From local volatility to local Lévy models

By Peter P. Carr, Helyette Geman, Dilip B Madan 5, Marc Yor. 2004

91 citations recorded by the source corpus.

We define the class of local Lévy processes. These are Lévy processes time changed by an inhomogeneous local speed function. The local speed function is a deterministic function of time and the level of the process itself. We show how to reverse engineer the local speed function from traded option prices of all strikes and maturities. The local Lévy processes generalize the class of local volatility models. Closed forms for local speed functions for a variety of cases are also presented. Numerical methods for recovery are also described.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d5d6cd81d6de5e4de2117db7e03e86c552330069

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Finance

1. Dilip B Madan 5
2. Ernst Eberlein
3. Carol Alexander
4. Jim Gatheral
5. Leonardo M. Nogueira
6. P. Laurence
7. Tai-Ho Wang
8. Ouyang Cheng

In Mathematical Physics

1. Ernst Eberlein
2. Jim Gatheral
3. Dilip B Madan 5
4. Xiang-Dong Li
5. Feng-Yu Wang
6. Emanuel Milman
7. Djalil CHAFAÏ

Citations in the corpus

• An Explicit Martingale Version of the One-Dimensional Brenier's Theorem with Full Marginals Constraint (2016, unknown): “In particular, it provides a new construction of the martingale peacock process (PCOC “Processus Croissant pour l’Ordre Convexe,” see Hirsch, Profeta, Roynette and Yor [24]), and a new remarkable example of discontinuous fake Brownian motions.”
• ASYMPTOTICS OF IMPLIED VOLATILITY IN LOCAL VOLATILITY MODELS (2012, methodology): “Recent years have seen an explosion of models using the method of stochastic time changes to produce ever more versatile models, see for instance Carr et al [8] and Carr and Wu [10].”
• Model-free hedge ratios and scale-invariant models (2007, background): “Hybrid models that mix local volatility with stochastic volatility or jumps (e.g. Hagan et al., 2002; Carr et al., 2004) are typically not scale-invariant because of the local volatility component.”
• Sato processes and the valuation of structured products (2009, methodology): “(2003), and non-parametric structures like local volatility (Derman and Kani 1994, Dupire 1994) and the local Lévy model (Carr et al. 2004).”
• Option Pricing Under a Double Exponential Jump Diffusion Model (2004, unknown): “[19].”