2010 – 2011
Calibrating stochastic models is a fundamental issue on financial markets. The aim of this talk is to propose a calibration methodology based on the knowledge of the asymptotic behaviour of the implied volatility. We focus on the general class of affine stochastic volatility models with jumps, which encompasses the Heston (with jumps) model, exponential Levy models, the Barndorff-Nielsen and Shephard model. Under mild conditions on the jump measures, we derive (semi) closed-form formulae for the implied volatility as the maturity gets large.
November 11, 2010
Large-time asymptotics for general stochastic volatility models
Dublin City University
We derive large-time asymptotics for the modified SABR model, and show that the estimates are consistent with the general result in Tehranchi09. We then discuss large-time asymptotics for a general uncorrelated model, using Donsker&Varadhan’s large deviation principle for the occupation measure of an ergodic process. This is related to the recent work of Feng, Fouque & Kumar using viscosity solutions and non-linear homogenization theory.
November 23, 2010
Long-term Behaviors and Implied Volatilities for General Affine Diffusions
For the past several years, studies on affine processes have been worked out by many researchers, regarding moment explosions, implied volatilities, and long-term behaviors. Recently, Glasserman and Kim, and Keller-Ressel investigated the moment explosions of the canonical affine models of Dai and Singleton, and general two-factor affine stochastic volatility models, respectively. They also presented the long-term behaviors of such processes. On the other hand, Benaim and Friz, and Lee showed that implied volatilities at extreme strikes are linked to the moment explosions of stock prices at given option maturities. In this work, we characterize the regions in which moment explosions happen for some time or at a given time, and relate them to the long-term behavior of stock prices and to implied volatilities, extending previous works on moment explosions for affine processes. (This is a joint work with Rudra P. Jena and Hao Xing.)
March 3, 2011
Asymptotic behavior of the implied volatility in stochastic asset price models with and without moment explosions
The main results discussed in the talk concern asymptotic formulas with error estimates fot the implied volatility at extreme strikes in various stochastic asset price models. These formulas are valid for practically any such model. It will be shown that the new formulas imply several known results, including Roger Lee’s moment formulas and the tail-wing formulas due to Shalom Benaim and Peter Friz. We will also provide necessary and sufficient conditions for the validity of asymptotic equivalence in Lee’s moment formulas. Applications will be given to various stochastic volatility models (Hull-White, Stein-Stein, and Heston). These are all models with moment explosions. For stochastic asset price models without moment explosions, the general formulas for the implied volatility can be given an especially simple form. Using these simplifications, we prove a modified version of Piterbarg’s conjecture. The asymptotic formula suggested by Vladimir Piterbarg may be considered as a substitute for Lee’s moment formula for the implied volatility at large strikes in the case of models without moment explosions. We will also discuss the asymptotic behavior of the implied volatility in several special asset price models without moment explosions, e.g., Rubinstein’s displaced diffusion model, the CEV model, the finite moment log-stable model of Carr and Wu, and SV1 and SV2 models of Rogers and Veraart.
March 4, 2011
Toxicity and Volatility in High Frequency Markets
March 9, 2011
Volatility Forecasting Models
European University Vladrina
The current research suggests a sequential procedure for monitoring validity of the volatility model. A state space representation describes dynamics of the daily integrated volatility. The observation equation relates the integrated volatility to its measures, such as the realized volatility or bipower variation. A control procedure, based on the corresponding forecasting errors, allows to decide, whether the chosen representation remains correctly specified. A signal indicates that the assumed volatility model may not be valid anymore. The performance of our approach is analyzed within a Monte Carlo study and illustrated in an empirical study for selected U.S. stocks.
March 10, 2011
Properties of Hierarchical Archimedean Copulas (Paper)
Humboldt University Vladrina
In this paper we analyse the properties of hierarchical Archimedean copulas. This class is a generalisation of the Archimedean copulas and allows for general non-exchangeable dependency structures. We show that the structure of the copula can be uniquely recovered from all bivariate margins. We derive the distribution of the copula value, which is particularly useful for tests and constructing confidence intervals. Furthermore, we analyse dependence orderings, multivariate dependence measures and extreme value copulas. Special attention we pay to the tail dependencies and derive several tail dependence indices for general hierarchical Archimedean copulas.
March 31, 2011
Near-expiration behavior of implied volatility for exponential Levy models
Implied volatility is the market’s measure of choice to summarize the risk of an underlying asset as reflected by its option prices. The asymptotic behavior of option prices near expiration is a problem of current interest with important practical consequences. In this talk, I will present a near-expiration expansion for out-of-the-money call options under an exponential Levy model, using a change of numeraire technique via the Esscher transform. Using this result, a small-time expansion for the implied volatility is obtained for both exponential Levy models and time-changed Levy models. Numerical implementation of our results shows that the second order approximation can significantly outperform the first order approximation. This talk is based on a joint work with Martin Forde.
April 8, 2011
Financial Engineering and the Credit Crisis
Coming out of (for some, still being in) the worst financial crisis since the 1930s Great Depression, we all have to ask ourselves some serious questions. In particular, the guild of Financial Engineers has to answer criticisms ranging from Michel Rocard, the former French Prime Minister’s “Quantitative models are a crime against humanity” to Felix Salmon’s “Recipe for Disaster: The formula that killed Wall Street”. A simple: “These are ridiculous statements” does not suffice. In this talk I will present my views on the above and discuss ways in which quantitative finance (financial engineering, financial and actuarial mathematics) has an important role to play going forward. Besides a brief discussion on some general points underlying the financial crisis, I will also discuss some more technical issues as there are (i) the relevance of micro-correlation for pricing CDO tranches, (ii) dependence modeling beyond linear correlation and (iii) model uncertainty.
This paper defines a new concept of market speed based on the time required for prices to be synchronized. A new methodology is developed which allows the time required to two-assets to be fully synchronized. The speed of a market can be estimated from ultra high-frequency data. This methodology is applied to the constituents of the S&P 500 using data form 1996 until 2009. I find that the time required to for markets to become synchronized has dropped from more than an hour to less than a minute. I explore factors which affect the speed of the market, and characteristics of firms which lead to slower synchronization times.
May 6, 2011
Modeling Financial Contagion Using Mutually Exciting Jump Processes (Paper)
As has become abundantly clear during the recent financial crisis, adverse shocks to stock markets propagate across the world, with a jump in one region of the world seemingly causing an increase in the likelihood of a different jump in another region of the world. To capture this effect, we propose a model for asset return dynamics with a drift component, a stochastic volatility component and mutually exciting jumps known as Hawkes processes. In the model, a jump in one region of the world increases the intensity of jumps occurring both in the same region (self-excitation) as well as in other regions (cross-excitation), generating jump clustering. Jump intensities then mean-revert until the next jump. We develop and implement a GMM-based estimation procedure for this model, and show that the model fits the data well. The estimates provide evidence for self-excitation both in the US and the other world markets, and for asymmetric cross-excitation. Implications of the model for measuring market stress, risk management and optimal portfolio choice are also investigated.
June 8, 2011
Conditional moment models under weak identification
Simone Fraser University
June 9, 2011
Generalized Method of Moments with Tail Trimming (Paper)
University of North Carolina at Chapel Hill
We develop a GMM estimator for stationary heavy tailed data by trimming an asymptotically vanishing sample portion of the estimating equations. Trimming ensures the estimator is asymptotically normal, and self-normalization implies we do not need to know the rate of convergence. Tailtrimming, however, ensures asymmetric models are covered under rudimentary assumptions about the thresholds, and it implies possibly heterogeneous convergence rates below, at or above pT. Further, it implies super-pT-consistency is achievable depending on regressor and error tail thickness and feedback, with a rate arbitrarily close to the largest possible rate amongst untrimmed minimum distance estimators for linear models with iid errors, and a faster rate than QML for heavy tailed GARCH. In the latter cases the optimal rate is achieved with the e¢ cient GMM weight, and by using simple rules of thumb for choosing the number of trimmed equations. Simulation evidence shows the new estimator dominates GMM and QML when these estimators are not or have not been shown to be asymptotically normal, and for asymmetric GARCH models dominates a heavy tail robust weighted version of QML. (Joint with Jonathan B. Hill.)
June 10, 2011
Factors on Demand: Building a Platform for Portfolio Managers, Risk Managers and Traders
Kepos Capital LP and Baruch College
June 15 – 17, 2011
Fourth Annual SoFie Conference
Joint Conference: Society of Financial Econometrics (SoFiE) and Stevanovich Center