Class Of Stochastic Volatility Models Research Articles

Multivariate Stochastic Volatility Modeling via Integrated Nested Laplace Approximations: A Multifactor Extension

This study introduces a multivariate extension to the class of stochastic volatility models, employing integrated nested Laplace approximations (INLA) for estimation. Bayesian methods for estimating stochastic volatility models through Markov Chain Monte Carlo (MCMC) can become computationally burdensome or inefficient as the dataset size and problem complexity increase. Furthermore, issues related to chain convergence can also arise. In light of these challenges, this research aims to establish a computationally efficient approach for estimating multivariate stochastic volatility models. We propose a multifactor formulation estimated using the INLA methodology, enabling an approach that leverages sparse linear algebra and parallelization techniques. To evaluate the effectiveness of our proposed model, we conduct in-sample and out-of-sample empirical analyses of stock market index return series. Furthermore, we provide a comparative analysis with models estimated using MCMC, demonstrating the computational efficiency and goodness of fit improvements achieved with our approach.

Inference on the maximal rank of time-varying covariance matrices using high-frequency data

We study the rank of the instantaneous or spot covariance matrix ΣX(t) of a multidimensional process X(t). Given high-frequency observations X(i/n), i=0,…,n, we test the null hypothesis rank(ΣX(t))≤r for all t against local alternatives where the average (r+1)st eigenvalue is larger than some signal detection rate vn. A major problem is that the inherent averaging in local covariance statistics produces a bias that distorts the rank statistics. We show that the bias depends on the regularity and spectral gap of ΣX(t). We establish explicit matrix perturbation and concentration results that provide nonasymptotic uniform critical values and optimal signal detection rates vn. This leads to a rank estimation method via sequential testing. For a class of stochastic volatility models, we determine data-driven critical values via normed p-variations of estimated local covariance matrices. The methods are illustrated by simulations and an application to high-frequency data of U.S. government bonds.

Optional projection under equivalent local martingale measures

We study optional projections of {mathbb{G}}-adapted strict local martingales on a smaller filtration {mathbb{F}} under changes of equivalent martingale measures. General results are provided as well as a detailed analysis of two specific examples given by the inverse Bessel process and a class of stochastic volatility models. This analysis contributes to clarify the absence of arbitrage opportunities of market models under restricted information.

Simple Factor Realized Stochastic Volatility Models

Abstract This paper considers the use of multiple noisy daily realized variance measures to extract a denoised latent variance process. The class of stochastic volatility models used for signal extraction has the important feature that they can be written as a linear state space model. As a result, prediction of the denoised latent variance and likelihood evaluation can be carried out efficiently using the Kalman filter. This is in contrast to stochastic models that jointly model the return and variance, which require computationally expensive nonlinear filtering for prediction and inference. The gain from using multiple noisy daily variance measures is examined empirically for the S&P 500 index using daily OHLC (open-high-low-close) data.

Nonparametric Pricing and Hedging of Volatility Swaps in Stochastic Volatility Models

In this paper the zero vanna implied volatility approximation for the price of freshly minted volatility swaps is generalised to seasoned volatility swaps. We also derive how volatility swaps can be hedged using a strip of vanilla options with weights that are directly related to trading intuition. Additionally, we derive first and second order hedges for volatility swaps using only variance swaps. As dynamically trading variance swaps is in general cheaper and operationally less cumbersome compared to dynamically rebalancing a continuous strip of options, our result makes the hedging of volatility swaps both practically feasible and robust. Within the class of stochastic volatility models our pricing and hedging results are model-independent and can be implemented at almost no computational cost.

Jacobi stochastic volatility factor for the LIBOR market model

We propose a new method to efficiently price swap rate derivatives under the LIBOR market model with stochastic volatility and displaced diffusion. This method applies series expansion techniques built around Gaussian (Gram–Charlier) or Gaussian mixture densities to polynomial processes. The standard pricing method for the considered model relies on dynamics freezing to recover a Heston-type model for which analytical formulas are available. This approach is time-consuming, and efficient approximations based on Gram–Charlier expansions have been proposed recently. In this article, we first discuss the fact that for a class of stochastic volatility model, including the Heston one, the classical sufficient condition ensuring the convergence of Gram–Charlier series is not satisfied. Then we propose an approximating model based on a Jacobi process for which we can prove the stability of Gram–Charlier-type expansions. For this approximation, we have been able to prove a strong convergence towards the original model; moreover, we give an estimate of the convergence rate. We also prove a new result on the convergence of the Gram–Charlier series when the volatility factor is not bounded from below. We finally illustrate our convergence results with numerical examples.

Pricing Options with Vanishing Stochastic Volatility

In the past years, there has been an extensive investigation of the class of stochastic volatility models for the evaluation of options and complex derivatives. These models have proven to be extremely useful in generalizing the classic Black–Scholes economy and accounting for discrepancies between observation and predictions in the simple log-normal, constant-volatility model. In this paper, we study the structure of an options market with a stochastic volatility that will eventually vanish (i.e., reaches zero) for very short periods of time with probability of one. We investigate the form of pricing measures in this situation, first in a simple binomial case, and then for a diffusion model, by constructing a weak approximation in discrete space and continuous time. The market described allows fleeting arbitrage opportunities, since a vanishing volatility prevents the construction of an equivalent measure, so that pricing contingent claims are, a priori, not obvious. Nevertheless, we can still produce a fair pricing equation. Let us note that this issue is not only of theoretical relevance, as the phenomenon of very low volatility has indeed been observed in the financial markets and the economy for quite a long time in the recent past.

Pricing European Options in the Heston and the Double Heston models

Two models of pricing European options are presented and compared in this paper, i.e. the Heston model and the double Heston model. As the models belong to the class of stochastic volatility models, particular attention is paid to the way the characteristic functions and their inverse Fourier transforms are determined. The aim of the study is to investigate computational efficiency of pricing European calls. The method applied is based on the assumption that the prices of the derivatives are evaluated by means of Gauss-Kronrod quadrature.

Pricing timer options: second-order multiscale stochastic volatility asymptotics

We study the pricing of timer options in a class of stochastic volatility models, where the volatility is driven by two diffusions—one fast mean-reverting and the other slowly varying. Employing singular and regular perturbation techniques, full second-order asymptotics of the option price are established. In addition, we investigate an implied volatility in terms of effective maturity for the timer options, and derive its second-order expansion based on our pricing asymptotics. A numerical experiment shows that the price approximation formula has a high level of accuracy, and the implied volatility in terms of its effective maturity is illustrated. doi:10.1017/S1446181121000249

Analysis of VIX-linked fee incentives in variable annuities via continuous-time Markov chain approximation

We consider the pricing of variable annuities (VAs) with general fee structures under a class of stochastic volatility models which includes the Heston, Hull-White, Scott, α-Hypergeometric, 3/2, and 4/2 models. In particular, we analyze the impact of different VIX-linked fee structures on the optimal surrender strategy of a VA contract with guaranteed minimum maturity benefit (GMMB). Under the assumption that the VA contract can be surrendered before maturity, the pricing of a VA contract corresponds to an optimal stopping problem with an unbounded, time-dependent, and discontinuous payoff function. We develop efficient algorithms for the pricing of VA contracts using a two-layer continuous-time Markov chain approximation for the fund value process. When the contract is kept until maturity and under a general fee structure, we show that the value of the contract can be approximated by a closed-form matrix expression. We also provide a quick and simple way to determine the value of early surrenders via a recursive algorithm and give an easy procedure to approximate the optimal surrender surface. We show numerically that the optimal surrender strategy is more robust to changes in the volatility of the account value when the fee is linked to the VIX index.

PRICING TIMER OPTIONS: SECOND-ORDER MULTISCALE STOCHASTIC VOLATILITY ASYMPTOTICS

AbstractWe study the pricing of timer options in a class of stochastic volatility models, where the volatility is driven by two diffusions—one fast mean-reverting and the other slowly varying. Employing singular and regular perturbation techniques, full second-order asymptotics of the option price are established. In addition, we investigate an implied volatility in terms of effective maturity for the timer options, and derive its second-order expansion based on our pricing asymptotics. A numerical experiment shows that the price approximation formula has a high level of accuracy, and the implied volatility in terms of its effective maturity is illustrated.

Application of iterated filtering to stochastic volatility models based on non-Gaussian Ornstein-Uhlenbeck process

Abstract Barndorff-Nielsen and Shephard (2001) proposed a class of stochastic volatility models in which the volatility follows the Ornstein–Uhlenbeck process driven by a positive Levy process without the Gaussian component. The parameter estimation of these models is challenging because the likelihood function is not available in a closed-form expression. A large number of estimation techniques have been proposed, mainly based on Bayesian inference. The main aim of the paper is to present an application of iterated filtering for parameter estimation of such models. Iterated filtering is a method for maximum likelihood inference based on a series of filtering operations, which provide a sequence of parameter estimates that converges to the maximum likelihood estimate. An application to S&P500 index data shows the model perform well and diagnostic plots for iterated filtering ensure convergence iterated filtering to maximum likelihood estimates. Empirical application is accompanied by a simulation study that confirms the validity of the approach in the case of Barndorff-Nielsen and Shephard’s stochastic volatility models.

Stochastic volatility models with ARMA innovations: An application to G7 inflation forecasts

We introduce a new class of stochastic volatility models with autoregressive moving average (ARMA) innovations. The conditional mean process has a flexible form that can accommodate both a state space representation and a conventional dynamic regression. The ARMA component introduces serial dependence, which results in standard Kalman filter techniques not being directly applicable. To overcome this hurdle, we develop an efficient posterior simulator that builds on recently developed precision-based algorithms. We assess the usefulness of these new models in an inflation forecasting exercise across all G7 economies. We find that the new models generally provide competitive point and density forecasts compared to standard benchmarks, and are especially useful for Canada, France, Italy, and the U.S.

STOCHASTIC VOLATILITY MODEL WITH CORRELATED JUMP SIZES AND INDEPENDENT ARRIVALS

In light of recent empirical research on jump activity, this article study the calibration of a new class of stochastic volatility models that include both jumps in return and volatility. Specifically, we consider correlated jump sizes and both contemporaneous and independent arrival of jumps in return and volatility. Based on the specifications of this model, we derive a closed-form relationship between the VIX index and latent volatility. Also, we propose a closed-form logarithmic likelihood formula by using the link to the VIX index. By estimating alternative models, we find that the general counting processes setting lead to better capturing of return jump behaviors. That is, the part where the return and volatility jump simultaneously and the part that jump independently can both be captured. In addition, the size of the jumps in volatility is, on average, positive for both contemporaneous and independent arrivals. However, contemporaneous jumps in the return are negative, but independent return jumps are positive. The sub-period analysis further supports above insight, and we find that the jumps in return and volatility increased significantly during the two recent economic crises.

VOLATILITY INFERENCE AND RETURN DEPENDENCIES IN STOCHASTIC VOLATILITY MODELS

Stochastic volatility models describe stock returns [Formula: see text] as driven by an unobserved process capturing the random dynamics of volatility [Formula: see text]. The present paper quantifies how much information about volatility [Formula: see text] and future stock returns can be inferred from past returns in stochastic volatility models in terms of Shannon’s mutual information. In particular, we show that across a wide class of stochastic volatility models, including a two-factor model, returns observed on the scale of seconds would be needed to obtain reliable volatility estimates. In addition, we prove that volatility forecasts beyond several weeks are essentially impossible for fundamental information theoretic reasons.

From (Martingale) Schrodinger Bridges to a New Class of Stochastic Volatility Model

Following closely the construction of the Schrodinger bridge, we build a new class of Stochastic Volatility Models exactly calibrated to market instruments such as for example Vanillas and options on realized variance. These models differ strongly from the well-known local stochastic volatility models, in particular the instantaneous volatility-of-volatility of the associated naked SVMs is not modified, once calibrated to market instruments. They can be interpreted as a martingale version of the Schrodinger bridge. The numerical calibration is performed using a dynamic-like version of the Sinkhorn algorithm. We finally highlight a striking relation with Dyson non-colliding Brownian motions.

Filtering and Estimation for a Class of Stochastic Volatility Models with Intractable Likelihoods

We introduce a new approach to latent state filtering and parameter estimation for a class of stochastic volatility models (SVMs) for which the likelihood function is unknown. The α-stable stochastic volatility model provides a flexible framework for capturing asymmetry and heavy tails, which is useful when modeling financial returns. However, the α-stable distribution lacks a closed form for the probability density function, which prevents the direct application of standard Bayesian filtering and estimation techniques such as sequential Monte Carlo and Markov chain Monte Carlo. To obtain filtered volatility estimates, we develop a novel approximate Bayesian computation (ABC) based auxiliary particle filter, which provides improved performance through better proposal distributions. Further, we propose a new particle based MCMC (PMCMC) method for joint estimation of the parameters and latent volatility states. With respect to other extensions of PMCMC, we introduce an efficient single filter particle Metropolis-within-Gibbs algorithm which can be applied for obtaining inference on the parameters of an asymmetric α-stable stochastic volatility model. We show the increased efficiency in the estimation process through a simulation study. Finally, we highlight the necessity for modeling asymmetric α-stable SVMs through an application to propane weekly spot prices.

Finite Element Methods for Partial Differential Equations for Option Pricing

This manuscript gives introduction to parabolic Partial Differential Equations (PDE) in one-dimensional case. It recalls the notion and main limitations of Black-Scholes model and establishes the Black-Scholes formula. It recalls the standard machinery of solving the parabolic PDE which is well-known in standard literature such as text-book of Bass (1999). Here we recall the notion of infinitesimal generator of the diffusion process and recall some basic well-known results on the infinitesimal generator of the Markov process. We avoid giving complete proofs of known results and provide reader with a short hint of proofs. Here, we recall again the theory of solving the parabolic PDE, we recall machinery of changes of variables and unknown functions. Then, we derive the Black-Scholes formula and give some intuition of application to some classes of options, such as a Vanilla European Call and Put. The next section is devoted to giving some notions of the classical solution to the parabolic PDE. It is immediately followed by the variational framework. The introduction to the reader of weighted Sobolev spaces on which the solution to the class of parabolic PDE under study lives is a subject to the next section in this Chapter. Straight after that we give a notion of weak formulation of the Black-Scholes equation followed by proving the regularity property of the weak solutions, application of the maximum principle for weak solutions which is a powerful tool to provide the solution to parabolic PDE and work further on derivation of various bounds. We give more insight into the localization procedure. We apply this methodology to option pricing, via Put-Call parity relation we derive a payoff function for a European Vanilla Put and Call. Among the further examples are General European Options, Barrier Options, European Options on a Basket of Asset, American Options, Asian Options. The chapter continues with giving some introduction to application of Finite Element methods where we recall the main steps of using this machinery. We recall as well more innovative Finite Element methods for the class of stochastic volatility models.

Zastosowanie filtru Kalmana do modeli stochastycznej zmienności typu Ornsteina‑Uhlenbecka

O. E. Barndorff‑Nielsen i N. Shephard (2001) zaproponowali klasę modeli stochastycznej zmienności typu Ornsteina‑Uhlenbecka, opartych na procesie Lévy’ego bez składnika Gaussowskiego. Estymacja parametrów modeli tego typu jest trudna, ponieważ nie można wyznaczyć odpowiedniej funkcji wiarygodności w postaci jawnego wzoru. W artykule zaprezentowana zostanie propozycja zastosowania filtru Kalmana do wyznaczania estymatorów parametrów w przypadku złożenia kilku procesów zmienności. Podejście to zostanie wykorzystane do modelowania kursu EUR/PLN. Empiryczny przykład uzupełnia eksperyment symulacyjny mający na celu zbadanie własności tak otrzymanych estymatorów.

Isogeometric analysis in option pricing

ABSTRACTIsogeometric analysis is a recently developed computational approach that integrates finite element analysis directly into design described by non-uniform rational B-splines (NURBS). In this paper, we show that price surfaces that occur in option pricing can be easily described by NURBS surfaces. For a class of stochastic volatility models, we develop a methodology for solving corresponding pricing partial integro-differential equations numerically by isogeometric analysis tools and show that a very small number of space discretization steps can be used to obtain sufficiently accurate results. Presented solution by finite element method is especially useful for practitioners dealing with derivatives where closed-form solution is not available.