The failure of the Black-Scholes model to fit markets in vanilla options has led to the specification of an implied volatility for each option. This, in turn, has led to the creation of implied volatility surfaces – interpolations and extrapolations of volatilities across all strikes and expiries.
However, even these complex data structures have not been enough to describe the variation of market prices and practitioners have been required to develop dynamics for these surfaces, i.e., a description of how the implied volatilities move as a function of the state variables of the market (see references  and  in the full paper at the bottom of this page). Most prominent has been the dynamics of the implied volatilities as a function of the underlying asset price – the so-called vol/spot dynamics (see e.g. ).
In this empirical analysis, we will consider the realized variation in Equity implied volatilities between one trading day and the next and see how well the most common vol/spot dynamics describe these moves. We will consider these dynamics across a range of major Equity indices both in normal and in volatile markets. Specifically, we will focus on the daily variation in implied volatilities in two periods – from 1st Nov 2019 to 25th Feb 2020 and from 26th Feb 2020 to 6th July 2020. The first period represents a period of well balanced markets and relatively low realized volatility (“Low Vol”). The second period represents a period of high realized volatility (“High Vol”).
Within Equity derivatives, there has been a long reliance on Sticky Strike dynamics. This analysis aims to test how well served we are by this old friend and whether two other dynamics, Sticky Delta and Sticky Surface, might be more appropriate. We will also examine the variation in our dynamics across the surface – by strike and by
maturity. We will consider whether the performance of a dynamic is influenced by the direction of the market moves or their size.
Performance will be measured in two ways:
• Measure #1: The ability to minimize the variance of P&L due to movements in the implied volatility surface. This approach could be viewed as most appropriate for traders who wish to minimize overall risk.
• Measure #2: The ability to make our volatility P&L independent of movements in the underlying asset. Our dynamic may not then necessarily minimize the variance of P&L. This approach could be viewed as most appropriate for traders who wish to take on a specific risk (e.g. vega) but want to be insulated from others (e.g. spot).
In the following sections, we present our analysis on the three vol/spot dynamics. From the details given in Sections 3 to 7, we make the following conclusions:
• Sticky Surface gives the best performance across almost all indices for Short (1w to 3m) and Medium (6m to 2y) maturities both in the Low and High Vol periods.
This is true using either measures of performance. Sticky Strike comes a close second – typically within the standard deviation of the sample means. This is independent of the size of the absolute spot move considered or its direction.
• For non-Asia/Pacific (APAC) indices, Sticky Delta proves a poor choice of dynamic even for Long (3y and beyond) maturities.
• For APAC indices, Sticky Delta is the best performer for Long maturities but Sticky Surface comes a close second – within the standard deviation of the sample means. However, even for the Low Vol period, P&L variance reduction is poor for all three dynamics.
• All three vol/spot dynamics do a poor job of minimizing P&L variation across the High Vol period. Performance on reducing P&L / spot correlation is reasonable on some assets (e.g. Nasdaq 100) but is generally worse than performance across the Low Vol period.
• Filtering by direction of movement does yield some interesting results on APAC indices but not on others. Filtering by size of move does not yield any obvious conclusions.
Overall, we see the need both for dynamics to be specific to a given underlying and to be a function of maturity. For non-APAC underlyings, we would recommend Sticky Surface vol/spot dynamics for all maturities. For APAC underlyings, we would recommend using Sticky Surface vol/spot dynamics for maturities up to 5y and Sticky Delta for longer. However there is clearly scope for a new vol/spot dynamic for APAC indices if the goal is to minimize P&L variance successfully. In periods of high volatility and high vol-of-vol, vol/spot dynamics are only one small part of hedging to minimize P&L variance and need to be supplemented with option hedges.
Our choice of vol/spot dynamic can be incorporated into our risk management in three main ways. Firstly, we can incorporate it into our scenarios to give more realistic estimations of our market data under perturbation of our state variables. Secondly, we can incorporate it into our Greeks. Doing this, we need to be comfortable with hedging volatility risk using stock/index futures. Thirdly, we can incorporate it into our P&L Explain to decide on how we partition our P&L between spot and vol factors. These decisions have a large subjective component to them but it is hoped that this analysis provides tools with which to assess existing and new vol/spot dynamics and show where maturity-based and regional flexibility is required.