Exponential Levy Models Research Articles

Risk Adjustments of Option Prices under Time-Changed Dynamics

We derive a closed-form expansion of option prices in terms of Black-Scholes prices and higher-order Greeks. We show how the true price of an option less its Black-Scholes price is given by a series of premiums on higher-order risks that are not priced under the Black-Scholes model assumptions. The expansion can be used for a broad class of option pricing models with dynamics governed by time-changed Brownian motions. Specifically, we study expansions for exponential Levy models such as the Variance Gamma and the Normal Inverse Gaussian models as well as their stochastic volatility counterparts, e.g., the VGSV and NIGSV models. Moreover, we consider extensions of the expansion to a more general subclass of affine jump-diffusion models for which the pricing transform may not be known in closed form.

Asymptotics of Forward Implied Volatility

We prove here a general closed-form expansion formula for forward-start options and the forward implied volatility smile in a large class of models, including the Heston stochastic volatility and time-changed exponential Levy models. This expansion applies to both small and large maturities and is based solely on the properties of the forward characteristic function of the underlying process. The method is based on sharp large deviations techniques and allows us to recover (in particular) many results for the spot implied volatility smile. In passing we (i) show that the forward-start date has to be rescaled in order to obtain nontrivial small-maturity asymptotics, (ii) prove that the forward-start date may influence the large-maturity behavior of the forward smile, and (iii) provide some examples of models with finite quadratic variation where the small-maturity forward smile does not explode.

Minimal Variance Hedging of Natural Gas Derivatives in Exponential Levy Models: Theory and Empirical Performance

We consider the problem of hedging European options written on natural gas futures, in a market where prices of traded assets exhibit jumps, by trading in the underlying asset. We provide a general expression for the hedging strategy which minimizes the variance of the terminal hedging error, in terms of stochastic integral representations of the payoffs of the options involved. This formula is then applied to compute hedge ratios for common options in various models with jumps, leading to easily computable expressions. As a benchmark we take the standard Black-Scholes-Merton delta hedges. We show that in natural gas option markets minimal variance hedging with underlying consistently outperform the benchmarks by quite a margin.

Polynomial processes and their applications to mathematical finance

We introduce a class of Markov processes, called m-polynomial, for which the calculation of (mixed) moments up to order m only requires the computation of matrix exponentials. This class contains affine processes, processes with quadratic diffusion coefficients, as well as Levy-driven SDEs with affine vector fields. Thus, many popular models such as exponential Levy models or affine models are covered by this setting. The applications range from statistical GMM estimation procedures to new techniques for option pricing and hedging. For instance, the efficient and easy computation of moments can be used for variance reduction techniques in Monte Carlo methods.

Exponential Levy Models Extended by a Jump to Default

This paper proposes a new dynamically consistent framework for joint valuation of equity derivatives and credit products, in which uncertainness of the economy is represented by Levy processes. In the framework, the pre-default stock price of a given firm follows an extended exponential Levy model, while the default arrival rate is given by the Cox proportional hazard model with stochastic covariates driven by Levy processes. Under the models, we find the solution of the pricing generator for evaluating equity and credit derivatives, and then derive the pricing formulas of equity call options and credit default swaps by utilizing the pricing generator. In the numerical examples, setting the variance gamma process and the Brownian motion as driving factors of the models, we compute term structure of credit default swaps and equity implied volatility skews. We also examine the impact of the convexity adjustment on term structure of credit spreads both analytically and numerically.

Statistical estimation of Lévy-type stochastic volatility models

Volatility clustering and leverage are two of the most prominent stylized features of the dynamics of asset prices. In order to incorporate these features as well as the typical fat-tails of the log return distributions, several types of exponential Levy models driven by random clocks have been proposed in the literature. These models constitute a viable alternative to the classical stochastic volatility approach based on SDEs driven by Wiener processes. This paper has two main objectives. First, using threshold type estimators based on high-frequency discrete observations of the process, we consider the recovery problem of the underlying random clock of the process. We show consistency of our estimator in the mean-square sense, extending former results in the literature for more general Levy processes and for irregular sampling schemes. Secondly, we illustrate empirically the estimation of the random clock, the Blumenthal-Geetor index of jump activity, and the spectral Levy measure of the process using real intraday high-frequency data.

On Continuity Properties for Option Prices in Exponential Lévy Models

For a converging sequence of exponential Levy models, we give conditions under which the associated sequence of option prices converges. We also study the behavior of the prices when no such convergence holds. We then consider two special cases: first when the martingale measure is chosen by minimization of entropy, and then when it minimizes Hellinger integrals.

Computing exponential moments of the discrete maximum of a Lévy process and lookback options

We present a fast and accurate method to compute exponential moments of the discretely observed maximum of a Levy process. The method involves a sequential evaluation of Hilbert transforms of expressions involving the characteristic function of the (Esscher-transformed) Levy process. It can be discretized with exponentially decaying errors of the form O(exp (−aM b )) for some a,b>0, where M is the number of discrete points used to compute the Hilbert transform. The discrete approximation can be efficiently implemented using the Toeplitz matrix–vector multiplication algorithm based on the fast Fourier transform, with total computational cost of O(NMlog (M)), where N is the number of observations of the maximum. The method is applied to the valuation of European-style discretely monitored floating strike, fixed strike, forward start and partial lookback options (both newly written and seasoned) in exponential Levy models.

NO-FREE-LUNCH EQUIVALENCES FOR EXPONENTIAL LÉVY MODELS UNDER CONVEX CONSTRAINTS ON INVESTMENT

We provide equivalence of numerous no-free-lunch type conditions for financial markets where the asset prices are modeled as exponential Levy processes, under possible convex constraints in the use of investment strategies. The general message is the following: if any kind of free lunch exists in these models it has to be of the most egregious type, generating an increasing wealth. Furthermore, we connect the previous to the existence of the numeraire portfolio, both for its particular expositional clarity in exponential Levy models and as a first step in obtaining analogues of the no-free-lunch equivalences in general semimartingale models, a task that is taken on in Karatzas and Kardaras (2007).

Pricing Forward Start Options in Models Based on (Time-Changed) Levy Processes

Options depending on the forward skew are very popular. One such option is the forward starting call option - the basic building block of a cliquet option. Widely applied models to account for the forward skew dynamics to price such options include the Heston model, the Heston-Hull-White model and the Bates model. Within these models solutions for options including forward start features are available using (semi) analytical formulas. Today exponential (subordinated) Levy models being increasingly popular for modelling the asset dynamics. While the simple exponential Levy models imply the same forward volatily surface for all future times the subordinated models do not. Depending on the subordinator the dynamic of the forward volatility surface and therefore stochastic volatility can be modelled. Analytical pricing formulas based on the characteristic function and Fourier transform methods are available for the class of these models. We extend the applicability of analytical pricing to options including forward start features. To this end we derive the forward characteristic functions which can be used in Fourier transform based methods. As examples we consider the Variance Gamma model and the NIG model subordinated by a Gamma Ornstein Uhlenbeck process and respectively by an Cox-Ingersoll-Ross process. We check our analytical results by applying Monte Carlo methods. These results can for instance be applied to calibration of the forward volatility surface.

A Note on Monte Carlo Greeks using the Characteristic Function

We consider the derivation of generic Monte Carlo estimators for Greeks for (path-dependent) options with discontinuous payoffs in the case where only the characteristic function is known. In Kienitz (2008) we have shown how to derive such Greeks for a wide range of models under the assumption that the transition probability is known in closed form. Unfortunately, this is not always the case. For example when considering exponential Levy models with stochastic volatility such as the Variance Gamma model with a Gamma Ornstein-Uhlenbeck or CIR stochastic clock. The characteristic function in this case the density is only given through its characteristic function. We give an algorithm to compute the probability density from the characteristic function and show that computing the transition density in this way gives the same results as in Kienitz (2008) but works for very general models. In this paper we focus on the Variance Gamma model and the same model with a Gamma Ornstein-Uhlenbeck stochastic clock. Since the methods are very general we can cope with other complex models like the Normal Inverse Gaussian model, considering other types of stochastic clocks or other classes of models where the characteristic function is known.

On q-optimal martingale measures in exponential Lévy models

We give a sufficient condition to identify the q-optimal signed and the q-optimal absolutely continuous martingale measures in exponential Levy models. As a consequence, we find that in the one-dimensional case, the q-optimal equivalent martingale measures may exist only if the tails for upward jumps are extraordinarily light. Moreover, we derive the convergence of q-optimal signed, resp. absolutely continuous, martingale measures to the minimal entropy martingale measure as q approaches one. Finally, some implications for portfolio optimization are discussed.

A Fast and Accurate FFT-Based Method for Pricing Early-Exercise Options under Lévy Processes

A fast and accurate method for pricing early exercise and certain exotic options in computational finance is presented. The method is based on a quadrature technique and relies heavily on Fourier transformations. The main idea is to reformulate the well-known risk-neutral valuation formula by recognizing that it is a convolution. The resulting convolution is dealt with numerically by using the fast Fourier transform. This novel pricing method, which we dub the convolution method, is applicable to a wide variety of payoffs and requires only the knowledge of the characteristic function of the model. As such, the method is applicable within many regular affine models, among which is the class of exponential Levy models. For an $M$-times exercisable Bermudan option, the overall complexity is $O(MN\log_2(N))$, with $N$ grid points used to discretize the price of the underlying asset. American options are priced efficiently by applying Richardson extrapolation to the prices of Bermudan options.

Convex ordering criteria for Lévy processes

Modelling financial and insurance time series with Levy processes or with exponential Levy processes is a relevant actual practice and an active area of research. It allows qualitatively and quantitatively good adaptation to the empirical statistical properties of asset returns. Due to model incompleteness it is a problem of considerable interest to determine the dependence of option prices in these models on the choice of pricing measures and to establish nontrivial price bounds. In this paper we review and extend ordering results of stochastic and convex type for this class of models. We also extend the ordering results to processes with independent increments (PII) and present several examples and applications as to α-stable processes, NIG-processes, GH-distributions, and others. Criteria are given for the Levy measures which imply corresponding comparison results for European type options in (exponential) Levy models.

How to Price Efficiently European Options in Some Geometric Lévy Processes Models?

This paper presents the implementation to the class of jump diffusion models of the approach used by Boyarchenko and Levendorskii (2002) in the case of exponential Levy models. We show that this approach is more computationally efficient than the semi closed form solutions derived by Kou (2002), especially compared to the latter case. A brand new model is then presented. It extends and generalizes Kou model.

Integro-differential equations for option prices in exponential Lévy models

We explore the precise link between option prices in exponential Levy models and the related partial integro-differential equations (PIDEs) in the case of European options and options with single or double barriers. We first discuss the conditions under which options prices are classical solutions of the PIDEs. We show that these conditions may fail in pure jump models and give examples of lack of smoothness of option prices with respect to the underlying. We give sufficient conditions on the Levy triplet for the prices of barrier options to be continuous with respect to the underlying and show that, in a general setting, option prices in exp-Levy models correspond to viscosity solutions of the pricing PIDE.

On the Martingale Measures in Exponential Lévy Models

We study the existence and uniqueness of martingale measures in the exponential Levy models of the form $S_t=e^{X_t}$, $S_t=e^{X_{\tau_t}}$, where X is a Levy process and $\tau$ is an independent increasing process.

On the Martingale Measures in Exponential Levy Models

On the Martingale Measures in Exponential Levy Models