Mitya Boyarchenko

Prices and Sensitivities of Barrier and First-Touch Digital Options in Levy-Driven Models

We present a fast and accurate FFT-based method of computing the prices and sensitivities of barrier options and first-touch digital options on stocks whose log-price follows a Levy process. The numerical results obtained via our approach are demonstrated to be in good agreement with the results obtained using other (sometimes fundamentally different) approaches that exist in the literature. However, our method is computationally much faster (often, dozens of times faster). Moreover, our technique has the advantage that its application does not entail a detailed analysis of the underlying Levy process: one only needs an explicit analytic formula for the characteristic exponent of the process. Thus our algorithm is very easy to implement in practice. Finally, our method yields accurate results for a wide range of values of the spot price, including those that are very close to the barrier, regardless of whether the maturity period of the option is long or short.

Valuation of Continuously Monitored Double Barrier Options and Related Securities

In this article we apply Carrs randomization approximation and the operator form of the Wiener-Hopf method to double barrier options in continuous time. Each step in the resulting backward induction algorithm is solved using a simple iterative procedure that reduces the problem of pricing options with two barriers to pricing a sequence of certain contingent claims with first-touch single barrier features. This procedure admits a clear financial interpretation that can be formulated in the language of embedded options.Our approach results in a fast and accurate pricing method that can be used in a rather wide class of Levy-driven models including Variance Gamma processes, Normal Inverse Gaussian processes and KoBoL processes (a.k.a. the CGMY model). At the same time, our work gives new insight into the known explicit formulas obtained by other authors in the setting of the Black-Scholes model. The operator form of the Wiener-Hopf method is generalized for wide classes of processes including the important class of Variance Gamma processes.Our method can be applied to double barrier options with arbitrary bounded terminal payoff functions, which, in particular, allows us to price knock-out double barrier put/call options as well as double-no-touch options.

A duality formalism in the spirit of Grothendieck and Verdier

We study monoidal categories that enjoy a certain weakening of the rigidity property, namely, the existence of a dualizing object in the sense of Grothendieck and Verdier. We call them Grothendieck-Verdier categories. Notable examples include the derived category of constructible sheaves on a scheme (with respect to tensor product) as well as the derived and equivariant derived categories of constructible sheaves on an algebraic group (with respect to convolution). We show that the notions of pivotal category and ribbon category, which are well known in the setting of rigid monoidal categories, as well as certain standard results associated with these notions, have natural analogues in the world of Grothendieck-Verdier categories.

Refined and Enhanced Fast Fourier Transform Techniques, with an Application to the Pricing of Barrier Options

The fast Fourier transform (FFT) technique is now a standard tool for the numerical calculation of prices of derivative securities. Unfortunately, in many important situations, such as the pricing of contingent claims of European type near expiry, and the pricing of barrier options close to the barrier, the standard implementation of this technique leads to serious systematic errors. We propose a new, fast and efficient, variant of the FFT technique, which is free of these problems, and is as easy to implement as the most common version of FFT. As an example, we show how our method leads to a pricing algorithm for down-and-out barrier put options that is the most efficient one to date, both in terms of the speed and in terms of the accuracy of the computations.

DOUBLE BARRIER OPTIONS IN REGIME-SWITCHING HYPER-EXPONENTIAL JUMP-DIFFUSION MODELS

We present a very fast and accurate algorithm for calculating prices of finite lived double barrier options with arbitrary terminal payoff functions under regime-switching hyper-exponential jump-diffusion (HEJD) models, which generalize the double-exponential jump-diffusion model pioneered by Kou and Lipton. Numerical tests demonstrate an excellent agreement of our results with those obtained using other methods, as well as a significant increase in computation speed (sometimes by a factor of 5). The first step of our approach is Carrs randomization, whose convergence we prove for barrier and double barrier options under strong Markov processes of a wide class. The resulting sequence of perpetual option pricing problems is solved using an efficient iteration algorithm and the Wiener-Hopf factorization.

PRICES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS IN LÉVY-DRIVEN MODELS, NEAR BARRIER

We calculate the leading term of asymptotics of the prices of barrier options and first-touch digitals near the barrier for wide classes of Levy processes with exponential jump densities, including the Variance Gamma model, the KoBoL (a.k.a. CGMY) model and Normal Inverse Gaussian processes. In the case of processes of infinite activity and finite variation, with the drift pointing from the barrier, we prove that the price is discontinuous at the boundary. This observation can serve as the basis for a simple robust test of the type of processes observed in real financial markets. In many cases, we calculate the second term of asymptotics as well. By comparing the exact asymptotic results for prices with those of Carrs randomization approximation, we conclude that the latter is very accurate near the barrier. We illustrate this by including numerical results for several types of Levy processes commonly used in option pricing.

Fast Simulation of Levy Processes

We present a robust method for simulating an increment of a Levy process, based on decomposing the jump part of the process into the sum of its positive and negative jump components. The characteristic exponent of a spectrally one-sided Levy process has excellent analytic properties, which we exploit to design a fast and accurate algorithm for calculating the cumulative distribution function of an increment of such a process. That algorithm is based on the parabolic inverse Fourier transform method introduced by S. Boyarchenko and S. Levendorskii, while the method of simulating a random variable using the values of its cumulative distribution function goes back to the work of P. Glasserman and Z. Liu.C code based on the simulation algorithm described in this article is available on the authors website for Levy processes of class KoBoL (a.k.a. the CGMY model). It can be used as a building block of any Monte-Carlo program for pricing derivative securities under a KoBoL process. Our method typically performs faster than the CGMY simulation method introduced by Madan and Yor by a factor of 10--100, and sometimes even higher, depending on the type of the option.

User's Guide to Pricing Double Barrier Options. Part I: Kou's Model and Generalizations

We present a very accurate algorithm for calculating prices of double barrier options, together with a simple set of detailed step-by-step instructions for implementing it in practice. Our algorithm works 5-10 times faster than any other known algorithm. At the same time, it involves no complicated technical tools, and can therefore be easily implemented in any programming language that supports elementary operations on real numbers.Our method applies to pricing double barrier options with arbitrary terminal payoff functions under Kous model (a.k.a. the double-exponential jump-diffusion model), as well as generalizations of Kous model that are referred to as hyper-exponential jump-diffusion (HEJD) models. Extensive numerical tests demonstrate excellent agreement of our results with those obtained using other approaches.

Pricing Barrier Options and Credit Default Swaps (CDS) in Spectrally One-Sided Levy Models: The Parabolic Laplace Inversion Method

Recently, the advantages of conformal deformations of the contours of integration in pricing formulas were demonstrated in the context of wide classes of Levy models and the Heston model. In the present paper we construct efficient conformal deformations of the contours of integration in the pricing formulas for barrier options and CDS in the setting of spectrally one-sided Levy models, taking advantage of Rogerss trick (J. Appl. Prob. 2000) that greatly simplifies calculation of the Wiener-Hopf factors. We extend the trick to wide classes of Levy processes of infinite variation with zero diffusion component. In the resulting formulas (both in the finite variation and the infinite variation cases), we make quasi-parabolic deformations as in S. Boyarchenko and Levendorskii (IJTAF 2013), which greatly increase the rate of convergence of the integrals. We demonstrate that the proposed method is more accurate than the standard realization of Laplace inversion in many cases. We also exhibit examples in which the standard realization is so unstable that it cannot be used for any choice of the error control parameters.