Large Deviations and Asymptotic Methods in Finance

✍ Scribed by Peter K. Friz, Jim Gatheral, Archil Gulisashvili, Antoine Jacquier, Josef Teichmann (eds.)

• Publisher: Springer International Publishing
• Year: 2015
• Tongue: English
• Leaves: 590
• Series: Springer Proceedings in Mathematics & Statistics 110
• Edition: 1
• Category: Library

✦ Synopsis

Topics covered in this volume (large deviations, differential geometry, asymptotic expansions, central limit theorems) give a full picture of the current advances in the application of asymptotic methods in mathematical finance, and thereby provide rigorous solutions to important mathematical and financial issues, such as implied volatility asymptotics, local volatility extrapolation, systemic risk and volatility estimation. This volume gathers together ground-breaking results in this field by some of its leading experts.

Over the past decade, asymptotic methods have played an increasingly important role in the study of the behaviour of (financial) models. These methods provide a useful alternative to numerical methods in settings where the latter may lose accuracy (in extremes such as small and large strikes, and small maturities), and lead to a clearer understanding of the behaviour of models, and of the influence of parameters on this behaviour.

Graduate students, researchers and practitioners will find this book very useful, and the diversity of topics will appeal to people from mathematical finance, probability theory and differential geometry.

✦ Table of Contents

Front Matter....Pages i-ix
Probability Distribution in the SABR Model of Stochastic Volatility....Pages 1-35
Asymptotic Implied Volatility at the Second Order with Application to the SABR Model....Pages 37-69
Unifying the BGM and SABR Models: A Short Ride in Hyperbolic Geometry....Pages 71-88
Second Order Expansion for Implied Volatility in Two Factor Local Stochastic Volatility Models and Applications to the Dynamic (\lambda ) -Sabr Model....Pages 89-136
General Asymptotics of Wiener Functionals and Application to Implied Volatilities....Pages 137-173
Implied Volatility of Basket Options at Extreme Strikes....Pages 175-212
Small-Time Asymptotics for the At-the-Money Implied Volatility in a Multi-dimensional Local Volatility Model....Pages 213-237
A Remark on Gatheral’s ‘Most-Likely Path Approximation’ of Implied Volatility....Pages 239-245
Implied Volatility from Local Volatility: A Path Integral Approach....Pages 247-271
Extrapolation Analytics for Dupire’s Local Volatility....Pages 273-286
The Gärtner-Ellis Theorem, Homogenization, and Affine Processes....Pages 287-320
Asymptotics for (d) -Dimensional Lévy-Type Processes....Pages 321-343
Asymptotic Expansion Approach in Finance....Pages 345-411
On Small Time Asymptotics for Rough Differential Equations Driven by Fractional Brownian Motions....Pages 413-438
On Singularities in the Heston Model....Pages 439-448
On the Probability Density Function of Baskets....Pages 449-472
On Small-Noise Equations with Degenerate Limiting System Arising from Volatility Models....Pages 473-505
Long Time Asymptotics for Optimal Investment....Pages 507-528
Systemic Risk and Default Clustering for Large Financial Systems....Pages 529-557
Estimation of Volatility Functionals: The Case of a (\sqrt{n}) Window....Pages 559-590

✦ Subjects

Quantitative Finance; Probability Theory and Stochastic Processes; Approximations and Expansions; Differential Geometry