Pricing currency derivatives with Markov-modulated Lévy dynamics

Using a Lévy process we generalize formulas in Bo et al. (2010) for the Esscher transform parameters for the log-normal distribution which ensure that the martingale condition holds for the discounted foreign exchange rate. Using these values of the parameters we find a risk-neural measure and provide new formulas for the distribution of jumps, the mean jump size, and the Poisson process intensity with respect to this measure. The formulas for a European call foreign exchange option are also derived. We apply these formulas to the case of the log-double exponential distribution of jumps. We provide numerical simulations for the European call foreign exchange option prices with different parameters.     

Dependence properties and comparison results for Lévy processes

In this paper we investigate dependence properties and comparison results for multidimensional Lévy processes. In particular we address the questions, whether or not dependence properties and orderings of the copulas of the distributions of a Lévy process can be characterized by corresponding properties of the Lévy copula, a concept which has been introduced recently in Cont and Tankov (Financial modelling with jump processes. Chapman & Hall/CRC, Boca Raton, 2004) and Kallsen and Tankov (J Multivariate Anal 97:1551–1572, 2006). It turns out that association, positive orthant dependence and positive supermodular dependence of Lévy processes can be characterized in terms of the Lévy measure as well as in terms of the Lévy copula. As far as comparisons of Lévy processes are concerned we consider the supermodular and the concordance order and characterize them by orders of the Lévy measures and by orders of the Lévy copulas, respectively. An example is given that the Lévy copula does not determine dependence concepts like multivariate total positivity of order 2 or conditionally increasing in sequence. Besides these general results we specialize our findings for subfamilies of Lévy processes. The last section contains some applications in finance and insurance like comparison statements for ruin times, ruin probabilities and option prices which extends the current literature. Anja Blatter was supported by the Deutsche Forschungsgemeinschaft (DFG).     

Corrigendum and addendum to ‘hyperfinite Lévy processes’ by Tom Lindstrøm (Stochastics 76(6):517–548, 2004)

A jump diffusion decomposition theorem for hyperfinite Lévy processes is proven; a counterexample to a previous attempt to phrase such a theorem is provided.     

Computation of Greeks in LIBOR models driven by time–inhomogeneous Lévy processes

The aim of this article is to compute Greeks, i.e. price sensitivities in the framework of the Lévy LIBOR model. Two approaches are discussed. The first approach is based on the integration-by-parts formula, which lies at the core of the application of the Malliavin calculus to finance. The second approach consists of using Fourier-based methods for pricing derivatives. We illustrate the result by applying the formula to a caplet price where the jump part of the driving process of the underlying model is given by a time–inhomogeneous Gamma process and alternatively by a Variance Gamma process.     

A Large Deviation Principle of Retarded Ornstein-Uhlenbeck Processes Driven by Lévy Noise

In this article, we develop a large deviation principle (LDP) for a class of retarded Ornstein-Uhlenbeck processes driven by Lévy processes. We first present a LDP result for time delay systems driven by cylindrical Wiener processes based on the large deviations of Gaussian processes. By using a contraction technique and passing on a finite-dimensional approximation, an LDP is obtained for stochastic time delay evolution equations driven by additive Lévy noise, whose solutions are generally not Lévy processes any more.     

A Note on the Suboptimality of Path-Dependent Pay-Offs in Lévy Markets

Abstract

Cox and Leland used techniques from the field of stochastic control theory to show that, in the particular case of a Brownian motion for the asset log-returns, risk-averse decision makers with a fixed investment horizon prefer path-independent pay-offs over path-dependent pay-offs. In this note we provide a novel and simple proof for the Cox and Leland result and we will extend it to general Lévy markets where pricing is based on the Esscher transform (exponential tilting). It is also shown that, in these markets, optimal path-independent pay-offs are increasing with the underlying final asset value. We provide examples that allow explicit verification of our theoretical findings and also show that the inefficiency cost of path-dependent pay-offs can be significant. Our results indicate that path-dependent investment pay-offs, the use of which is widespread in financial markets, do not offer good value from the investor's point of view.     

On Anticipative Girsanov Transformations for Lévy Processes

The smooth approach to Malliavin calculus for Lévy processes in (Osswald in J. Theor. Probab., 2008) is used to study time-anticipative Girsanov transformations for a large class of Lévy processes by means of the substitution rule in finite-dimensional analysis. Dedicated to Wolfram Pohlers on the occasion of his 65th birthday.     

A note on arbitrage‐free pricing of forward contracts in energy markets

Arbitrage theory is used to price forward (futures) contracts in energy markets, where the underlying assets are non‐tradeable. The method is based on the so‐called ‘fitting of the yield curve’ technique from interest rate theory. The spot price dynamics of Schwartz is generalized to multidimensional correlated stochastic processes with Wiener and Lévy noise. Findings are illustrated with examples from oil and electricity markets.     

Fluctuation theory for level-dependent Lévy risk processes

A level-dependent Lévy process solves the stochastic differential equation $dU\left(t\right)=dX\left(t\right)??\left(U\left(t\right)\right)dt$, where $X$ is a spectrally negative Lévy process. A special case is a multi-refracted Lévy process with ${?}_k\left(x\right)={\sum }_{j=1}^k{\delta }_j{\mathbf{1}}_{\{x\ge b_j\}}$. A general rate function $?$ that is non-decreasing and locally Lipschitz continuous is also considered. We discuss solutions of the above stochastic differential equation and investigate the so-called scale functions, which are counterparts of the scale functions from the theory of Lévy processes. We show how fluctuation identities for $U$ can be expressed via these scale functions. We demonstrate that the derivatives of the scale functions are solutions of Volterra integral equations.     

Operator Semi-Stable Distributions

Abstract

Jajte introduced the operator semi-stable distributions on R ⁿ in [2 Jajte , R. 1977 . Semi-stable probability measures on R ^N . Studia Math. 61 : 29 – 39 . [Google Scholar]] and proved an important fact: A full distribution μ is operator semi-stable, if and only if, there exist a number c(0 < c < 1), a vector h ∈ R ⁿ , and a nonsingular linear operator B in R ⁿ such that the formula μ ^c  = Bμ*δ(h) holds. In this paper, we make use of the eigenvalue of the matrix B to give a necessary and sufficient condition for ∫_|x|≤1|x| ^r M(dx) < ∞, where M is the Lévy measure of μ. Also, we use the symmetric group of μ to characterize the operators B in (1).     

Lowest priority waiting time distribution in an accumulating priority Lévy queue

We derive the waiting time distribution of the lowest class in an accumulating priority (AP) queue with positive Lévy input. The priority of an infinitesimal customer (particle) is a function of their class and waiting time in the system, and the particles with the highest AP are the next to be processed. To this end we introduce a new method that relies on the construction of a workload overtaking process and solving a first-passage problem using an appropriate stopping time.     

Nonlinear stochastic integrals for hyperfinite Lévy processes

I develop a notion of nonlinear stochastic integrals for hyperfinite Lévy processes and use it to find exact formulas for expressions which are intuitively of the form and , where l is a Lévy process. These formulas are then applied to geometric Lévy processes, infinitesimal transformations of hyperfinite Lévy processes, and to minimal martingale measures. Some of the central concepts and results are closely related to those found in S. Cohen’s work on stochastic calculus for processes with jumps on manifolds, and the paper may be regarded as a reworking of his ideas in a different setting and with totally different techniques.     

A note on convergence of option prices and their Greeks for Lévy models

We study the robustness of option prices to model variation after a change of measure where the measure depends on the model choice. We consider geometric Lévy models in which the infinite activity of the small jumps is approximated by a scaled Brownian motion. For the Esscher transform, the minimal entropy martingale measure, the minimal martingale measure and the mean variance martingale measure, we show that the option prices and their corresponding deltas converge as the scaling of the Brownian motion part tends to zero. We give some examples illustrating our results.     

Optimal importance sampling for Lévy processes

We develop importance sampling estimators for Monte Carlo pricing of European and path-dependent options in models driven by Lévy processes. Using results from the theory of large deviations for processes with independent increments, we compute an explicit asymptotic approximation for the variance of the pay-off under a time-dependent Esscher-style change of measure. Minimizing this asymptotic variance using convex duality, we then obtain an importance sampling estimator of the option price. We show that our estimator is logarithmically optimal among all importance sampling estimators. Numerical tests in the variance gamma model show consistent variance reduction with a small computational overhead.     

Credit Derivatives Pricing Based on Lévy Field Driven Term Structure

We use Lévy random fields to model the term structure of forward default intensity, which allows to describe the contagion risks. We consider the pricing of credit derivatives, notably of defaultable bonds in our model. The main result is to prove the pricing kernel as the unique solution of a parabolic integro-differential equation by constructing a suitable contractible operator and then considering the limit case for an unbounded terminal condition. Finally, we illustrate the impact of contagious jump risks on the defaultable bond price by numerical examples.