Background

This research project was collaborated during my study at the University of Chicago, back in the spring of 2019. Reposting it here at the end of 2020 just for fun.

Content

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.

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

So in conclusion we have:

The goal 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 $g(x)$:

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.