In this research, we aim to extract 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 calibrate 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 start by asssuming that the local volatility as a function of the log-in-the-moneyness $x$ for a fixed maturity window of $T$ business days.

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

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

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

We can approximate the second-order derivative numerically:

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

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

So in conclusion we have:

The goal is 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 $g(x)$:

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 can observe that overall as the time-window increases, the skewness decreases (in absolute values). We also see that the skewness hit a local minimum (in absolute values) at $T=100$, or 6 month time. This phenomenon is also observed in future trading, and could potentially be explained by the fact that market tends to recover from the left-skewed losses in 6 month time on average.

Click on this link to see the jupyter notebook.