days.

$$\sigma(x) = g(x) \\ \text{ where } x=log(K/S_0)$$

Under the Black-Scholes framework with risk-free rate $r=0$, a call option premium for $S_0=1$ is:

$$C(K, T) = \Phi(d1) - K\Phi(d2) \\ d_{\text{1, 2}} = \dfrac{log(1/K) \pm (\sigma^2/2)T}{\sigma\sqrt{T}}$$

If we assume that there exists a probability density function $f_{S_T}$, we have:

$$C(K, T) = \int^{\infty}_K(S_T-K)f_{S_T}(S_T)dS_T$$

Taking partial derivative w.r.t. $K$, we get the Dupire formula:

$$\dfrac{\partial C}{\partial K} = \int^{\infty}_K -f_{S_T}(S_T)dS_T \\ \dfrac{\partial^2 C}{\partial K^2} = f_{S_T}(K)$$

We can approximate the second-order derivative numerically:

$$f_{S_T}(K) \approx \dfrac{C(K + \Delta K) - 2C(K) + C(K - \Delta K)}{(\Delta K)^2}$$

We previously defined that $x = log(K)$. Given any $\Delta K$, let us define $\Delta x$ such that:

$$\Delta x = \dfrac{\Delta K}{K} \\ \text{therefore, } \Delta K = e^{x}\Delta x$$

We defined $r_T$ as the $T$ period log return, and that $r_T=log(S_T)$. Based on chain rule:

$$f_{r_T}(x) = f_{S_T}(K) \times K$$

So in conclusion we have:

$$f_{r_T}(x) = f_{S_T}(K) \times K \approx \dfrac{C(e^x + e^x\Delta x) - 2C(e^x) + C(e^x - e^x\Delta x)}{e^x(\Delta x)^2} \\ \text{where } C(x) = \Phi(\dfrac{-x + (\sigma(x)^2/2)T}{\sigma(x)\sqrt{T}}) - e^x\Phi(\dfrac{-x - (\sigma(x)^2/2)T}{\sigma(x)\sqrt{T}}) \text{, and } \sigma(x) = g(x)$$

The goal was to find the optimal volatility function $g(x)$ such that $f_{r_T}(x)$ fits to the historical distribution of $r_T$. Here we use a quadratic form for $g(x)$:

$$g(x) = ax^2 + bx + c$$

Using sum of square as the objective function, we obtained a set of coefficients of $a$, $b$, and $c$ that fit the Dupire density to the empirical density of S&P 500 log returns. This following graph shows the skewness coefficient $b$ plotted against term windows.

We observed that overall as the time-window increases, the skewness decreases (in absolute values), and that the skewness reaches a local minimum (in absolute values) at $T=100$, or 6 month time. Coincidentally, this is observed in futures trading - potentially explained by the fact that market tends to recover from the left-skewed losses in 6 month time on average.