Comment on ‘Correcting for Simulation Bias in Monte Carlo Methods to Value Exotic Options in Models Driven by Lévy Processes’ by C. Ribeiro and N. Webber

Abstract

This paper proposes a pricing method for path-dependent derivatives with discrete monitoring when an underlying asset price is driven by a time-changed Lévy process. The key to our method is to derive a backward recurrence relation for computing the multivariate characteristic function of the intertemporal joint distribution of the time-changed Lévy process. Using the derived representation of the characteristic function, we obtain semi-analytical pricing formulas for geometric Asian, forward start, barrier, fader and lookback options, all of which are discretely monitored.     

Approximate indifference pricing in exponential Lévy models

Financial markets based on Lévy processes are typically incomplete and option prices depend on risk attitudes of individual agents. In this context, the notion of utility indifference price has gained popularity in the academic circles. Although theoretically very appealing, this pricing method remains difficult to apply in practice, due to the high computational cost of solving the non-linear partial integro-differential equation associated to the indifference price. In this work, we develop closed-form approximations to exponential utility indifference prices in exponential Lévy models. To this end, we first establish a new non-asymptotic approximation of the indifference price which extends earlier results on small risk aversion asymptotics of this quantity. Next, we use this formula to derive a closed-form approximation of the indifference price by treating the Lévy model as a perturbation of the Black–Scholes model. This extends the methodology introduced in a recent paper for smooth linear functionals of Lévy processes (?erný et al. 2013) to non-linear and non-smooth functionals. Our formula represents the indifference price as the linear combination of the Black–Scholes price and correction terms which depend on the variance, skewness and kurtosis of the underlying Lévy process, and the derivatives of the Black–Scholes price. As a by-product, we obtain a simple approximation for the spread between the buyer’s and the seller’s indifference price. This formula allows to quantify, in a model-independent fashion, how sensitive a given product is to jump risk when jump size is small.     

Option Pricing for Time-Change Exponential Levy Model Under Memm

The purpose of this article is to study the rational evaluation of European options price when the underlying price process is described by a time-change Levy process. European option pricing formula is obtained under the minimal entropy martingale measure （MEMM） and applied to several examples of particular time-change Levy processes. It can be seen that the framework in this paper encompasses the Black-Scholes model and almost all of the models proposed in the subordinated market.     

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).     

Convex ordering criteria for Lévy processes

Modelling financial and insurance time series with Lévy processes or with exponential Lévy 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 Lévy measures which imply corresponding comparison results for European type options in (exponential) Lévy models.     

A note on the mean correcting martingale measure for geometric Lévy processes

A martingale measure is constructed by using a mean correcting transform for the geometric Lévy processes model. It is shown that this measure is the mean correcting martingale measure if and only if, in the Lévy process, there exists a continuous Gaussian part. Although this measure cannot be equivalent to a physical probability for a pure jump Lévy process, we show that a European call option price under this measure is still arbitrage free.     

Robust barrier option pricing by frame projection under exponential Lévy dynamics

We present an efficient method for robustly pricing discretely monitored barrier and occupation time derivatives under exponential Lévy models. This includes ordinary barrier options, as well as (resetting) Parisian options, delayed barrier options (also known as cumulative Parisian or Parasian options), fader options and step options (soft-barriers), all with single and double barriers, which have yet to be priced with more general Lévy processes, including KoBoL (CGMY), Merton’s jump diffusion and NIG. The method’s efficiency is derived in part from the use of frame-projected transition densities, which transform the problem into the Fourier domain and accelerate the convergence of intermediate expectations. Moreover, these expectations are approximated by Toeplitz matrix-vector multiplications, resulting in a fast implementation. We devise an augmentation approach that contributes to the method’s robustness, adding protection against mis-specifying a proper truncation support of the transition density. Theoretical convergence is verified by a series of numerical experiments which demonstrate the method’s efficiency and accuracy.     

Valuing continuous-installment options

Installment options are path-dependent contingent claims in which the premium is paid discretely or continuously in installments, instead of paying a lump sum at the time of purchase. This paper deals with valuing European continuous-installment options written on dividend-paying assets in the standard Black–Scholes–Merton framework. The valuation of installment options can be formulated as a free boundary problem, due to the flexibility of continuing or stopping to pay installments. On the basis of a PDE for the initial premium, we derive an integral representation for the initial premium, being expressed as a difference of the corresponding European vanilla value and the expected present value of installment payments along the optimal stopping boundary. Applying the Laplace transform approach to this PDE, we obtain explicit Laplace transforms of the initial premium as well as its Greeks, which include the transformed stopping boundary in a closed form. Abelian theorems of Laplace transforms enable us to characterize asymptotic behaviors of the stopping boundary close and at infinite time to expiry. We show that numerical inversion of these Laplace transforms works well for computing both the option value and the optimal stopping boundary.     

Perpetual Bermudan Continuity Corrections and a Multi-Dimensional Wiener–Hopf Type Result

Abstract

In this article, Wiener–Hopf type results by Feller are generalized to higher dimensions, in order to derive bounds on the continuity corrections at the exercise boundary for certain perpetual Bermudan options on multiple assets. We assume that the vector of logarithmic price processes of the underlyings is a Lévy process under some risk-neutral measure. In addition, we have to impose the condition that the payoff functions of these perpetual Bermudans have been chosen in such a way that the corresponding optimal exercise regions of the Bermudan options are, up to translation, a half-space. (This is, of course, a fairly restrictive assumption for higher dimensions, but none for dimension one.)     

Jump tail dependence in Lévy copula models

This paper investigates the dependence of extreme jumps in multivariate Lévy processes. We introduce a measure called jump tail dependence, defined as the probability of observing a large jump in one component of a process given a concurrent large jump in another component. We show that this measure is determined by the Lévy copula alone and that it is independent of marginal Lévy processes. We derive a consistent nonparametric estimator for jump tail dependence and establish its asymptotic distribution. Regarding the economic relevance of the measure, a simulation study illustrates that jump tail dependence has a substantial impact on financial portfolio distributions and optimal portfolio weights.     

An efficient algorithm for Bermudan barrier option pricing

An efficient option pricing method based on Fourier-cosine expansions was presented by Fang and Oosterlee for European options in 2008,and later,this method was also used by them to price early-exercis...     

On the infimum attained by a reflected Lévy process

This paper considers a Lévy-driven queue (i.e., a Lévy process reflected at 0), and focuses on the distribution of M(t), that is, the minimal value attained in an interval of length t (where it is assumed that the queue is in stationarity at the beginning of the interval). The first contribution is an explicit characterization of this distribution, in terms of Laplace transforms, for spectrally one-sided Lévy processes (i.e., either only positive jumps or only negative jumps). The second contribution concerns the asymptotics of ℙ(M(T _u )>u) (for different classes of functions T _u and u large); here we have to distinguish between heavy-tailed and light-tailed scenarios.     

The Lévy Swap Market Model

Models driven by Lévy processes are attractive since they allow for better statistical fitting than classical diffusion models. The dynamics of the forward swap rate process is derived in a semimartingale setting and a Lévy swap market model is introduced. In order to guarantee positive rates, the swap rates are modelled as ordinary exponentials. The model starts with the most distant rate, which is driven by a non‐homogeneous Lévy process. Via backward induction the remaining swap rates are constructed such that they become martingales under the corresponding forward swap measures. Finally it is shown how swaptions can be priced using bilateral Laplace transforms.     

A Radial Basis Function Scheme for Option Pricing in Exponential Lévy Models

Abstract

We use Radial Basis Function (RBF) interpolation to price options in exponential Lévy models by numerically solving the fundamental pricing PIDE (Partial integro-differential equations). Our RBF scheme can handle arbitrary singularities of the Lévy measure in 0 without introducing further approximations, making it simpler to implement than competing methods. In numerical experiments using processes from the CGMY-KoBoL class, the scheme is found to be second order convergent in the number of interpolation points, including for processes of unbounded variation.     

On pre-exit joint occupation times for spectrally negative Lévy processes

We adopt a new approach to find Laplace transforms of joint occupation times over disjoint intervals for spectrally negative Lévy processes. The Laplace transforms are expressed in terms of scale functions.     

Equivalence of floating and fixed strike Asian and lookback options

We prove a symmetry relationship between floating strike and fixed strike Asian options for assets driven by general Lévy processes using a change of numéraire and the characteristic triplet of the dual process. We apply the same technique to prove a similar relationship between floating strike and fixed strike lookback options.     

American Options in the Heston Model with Stochastic Interest Rate and Its Generalizations

Abstract

We consider the Heston model with the stochastic interest rate of Cox–Ingersoll–Ross (CIR) type and more general models with stochastic volatility and interest rates depending on two CIR-factors; the price, volatility and interest rate may correlate. Time-derivative and infinitesimal generator of the process for factors that determine the dynamics of the interest rate and/or volatility are discretized. The result is a sequence of embedded perpetual options arising in the time discretization of a Markov-modulated Lévy model. Options in this sequence are solved using an iteration method based on the Wiener–Hopf factorization. Typical shapes of the early exercise boundary are shown, and good agreement of option prices with prices calculated with the Longstaff–Schwartz method and Medvedev–Scaillet asymptotic method is demonstrated.     

PIDE and Solution Related to Pricing of Lévy Driven Arithmetic Type Floating Asian Options

We propose a stochastic model to develop a pricing partial integro-differential equation (PIDE) and its Fourier transform expression for floating Asian options based on the Itô-Lévy calculus. The stock price is driven by a class of infinite activity Lévy processes leading to the market inherently incomplete, and dynamic hedging is no longer risk free. We first develop a PIDE for floating Asian options, and apply the Fourier transform to derive a pricing expression. Our main contribution is to develop a PIDE with its closed form pricing expression for the contract. The procedure is easy to implement for all class of Lévy processes. Finally, the model is calibrated with the market data and its accuracy is presented.     

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.     

Pricing American currency options in an exponential Lévy model

In this article the problem of the American option valuation in a Lévy process setting is analysed. The perpetual case is first considered. Without possible discontinuities (i.e. with negative jumps in the call case), known results concerning the currency option value as well as the exercise boundary are obtained with a martingale approach. With possible discontinuities of the underlying process at the exercise boundary (i.e. with positive jumps in the call case), original results are derived by relying on first passage time and overshoot associated with a Lévy process. For finite life American currency calls, the formula derived by Bates or Zhang, in the context of a negative jump size, is tested. It is basically an extension of the one developed by Mac Millan and extended by Barone‐Adesi and Whaley. It is shown that Bates' model generates pretty good results only when the process is continuous at the exercise boundary.