arXiv subjects

Sergei Levendorskiĭ

Publications and source records attributed to Sergei Levendorskiĭ.

14 recordsLinked to original sources

arXiv2026

Fast reliable pricing and calibration of the rough Heston model

The paper is an extended and modified version of the preprint S.Boyarchenko and S.Levendorskiĭ ``Correct implied volatility shapes and reliable pricing in the rough Heston model". We combine a modification of the Adams method with the SINH-acceleration method S.Boyarchenko and S.Levendorskii (IJTAF 2019, v.22) of Fourier inversion (iFT) to price vanilla options under the rough Heston model. For moderate or long maturities and strikes near spot, thousands of prices are computed in several milliseconds (ms) in Matlab on a Mac with moderate specs, with relative errors $\lesssim 10^{-4}$. Even for options close to expiry and far-OTM, the pricing takes a few tens or hundreds of ms. We show that, for the calibrated parameters in El Euch and Rosenbaum (Math.Finance 2019, v.29), the model implied vol surface is much flatter and fits the market data poorly; thus the calibration in op.cit. is a case of ``ghost calibration'' (M.Boyarchenko and S.Levendorskiĭ, Quant. Finance 2015, v.15): numerical error and model specification error offset each other, creating an apparently good fit that vanishes when a more accurate pricer is used. We explain how such errors arise in popular iFT implementations that use fixed numerical parameters, yielding spurious smiles/skews, and provide numerical evidence that SINH acceleration is faster and more accurate than competing methods. Robust error control is ensured by a general Conformal Bootstrap principle that we formulate; the principle is applicable to many Fourier-pricing methods. We outline how this principle and our method enable accurate calibration procedures that are hundreds of times faster than approaches commonly used in the industry. Disclaimer: The views expressed herein are those of the authors only. No other representation should be attributed.

q-fin.CP↗

arXiv2025

Asymptotics of survival probabilities and lower tail probability problem

The present paper is an addendum to the paper ``Lévy models amenable to efficient calculations", where we introduced a general class of Stieltjes-Lévy processes (SL-processes) and signed SL processes defined in terms of certain Stieltjes-Lévy measures. We demonstrated that SL-processes enjoyed all properties that we used earlier to develop efficient methods for evaluation of expectations of functions of a Lévy process and its extremum processes, and proved that essentially all popular classes of Lévy processes are SL-processes; sSL-processes fail to possess one important property. In the present paper, we use the properties of (s)SL-processes to derive new formulas for the Wiener-Hopf factors $ϕ^\pm_q$ for small $q$ in terms of the absolute continuous components of SL-measures and their densities, and calculate the leading terms of the survival probability also in terms of the absolute continuous components of SL-measures and their densities. The lower tail probability is calculated for more general classes of SINH-regular processes constructed earlier.

math.PR↗

arXiv2024

Efficient inverse $Z$-transform and Wiener-Hopf factorization

We suggest new closely related methods for numerical inversion of $Z$-transform and Wiener-Hopf factorization of functions on the unit circle, based on sinh-deformations of the contours of integration, corresponding changes of variables and the simplified trapezoid rule. As applications, we consider evaluation of high moments of probability distributions and construction of causal filters. Programs in Matlab running on a Mac with moderate characteristics achieves the precision E-14 in several dozen of microseconds and E-11 in several milliseconds, respectively.

math.NA↗

arXiv2024

Alternative models for FX: pricing double barrier options in regime-switching Lévy models with memory

This paper is a supplement to our recent paper ``Alternative models for FX, arbitrage opportunities and efficient pricing of double barrier options in Lévy models". We introduce the class of regime-switching Lévy models with memory, which take into account the evolution of the stochastic parameters in the past. This generalization of the class of Lévy models modulated by Markov chains is similar in spirit to rough volatility models. It is flexible and suitable for application of the machine-learning tools. We formulate the modification of the numerical method in ``Alternative models for FX, arbitrage opportunities and efficient pricing of double barrier options in Lévy models", which has the same number of the main time-consuming blocks as the method for Markovian regime-switching models.

q-fin.PR↗

arXiv2023

Efficient evaluation of expectations of functions of a Lévy process and its extremum

We prove simple general formulas for expectations of functions of a Lévy process and its running extremum. Under additional conditions, we derive analytical formulas using the Fourier/Laplace inversion and Wiener-Hopf factorization, and discuss efficient numerical methods for realization of these formulas. As applications, the cumulative probability distribution function of the process and its running maximum and the price of the option to exchange the power of a stock for its maximum are calculated. The most efficient numerical methods use the sinh-acceleration technique and simplified trapezoid rule. The program in Matlab running on a Mac with moderate characteristics achieves the precision E-7 and better in several milliseconds, and E-14 - in a fraction of a second.

math.PR↗

arXiv2023

Efficient inverse $Z$-transform: sufficient conditions

We derive several sets of sufficient conditions for applicability of the new efficient numerical realization of the inverse $Z$-transform. For large $n$, the complexity of the new scheme is dozens of times smaller than the complexity of the trapezoid rule. As applications, pricing of European options and single barrier options with discrete monitoring are considered; applications to more general options with barrier-lookback features are outlined. In the case of sectorial transition operators, hence, for symmetric Lévy models, the proof is straightforward. In the case of non-symmetric Lévy models, we construct a non-linear deformation of the dual space, which makes the transition operator sectorial, with an arbitrary small opening angle, and justify the new realization. We impose mild conditions which are satisfied for wide classes of non-symmetric Stieltjes-Lévy processes.

math.PR↗

arXiv2022

Efficient evaluation of double-barrier options and joint cpdf of a Lévy process and its two extrema

In the paper, we develop a very fast and accurate method for pricing double barrier options with continuous monitoring in wide classes of Lévy models; the calculations are in the dual space, and the Wiener-Hopf factorization is used. For wide regions in the parameter space, the precision of the order of $10^{-15}$ is achievable in seconds, and of the order of $10^{-9}-10^{-8}$ - in fractions of a second. The Wiener-Hopf factors and repeated integrals in the pricing formulas are calculated using sinh-deformations of the lines of integration, the corresponding changes of variables and the simplified trapezoid rule. If the Bromwich integral is calculated using the Gaver-Wynn Rho acceleration instead of the sinh-acceleration, the CPU time is typically smaller but the precision is of the order of $10^{-9}-10^{-6}$, at best. Explicit pricing algorithms and numerical examples are for no-touch options, digitals (equivalently, for the joint distribution function of a Lévy process and its supremum and infimum processes), and call options. Several graphs are produced to explain fundamental difficulties for accurate pricing of barrier options using time discretization and interpolation-based calculations in the state space.

q-fin.CP↗

arXiv2022

Efficient evaluation of expectations of functions of a stable Lévy process and its extremum

Integral representations for expectations of functions of a stable Lévy process $X$ and its supremum $\bar X$ are derived. As examples, cumulative probability distribution functions (cpdf) of $X_T, \barX_T$, the joint cpdf of $X_T$ and $\barX_T$, and the expectation of $(\be X_T-\barX_T)_+$, $\be>1$, are considered, and efficient numerical procedures for cpdfs are developed. The most efficient numerical methods use the conformal acceleration technique and simplified trapezoid rule.

math.PR↗

arXiv2022

Efficient inverse $Z$-transform and pricing barrier and lookback options with discrete monitoring

We prove simple general formulas for expectations of functions of a random walk and its running extremum. Under additional conditions, we derive analytical formulas using the inverse $Z$-transform, the Fourier/Laplace inversion and Wiener-Hopf factorization, and discuss efficient numerical methods for realization of these formulas. As applications, the cumulative probability distribution function of the process and its running maximum and the price of the option to exchange the power of a stock for its maximum are calculated. The most efficient numerical methods use a new efficient numerical realization of the inverse $Z$-transform, the sinh-acceleration technique and simplified trapezoid rule. The program in Matlab running on a Mac with moderate characteristics achieves the precision E-10 and better in several dozen of milliseconds, and E-14 - in a fraction of a isecond.

math.NA↗

arXiv2022

Lévy models amenable to efficient calculations

In our previous publications (IJTAF 2019, Math. Finance 2020), we introduced a general class of SINH-regular processes and demonstrated that efficient numerical methods for the evaluation of the Wiener-Hopf factors and various probability distributions (prices of options of several types) in Lévy models can be developed using only a few general properties of the characteristic exponent $ψ$. Essentially all popular Lévy processes enjoy these properties. In the present paper, we define classes of Stieltjes-Lévy processes (SL-processes) as processes with completely monotone Lévy densities of positive and negative jumps, and signed Stieltjes-Lévy processes (sSL-processes) as processes with densities representable as differences of completely monotone densities. We demonstrate that 1) all crucial properties of $ψ$ are consequences of the representation $ψ(ξ)=(a^+_2ξ^2-ia^+_1ξ)ST(\cG_+)(-iξ)+(a^-_2ξ^2+ia^-_1ξ)ST(\cG_-)(iξ)+(\sg^2/2)ξ^2-iμξ$, where $ST(\cG)$ is the Stieltjes transform of the (signed) Stieltjes measure $\cG$ and $a^\pm_j\ge 0$; 2) essentially all popular processes other than Merton's model and Meixner processes areSL-processes; 3) Meixner processes are sSL-processes; 4) under a natural symmetry condition, essentially all popular classes of Lévy processes are SL- or sSL-subordinated Brownian motion.

math.PR↗

arXiv2021

SINH-acceleration for B-spline projection with Option Pricing Applications

We clarify the relations among different Fourier-based approaches to option pricing, and improve the B-spline probability density projection method using the sinh-acceleration technique. This allows us to efficiently separate the control of different sources of errors better than the FFT-based realization allows; in many cases, the CPU time decreases as well. We demonstrate the improvement of the B-spline projection method through several numerical experiments in option pricing, including European and barrier options, where the SINH acceleration technique proves to be robust and accurate.

q-fin.CP↗

arXiv2019

Gauge transformations in the dual space, and pricing and estimation in the long run in affine jump-diffusion models

We suggest a simple reduction of pricing European options in affine jump-diffusion models to pricing options with modified payoffs in diffusion models. The procedure is based on the conjugation of the infinitesimal generator of the model with an operator of the form $e^{iΦ(-i\dd_x)}$ (gauge transformation in the dual space). A general procedure for the calculation of the function $Φ$ is given, with examples. As applications, we consider pricing in jump-diffusion models and their subordinated versions using the eigenfunction expansion technique, and estimation of the extremely rare jumps component. The beliefs of the market about yet unobserved extreme jumps and pricing kernel can be recovered: the market prices allow one to see "the shape of things to come".

q-fin.CP↗

arXiv2018

SINH-acceleration: efficient evaluation of probability distributions, option pricing, and Monte-Carlo simulations

Characteristic functions of several popular classes of distributions and processes admit analytic continuation into unions of strips and open coni around $\mathbb{R}\subset \mathbb{C}$. The Fourier transform techniques reduces calculation of probability distributions and option prices to evaluation of integrals whose integrands are analytic in domains enjoying these properties. In the paper, we suggest to use changes of variables of the form $ξ=\sqrt{-1}ω_1+b\sinh (\sqrt{-1}ω+y)$ and the simplified trapezoid rule to evaluate the integrals accurately and fast. We formulate the general scheme, and apply the scheme for calculation probability distributions and pricing European options in Lévy models, the Heston model, the CIR model, and a Lévy model with the CIR-subordinator. We outline applications to fast and accurate calibration procedures and Monte Carlo simulations in Lévy models, regime switching Lévy models that can account for stochastic drift, volatility and skewness, and the Heston model. For calculation of quantiles in the tails using the Newton or bisection method, it suffices to precalculate several hundred of values of the characteristic exponent at points of an appropriate grid ({\em conformal principal components}) and use these values in formulas for cpdf and pdf.

q-fin.CP↗

arXiv2018

Conformal accelerations method and efficient evaluation of stable distributions, revisited

We introduce new efficient integral representations and methods for evaluation of pdfs, cpds and quantiles of stable distributions. For wide regions in the parameter space, absolute errors of order $10^{-15}$ can be achieved in 0.005-0.1 msec (Matlab implementation), even when the index of the distribution is small or close to 1. For the calculation of quantiles in wide regions in the tails using the Newton or bisection method, it suffices to precompute several hundred values of the characteristic exponent at points of an appropriate grid (conformal principal components) and use these values in formulas for cpdf and pdf, which require a fairly small number of elementary operations. The methods of the paper are applicable to other classes of integrals, especially highly oscillatory ones, and are typically faster than the popular methods.

math.NA↗