[Paper Review] Don't stay local - extrapolation analytics for Dupire's local volatility

Peter K. Friz, Stefan Gerhold|arXiv (Cornell University)|May 6, 2011

Stochastic processes and financial applications24 references3 citations

TL;DR

This paper presents a novel analytic formula for approximating local volatility in the extreme strike regime using saddle point methods and moment generating functions. It derives an explicit asymptotic expression for Dupire's local volatility in the Heston model, showing that as log-strike $k \to \infty$, $\sigma_{\text{loc}}^2(k,T)/k$ converges to a closed-form expression involving model parameters, enabling robust extrapolation beyond observed market strikes.

ABSTRACT

A robust implementation of a Dupire type local volatility model is an important issue for every option trading floor. Typically, this (inverse) problem is solved in a two step procedure : (i) a smooth parametrization of the implied volatility surface; (ii) computation of the local volatility based on the resulting call price surface. Point (i), and in particular how to extrapolate the implied volatility in extreme strike regimes not seen in the market, has been the subject of numerous articles, starting with Lee (Math. Finance, 2004). In the present paper we give direct analytic insights into the asymptotic behavior of local volatility at extreme strikes.

Motivation & Objective

• To address the challenge of robustly extrapolating local volatility in extreme strike regimes where market data is sparse or absent.
• To provide direct analytic insight into the asymptotic behavior of local volatility at large or small strikes.
• To derive a closed-form approximation for local volatility in the Heston model under the assumption that the moment generating function of log-returns blows up at the upper critical moment.
• To establish a general approximation formula (1.6) valid beyond the Heston model, applicable when saddle point methods are justified.
• To rigorously justify the asymptotic approximation using ODE comparison techniques applied to Riccati equations in the Heston framework.

Proposed method

• The method relies on a saddle point approximation of the local volatility formula, using the moment generating function (mgf) of the log-price under the pricing measure.
• The key approximation is given by $\sigma_{\text{loc}}^2(k,T) \approx \left. \frac{2\partial_T m(s,T)}{s(s-1)} \right|_{s=\hat{s}(k,T)}$, where $\hat{s}(k,T)$ solves $\partial_s m(s,T) = k$.
• The analysis assumes the mgf $M(s,T) = \mathbb{E}[\exp(sX_T)]$ is finite in an interval $ (s_-, s_+) $, with $ \lim_{s \uparrow s_+} M(s,T) = \infty $, which holds in the Heston model.
• For the Heston model, the authors derive explicit expressions for $ R_1 $ and $ R_2 $ in terms of model parameters $ c, \rho, b, s_+ $, leading to the asymptotic ratio in Theorem 1.
• The proof uses ODE comparison techniques on the Riccati equations governing the mgf, establishing bounds on auxiliary functions $ f(t) $ and $ g(t) $ to control the asymptotic behavior.
• The method is validated numerically and extended to non-Heston models via Karamata’s Tauberian theorem, showing robustness even when the saddle point method is not strictly applicable.

Experimental results

Research questions

• RQ1 How does local volatility behave asymptotically as the strike price tends to infinity in the Heston model?
• RQ2 Can a closed-form approximation be derived for local volatility in extreme strikes using only the moment generating function of the log-price?
• RQ3 What is the precise rate of growth of local volatility as a function of log-strike in the Heston model, and how does it depend on model parameters?
• RQ4 To what extent is the saddle point approximation (1.6) valid beyond the Heston model, and when can it be trusted even when the mgf blows up slowly?
• RQ5 How can the asymptotic behavior of local volatility be rigorously justified using differential equation analysis and comparison principles?

Key findings

• In the Heston model with $ \rho \leq 0 $, the local volatility satisfies $ \lim_{k \to \infty} \frac{\sigma_{\text{loc}}^2(k,T)}{k} = \frac{2}{T \, s_+(s_+ - 1) R_1 / R_2} $, where $ R_1 $ and $ R_2 $ are explicit functions of model parameters and $ s_+ $.
• The approximation formula (1.6) is shown to be asymptotically equivalent to the exact expression in the Heston model, providing a rigorous foundation for extrapolation.
• Even when the mgf blows up slowly, the saddle point formula (1.6) yields surprisingly accurate results, as demonstrated numerically in the variance gamma model.
• The authors establish that $ f(t) \in [-C_{3,T} y^2, -C_{1,T} y] $ and $ g(t) \in [-C_{4,T} y^3, C_{2,T} y] $, with explicit constants depending on model parameters, which are crucial for proving the asymptotic behavior.
• The proof relies on bounding solutions of Riccati equations via ODE comparison, showing that $ g(t) \geq 0 $ and deriving tight bounds on $ f $ and $ g $, which control the asymptotic dynamics.
• The method is general and applies to any model where the mgf is well-behaved and the saddle point method is applicable, offering a robust alternative to ad hoc extrapolation in local volatility modeling.