Asymptotics for Exponential Levy Processes and their Volatility Smile: Survey and New Results

Exponential L\'evy processes can be used to model the evolution of various financial variables such as FX rates, stock prices, etc. Considerable efforts have been devoted to pricing derivatives written on underliers governed by such processes, and the corresponding implied volatility surfaces have been analyzed in some detail. In the non-asymptotic regimes, option prices are described by the Lewis-Lipton formula which allows one to represent them as Fourier integrals; the prices can be trivially expressed in terms of their implied volatility. Recently, attempts at calculating the asymptotic limits of the implied volatility have yielded several expressions for the short-time, long-time, and wing asymptotics. In order to study the volatility surface in required detail, in this paper we use the FX conventions and describe the implied volatility as a function of the Black-Scholes delta. Surprisingly, this convention is closely related to the resolution of singularities frequently used in algebraic geometry. In this framework, we survey the literature, reformulate some known facts regarding the asymptotic behavior of the implied volatility, and present several new results. We emphasize the role of fractional differentiation in studying the tempered stable exponential Levy processes and derive novel numerical methods based on judicial finite-difference approximations for fractional derivatives. We also briefly demonstrate how to extend our results in order to study important cases of local and stochastic volatility models, whose close relation to the L\'evy process based models is particularly clear when the Lewis-Lipton formula is used. Our main conclusion is that studying asymptotic properties of the implied volatility, while theoretically exciting, is not always practically useful because the domain of validity of many asymptotic expressions is small.Comment: 92 pages, 15 figure

From Discrete to Continuous: Modeling Volatility of the Istanbul Stock Exchange Market with GARCH and COGARCH

The objective of this paper is to model the volatility of Istanbul Stock Exchange market, ISE100 Index by ARMA and GARCH models and then take a step further into the analysis from discrete modeling to continuous modeling. Through applying unit root and stationary tests on the log return of the index, we found that log return of ISE100 data is stationary. Best candidate model chosen was found to be AR(1)~GARCH(1,1) by AIC and BIC criteria. Then using the parameters from the discrete model, COGARCH(1,1) was applied as a continuous model

From Discrete to Continuous: Modeling Volatility of the Istanbul Stock Exchange Market with GARCH and COGARCH

The objective of this paper is to model the volatility of Istanbul Stock Exchange market, ISE100 Index by ARMA and GARCH models and then take a step further into the analysis from discrete modeling to continuous modeling. Through applying unit root and stationary tests on the log return of the index, we found that log return of ISE100 data is stationary. Best candidate model chosen was found to be AR(1)~GARCH(1,1) by AIC and BIC criteria. Then using the parameters from the discrete model, COGARCH(1,1) was applied as a continuous model.ISE100,IMKB100,GARCH Modeling,COGARCH Modeling,discrete modeling,continuous modeling

Martingalized Historical approach for Option Pricing

In a discrete time option pricing framework, we compare the empirical performance of two pricing methodologies, namely the affine stochastic discount factor and the empirical martingale correction methodologies. Using a CAC 40 options dataset, the differences are found to be small : the higher order moment correction involved in the SDF approach may not be that essential to reduce option pricing errors.Generalized hyperbolic distribution, option pricing, incomplete market, CAC 40, Stochastic Discount Factor, martingale correction.

Martingalized Historical approach for Option Pricing

In a discrete time option pricing framework, we compare the empirical performance of two pricing methodologies, namely the affine stochastic discount factor (SDF) and the empirical martingale correction methodologies. Using a CAC 40 options dataset, the differences are found to be small: the higher order moment correction involved in the SDF approach may not be that essential to reduce option pricing errors. This paper puts into evidence the fact that an appropriate modelling under the historical measure associated with an adequate correction (that we call here a ”martingale correction”) permits to provide option prices which are close to market ones.Generalized Hyperbolic Distribution; Option pricing; Incomplete market; CAC40; Stochastic Discount Factor; Martingale Correction

A Heat Kernel Approach to Interest Rate Models

We construct default-free interest rate models in the spirit of the well-known Markov funcional models: our focus is analytic tractability of the models and generality of the approach. We work in the setting of state price densities and construct models by means of the so called propagation property. The propagation property can be found implicitly in all of the popular state price density approaches, in particular heat kernels share the propagation property (wherefrom we deduced the name of the approach). As a related matter, an interesting property of heat kernels is presented, too