# Model to Predict the Change in IV of an Option

I am looking for a model that would allow me to predict the change in the Implied Volatility of an option based on a hypothetical change in the market. The goal is to create a better simulation of possible risks in an options based portfolio when it is exposed to market shocks.

When market conditions change, the price of an option adjusts to reflect the risk perceived by the market. This is observed as a change in calculated Implied Volatility.

A common phenomenon is that when the price of a stock suddenly drops the associated options see an increase in risk premium and the calculated Implied Volatility.

My goal is to have a model that allows me to quantify the range of potential changes based on a hypothetical change in market conditions.

I realize that the goal is difficult and there may be no perfect answer but I need to find something that at least allows me to start the modeling process.

Does anyone know of any research papers or work in this area?

One approach that I have seen being used is to try to model the (joint) dynamics of the forward at-the-money volatility as well as its first one or two derivatives. The idea is to find a parametrization for each of these quantities that you can easily estimate from historical data. You will generally find that the sensitivities themselves have a significant term-structure. Here are a few references to get you started.

Forward At-the-Money

In the Hagan et al. (2002) SABR model, the parameter $\beta$ controls the dynamics of the forward at-the-money implied volatility, the so-called "backbone". The corresponding implied volatility approximation is given by

\begin{equation} \sigma \left( F_0(T), T \right) = \frac{\alpha}{F_0(T)^{1 - \beta}} \left( 1 + \left( \frac{(1 - \beta)^2 \alpha^2}{24 F_0(T)^{2(1 - \beta)}} + \frac{\rho \beta \nu \alpha}{4 F_0(T)^{1 - \beta}} + \frac{\left( 2 - 3 \rho^2 \right) \nu^2}{24} \right) T \right) \end{equation}

Within this model, we can approximate a change in the forward at-the-money volatility as

\begin{equation} \sigma \left( F_0(T) + \Delta, T \right) \approx \sigma \left( F_0(T), T \right) \left( \frac{F_0(T) + \Delta}{F_0(T)} \right)^{\beta - 1}. \end{equation}

I.e. you can estimate the parameter $\beta$ by regressing the change in the logarithmic at-the-money volatility on the change in the logarithmic forward. As mentioned before, you'll need to allow $\beta$ to be time-to-maturity dependent in practice.

Derivatives

Instead of trying to model the dynamics of the at-the-money derivatives of the implied volatility smile directly, it makes sense to first introduce a normalization. Let for example

\begin{equation} x(K, T) = \frac{\ln \left( K / F_0(T) \right)}{\sigma \left( T, F_0(T) \right) \sqrt{T}} \end{equation}

be the number of at-the-money standard deviations that the strike $K$ is away from the forward of maturity $T$. Now compute the first few derivatives of the implied volatility smile as a function of this new moneyness measure. These normalized moneyness measures have a much milder term-structure and level-dependence. See for example Tompkins (2001) or Klassen (2016) who both use this normalization though in slightly different contexts.

References

Hagan, Patrick S., Deep Kumar, Andrew S. Lesniewski, and Diana E. Woodward (2002) "Managing Smile Risk", Wilmott Magazine, pp. 84-108

Klassen, Timothy R. (2016) "Equity Implied Vols for All, Part 2: Implied Volatility Curve Design and Fitting", Presentation, Volar Technologies

Tompkins, Robert G. (2001) "Implied Volatility Surfaces: Uncovering Regularities for Options on Financial Futures", European Journal of Finance, Vol. 7, No. 3

• Thank you, I have worked with the SABR model before and this looks good. As I dig into it I will give some feedback. Dec 22, 2016 at 17:47

I would say that Derman's 99 paper on "Regimes of Volatility" (also called volatility "stickiness assumptions" by some practitioners) is an excellent place to start your investigations.

Here is the paper and a lecture around it. Also see this related SE question.

You may be interested in these papers by Dumas et al. (1998) and Goncalves and Guidolin (2006). Here the abstracts:

Black and Scholes (1973) implied volatilities tend to be systematically related to the option’s exercise price and time to expiration. Derman and Kani (1994), Dupire (1994), and Rubinstein (1994) attribute this behavior to the fact that the Black/Scholes constant volatility assumption is violated in practice. These authors hypothesize that the volatility of the underlying asset’s return is a deterministic function of the asset price and time. Since the volatility function in their model has an arbitrary specification, the deterministic volatility (DV) option valuation model has the potential of fitting the observed cross-section of option prices exactly. Using a sample of S&P 500 index options during the period June 1988 and December 1993, we attempt to evaluate the economic significance of the implied volatility function by examining the predictive and hedging performance of the DV option valuation model.

One key stylized fact in the empirical option pricing literature is the existence of an implied volatility surface (IVS). The usual approach consists of fitting a linear model linking the implied volatility to the time to maturity and the moneyness, for each cross section of options data. However, recent empirical evidence suggests that the parameters characterizing the IVS change over time. In this paper, we study whether the resulting predictability patterns in the IVS coefficients may be exploited in practice. We propose a two-stage approach to modeling and forecasting the S&P 500 index options IVS. In the first stage, we model the surface along the cross-sectional moneyness and time-to-maturity dimensions, similarly to Dumas, et. al., (1998). In the second-stage, we model the dynamics of the cross-sectional first-stage implied volatility surface coefficients by means of vector autoregression models. We find that not only the S&P 500 implied volatility surface can be successfully modeled, but also that its movements over time are highly predictable in a statistical sense. We then examine the economic significance of this statistical predictability with mixed findings. Whereas profitable delta-hedged positions can be set up that exploit the dynamics captured by the model under moderate transaction costs and when trading rules are selective in terms of expected gains from the trades, most of this profitability disappears when we increase the level of transaction costs and trade multiple contracts off wide segments of the IVS. This suggests that predictability of the time-varying S&P 500 implied volatility surface may be not inconsistent with market efficiency.

Pay attention that both paper “forget” to take into account the possibility of arbitrage in their estimation, so to do something meaningful you should implement some no arbitrage constraints while performing the regressions.

• Thank you. This also looks like a solid direction. As I dig through this I will give feedback on what I find. Dec 22, 2016 at 17:50