Vy Models Research Articles

Liquidity Premiums in a Levy Market

<p>This paper gives a theorem for the continuous time super-replication cost of European options where the stock price follows an exponential L\'{e}vy process.<br />Under a mild assumption on the legend transform of the trading cost function, the limit of the sequence of the discrete super-replication cost is proved to be greater than or equal to an optimal control problem.<br />The main tool is an approximation multinomial scheme based on a discrete grid on a finite time interval [0,1] for a pure jump L\'{e}vy model.<br />This multinomial model is constructed similar to that proposed by (Szimayer {\&} Maller, Stoch. Proce. {\&} Their Appl., 117, 1422-1447, 2007).<br />Furthermore, it is proved that the existence of a liquidity premium for the continuous-time model under a L\'{e}vy process.<br />This paper concentrates on the L\'{e}vy processes with infinitely many jumps in any finite time interval.<br />The approach overcomes some difficulties that can be encountered when the L\'{e}vy process has infinite activity.</p>

Perpetual American Options with Disconnected Exercise Regions in L vy Models

For wide classes of payoff functions and Levy processes, we solve perpetual restricted optimal stopping problems, when one exercise region is exogenously given (barrier like feature), and the other is chosen to optimize the option value, and straddle-like perpetual options, when the optimal exercise region is a union of two disjoint semi-infinite intervals.

Discrete endogenous variables in weakly separable models

Summary This paper contains an extension of the identification method proposed in Jun et al. (2011), hereafter JPX, which is based on a generated collection of sets, that is a ‘Dynkin system’. We demonstrate the usefulness of this extension in the context of the model proposed by Vytlacil and Yildiz (2007), hereafter VY. VY formulate a fully non-parametric model featuring a nested weakly separable structure in which an endogenous regressor is binary-valued. The extension of the JPX approach considered here allows for non-binary-valued discrete endogenous regressors and requires weaker support conditions than VY in the binary case, which substantially broadens the range of potential applications of the VY model. In this paper we focus on the binary case for which we provide several alternative simpler sufficient conditions and outline an estimation strategy.

LEVY-BASED HEATH-JARROW-MORTON INTEREST RATE DERIVATIVES: CHANGE OF TIME METHOD AND PIDES

In this paper, we show how to calculate the price of zero-coupon bonds Heath-Jarrow-Morton (HJM) L\'{e}vy model of forward interest rate $f(t,T)$ using the change of time method. We also derive partial integro-differential equations (PIDEs) for the values of swaps, caps, floors and options on them, swaptions, captions and floortions, respectively. We apply the change of time method to price the interest rate derivatives for the forward interest rates $f(t,T)$ described by the stochastic differential equation driven by $\alpha$-stable L\'{e}vy processes.

Statistical inference for time-changed Lévy processes via composite characteristic function estimation

In this article, the problem of semi-parametric inference on the parameters of a multidimensional L\'{e}vy process $L_t$ with independent components based on the low-frequency observations of the corresponding time-changed L\'{e}vy process $L_{\mathcal{T}(t)}$, where $\mathcal{T}$ is a nonnegative, nondecreasing real-valued process independent of $L_t$, is studied. We show that this problem is closely related to the problem of composite function estimation that has recently gotten much attention in statistical literature. Under suitable identifiability conditions, we propose a consistent estimate for the L\'{e}vy density of $L_t$ and derive the uniform as well as the pointwise convergence rates of the estimate proposed. Moreover, we prove that the rates obtained are optimal in a minimax sense over suitable classes of time-changed L\'{e}vy models. Finally, we present a simulation study showing the performance of our estimation algorithm in the case of time-changed Normal Inverse Gaussian (NIG) L\'{e}vy processes.

The Longstaff–Schwartz algorithm for Lévy models: Results on fast and slow convergence

We investigate the Longstaff--Schwartz algorithm for American option pricing assuming that both the number of regressors and the number of Monte Carlo paths tend to infinity. Our main results concern extensions, respectively, applications of results by Glasserman and Yu [Ann. Appl. Probab. 14 (2004) 2090--2119] and Stentoft [Manag. Sci. 50 (2004) 1193--1203] to several L\'{e}vy models, in particular the geometric Meixner model. A convenient setting to analyze this convergence problem is provided by the L\'{e}vy--Sheffer systems introduced by Schoutens and Teugels.