arXiv research

In quantitative finance, we often model asset prices as a noisy Ito semimartingale. As this model is not identifiable, approximating by a time-changed Levy process can be useful for generative modelling. We give a new estimate of the normalised volatility or time change in this model, which obtains minimax convergence …

In this paper we propose a general derivative pricing framework which employs decoupled time-changed (DTC) Lévy processes to model the underlying asset of contingent claims. A DTC Lévy process is a generalized time-changed Lévy process whose continuous and pure jump parts are allowed to follow separate random time scal…

The paper prices weather contracts using a complex temperature model.

Carr and Wu (2004), henceforth CW, developed a framework that encompasses almost all of the continuous-time models proposed in the option pricing literature. Their framework hinges on the stopping time property of the time changes. By analyzing the measurability of the time changes with respect to the underlying filtra…

Develops a new model for multi-currency volatility using CBI-time-changed Lévy processes.

New GLPs split Lévy bridges into non-overlapping subprocesses.

We compute the value of a variance swap when the underlying is modeled as a Markov process time changed by a Lévy subordinator. In this framework, the underlying may exhibit jumps with a state-dependent Lévy measure, local stochastic volatility and have a local stochastic default intensity. Moreover, the Lévy subordina…

The paper improves energy contract pricing models by incorporating jumps and varying parameters.

This paper proposes a new model for SPX and VIX derivatives markets.

This paper extends subordinated models to include stochastic time changes, improving financial modeling.

We develop a comprehensive mathematical framework for polynomial jump-diffusions in a semimartingale context, which nest affine jump-diffusions and have broad applications in finance. We show that the polynomial property is preserved under polynomial transformations and Lévy time change. We present a generic method for…

The present paper introduces a jump-diffusion extension of the classical diffusion default intensity model by means of subordination in the sense of Bochner. We start from the bi-variate process $(X,D)$ of a diffusion state variable $X$ driving default intensity and a default indicator process $D$ and time change it wi…

This paper studies subordinate Ornstein-Uhlenbeck (OU) processes, i.e., OU diffusions time changed by Lévy subordinators. We construct their sample path decomposition, show that they possess mean-reverting jumps, study their equivalent measure transformations, and the spectral representation of their transition semigro…

We derive asymptotic expansions for option data to detect infinite variation volatility.

We derive a small-time expansion for out-of-the-money call options under an exponential Levy model, using the small-time expansion for the distribution function given in Figueroa-Lopez & Houdre (2009), combined with a change of numéraire via the Esscher transform. In particular, we quantify find that the effect of a no…

We prove that the variance swap rate (fair strike) equals the price of a co-terminal European-style contract when the underlying is an exponential Markov process, time-changed by an arbitrary continuous stochastic clock, which has arbitrary correlation with the driving Markov process, provided that the payoff function …

Study uses AI to price exotic options with a new Levy process model.

Study shows subordinated Cramér-Lundberg model increases ruin probability.

For a given Markov process $X$ and survival function $\overline{H}$ on $\mathbb{R}^+$, the inverse first-passage time problem (IFPT) is to find a barrier function $b:\mathbb{R}^+\to[-\infty,+\infty]$ such that the survival function of the first-passage time $τ_b=\inf \{t\ge0:X(t)<b(t)\}$ is given by $\overline{H}$. In …

We consider a square-integrable semimartingale and investigate the convex order relations between its discrete, continuous and predictable quadratic variation. As the main results, we show that if the semimartingale has conditionally independent increments and symmetric jump measure, then its discrete realized variance…

Subordination is an often used stochastic process in modeling asset prices. Subordinated Levy price processes and local volatility price processes are now the main tools in modern dynamic asset pricing theory. In this paper, we introduce the theory of multiple internally embedded financial time-clocks motivated by beha…

Extends Gaussian process regression for non-Gaussian data.

We introduce a class of randomly time-changed fast mean-reverting stochastic volatility models and, using spectral theory and singular perturbation techniques, we derive an approximation for the prices of European options in this setting. Three examples of random time-changes are provided and the implied volatility sur…

We derive precise transformation formulas for synthetic lower Ricci bounds under time change. More precisely, for local Dirichlet forms we study how the curvature-dimension condition in the sense of Bakry-Emery will transform under time change. Similarly, for metric measure spaces we study how the curvature-dimension c…

We present a new and easy-to-implement sequential sampling method for CGMY processes with either finite or infinite variation, exploiting the time change representation of the CGMY model and a decomposition of its time change. We find that the time change can be decomposed into two independent components. While the fir…

Improved stochastic clocks for financial models without increasing trades.

We prove here a general closed-form expansion formula for forward-start options and the forward implied volatility smile in a large class of models, including the Heston stochastic volatility and time-changed exponential Lévy models. This expansion applies to both small and large maturities and is based solely on the p…

Motivated by the interplay between structural and reduced form credit models, we propose to model the firm value process as a time-changed Brownian motion that may include jumps and stochastic volatility effects, and to study the first passage problem for such processes. We are lead to consider modifying the standard f…

The Kelly rule fails to maximize growth in a time-changed return setting.

We propose an efficient method to evaluate callable and putable bonds under a wide class of interest rate models, including the popular short rate diffusion models, as well as their time changed versions with jumps. The method is based on the eigenfunction expansion of the pricing operator. Given the set of call and pu…

The paper approximates CARMA models for option pricing.

Recently Carr and Wu (2004, 2005) and also Huang and Wu (2004) show that most stochastic processes used in traditional option pricing models can be cast as special cases of time-changed Lévy processes. In particular these are models which can be tailored to exhibit correlated jumps in both the log price of assets and t…

We address the problem of parameter estimation for diffusion driven stochastic volatility models through Markov chain Monte Carlo (MCMC). To avoid degeneracy issues we introduce an innovative reparametrisation defined through transformations that operate on the time scale of the diffusion. A novel MCMC scheme which ove…

The paper optimizes RV estimation by efficient sampling in time-changed diffusion models.

The aim of this paper is to evaluate geometric Asian option by a mixed fractional subdiffusive Black-Scholes model. We derive a pricing formula for geometric Asian option when the underlying stock follows a time changed mixed fractional Brownian motion. We then apply the results to price Asian power options on the stoc…

New findings show independent subordination is not relevant for accurate option pricing.

We consider structural credit modeling in the important special case where the log-leverage ratio of the firm is a time-changed Brownian motion (TCBM) with the time-change taken to be an independent increasing process. Following the approach of Black and Cox, one defines the time of default to be the first passage time…

We consider flows, called $W^{\rm u}$ flows, whose orbits are the unstable manifolds of a codimension one Anosov flow. Under some regularity assumptions, we give a short proof of the strong mixing property of $W^{\rm u}$ flows and we show that $W^{\rm u}$ flows have purely absolutely continuous spectrum in the orthocom…

The accurate prediction of time-changing covariances is an important problem in the modeling of multivariate financial data. However, some of the most popular models suffer from a) overfitting problems and multiple local optima, b) failure to capture shifts in market conditions and c) large computational costs. To addr…

We address the so-called calibration problem which consists of fitting in a tractable way a given model to a specified term structure like, e.g., yield or default probability curves. Time-homogeneous jump-diffusions like Vasicek or Cox-Ingersoll-Ross (possibly coupled with compounded Poisson jumps, JCIR), are tractable…

The accurate prediction of time-changing variances is an important task in the modeling of financial data. Standard econometric models are often limited as they assume rigid functional relationships for the variances. Moreover, function parameters are usually learned using maximum likelihood, which can lead to overfitt…

No-arbitrage models of term structure have the feature that the return on zero-coupon bonds is the sum of the short rate and the product of volatility and market price of risk. Well known models restrict the behavior of the market price of risk so that it is not dependent on the type of asset being modeled. We show tha…

TCNF models SDEs using time deformation of Brownian motion.

Credit Valuation Adjustment (CVA) pricing models need to be both flexible and tractable. The survival probability has to be known in closed form (for calibration purposes), the model should be able to fit any valid Credit Default Swap (CDS) curve, should lead to large volatilities (in line with CDS options) and finally…

Study reveals finite-size effects and sensitivity to random numbers in Levy-Levy-Solomon model.

We present a new numerical method to price vanilla options quickly in time-changed Brownian motion models. The method is based on rational function approximations of the Black-Scholes formula. Detailed numerical results are given for a number of widely used models. In particular, we use the variance-gamma model, the CG…

Develops information geometry for Lévy processes in finance.

These lectures notes aim at introducing Lévy processes in an informal and intuitive way, accessible to non-specialists in the field. In the first part, we focus on the theory of Lévy processes. We analyze a `toy' example of a Lévy process, viz. a Lévy jump-diffusion, which yet offers significant insight into the distri…