[Paper Review] Regular Variation and Smile Asymptotics

Shalom Benaim, Peter K. Friz|ArXiv.org|Mar 6, 2006

Stochastic processes and financial applications4 references4 citations

TL;DR

This paper establishes a rigorous link between the tail behavior of risk-neutral asset returns and the asymptotic shape of the implied volatility smile using the theory of regular variation. It sharpens Roger Lee's moment formula by showing that the implied volatility wing behavior is directly determined by the regular variation index of the tail distribution, density, or option price, providing precise asymptotic formulas for both left- and right-wing implied volatility dynamics.

ABSTRACT

We consider risk-neutral returns and show how their tail asymptotics translate directly to asymptotics of the implied volatility smile, thereby sharpening Roger Lee's celebrated moment formula. The theory of regular variation provides the ideal mathematical framework to formulate and prove such results. The practical value of our formulae comes from the vast literature on tail asymptotics and our conditions are often seen to be true by simple inspection of known results.

Motivation & Objective

• To formalize the connection between the tail asymptotics of risk-neutral returns and the implied volatility smile's wing behavior.
• To refine Roger Lee's moment formula by incorporating regular variation theory for sharper asymptotic results.
• To provide practical, analytically tractable formulas for implied volatility wings based on known tail behaviors in financial models.
• To establish sufficient conditions under which the implied volatility variance grows sublinearly, with explicit asymptotic expressions.
• To unify and generalize existing parametrizations of the volatility smile by deriving their asymptotic foundations from distributional tail properties.

Proposed method

• Utilizes the theory of regular variation to characterize the asymptotic behavior of survival functions, densities, and option prices in the tails.
• Applies the function $ \psi[x] = 2 - 4[\sqrt{x^2 + x} - x] $ to map tail decay rates to implied volatility wing behavior.
• Derives asymptotic equivalence relations between $ -\log c(k)/k $, $ -\log \bar{F}(k)/k $, and $ -\log f(k)/k $ under regular variation assumptions.
• Employs saddlepoint approximations and bounds on the normal tail function to control error terms in the implied volatility derivation.
• Establishes that $ V(k)^2/k \sim \psi[-\log c(k)/k] $ under regular variation of the call price tail, with corrections for density and survival function asymptotics.
• Uses the inverse relation of the $ \psi $ function to derive sublinear asymptotics when tail decay rates grow unboundedly.

Experimental results

Research questions

• RQ1 How do the tail properties of risk-neutral returns determine the asymptotic shape of the implied volatility smile?
• RQ2 Under what conditions does the implied volatility variance grow sublinearly, and what is its precise rate?
• RQ3 Can Roger Lee’s moment formula be sharpened using regular variation theory to yield exact asymptotic expressions?
• RQ4 What is the functional relationship between the decay rate of the tail distribution and the implied volatility wing?
• RQ5 How do the left- and right-wing asymptotics of implied volatility relate to the left- and right-tail behavior of the return density and cumulative distribution?

Key findings

• The implied volatility wing behavior satisfies $ V(k)^2/k \sim \psi[-\log c(k)/k] $ when $ -\log c(k) \in R_\alpha $, linking option price tail decay directly to volatility shape.
• If $ -\log \bar{F}(k) \in R_\alpha $, then $ V(k)^2/k \sim \psi[-1 - \log \bar{F}(k)/k] $, showing that the survival function tail determines the smile wing.
• When $ -\log f(k) \in R_\alpha $, the implied volatility satisfies $ V(k)^2/k \sim \psi[-1 - \log f(k)/k] $, demonstrating that the density tail governs the smile.
• In the case of sublinear growth, $ V(k)^2/k \sim 1/(-2 \log f(k)/k) $, providing a simple asymptotic approximation when tail decay is fast.
• The limit $ \lim_{k \to \infty} V(k)^2/k $ exists if and only if $ -\log \bar{F}(k)/k \to \theta $, and in that case $ \theta > 1 $, ensuring integrability.
• For models like Black-Scholes, the formula recovers the flat volatility smile, confirming consistency with known results.