Restitution of the QuantMinds event

Restitution of the QuantMinds event

Restitution of the QuantMinds event 640 480 Quanteam

This article presents a summary of one of the topics discussed at Quantminds 2023: ATM volatility skew and rough volatility. From the perspective of option pricing, certain stochastic volatility models such as Hull White, Heston and SABR generate implied volatility surfaces whose shapes differ from those observed empirically. It has been observed that while the level and orientation of the volatility surface change over time, the overall shape does not. This may suggest modeling volatility as a price- and time-independent process. However, for a fixed maturity and smile, little can be said.

On the other hand, the smile's dependence on expiration time is closely linked to the underlying's dynamics. ATM skew is thus defined as :

then became a quantity of interest. Where K is the log strike and τ the time to expiration. For stochastic models, the calculation of skew ATM volatility can still be inconsistent, the conventional method of solving this is to introduce more volatility factors. Another method is to incorporate raw volatility into the models.

Skew Stickiness & Rough Volatility by J.Gatheral

Jim Gatheral's presentation introduces a calculation method for the indicator known as skew stickiness. He provides an approximation using several mathematical methods(Diamond Product, G-Theorem and Trees and Forests) presented in his book. Here, a Variance Forward model is used to make a general point. Indeed, any Markov model can be transformed into a variance forward model, and it's natural to specify a model in this form. Variance forwards are tradable assets.

So, given a stochastic model written in the form of the forward variance curve, we can easily expand implied volatility and calculate the forward structure of skew ATM volatility. The Bergomi-Guyon expansion provides the direct relationship between the smile and the explicit formulation of a model in variance forward form. Now, to indicate the rate of change of skew-predicted ATM volatility and price return, the skew stickiness ratio is introduced. This ratio R is a model-dependent quantity, informally :

This ratio links two quantities of interest: the first is the correlation between ATM implied volatility increments of maturity T and the underlying's logarithmic returns, while the second is the implied skew of the same maturity T. This ratio can be added to various models (SABR, Heston, etc.). It is always compatible with Heston's rough volatility. It depends on the forward variance curve and is highly sensitive to recent historical stock market returns.

Does the term of the equity ATM skew really follow a power law? by Julien Guyon

J.Guyon's motivation is to examine the ATM skew and find a way to parameterize it. First, he observes that it corresponds to the power law if we ignore the shortest maturities. The aim of his research is to find enough evidence to prove this. In this work, a variance curve model is also used. Three parameterizations are considered. For each of them, the initial skew ATM term structure is calculated, then an optimization is performed to get the exact value. Two fits are compared: two-exponential kernel and power low kernel. Other presentations took place during this Quantminds session on the subject of ATM skew volatility and roughness. The literature on some of them can be found in the bibliography.

an article written by...

Agathe DEVOULON

Quant Consultant at Quanteam

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