Conditional Characteristic Function Research Articles

Some bivariate options pricing in a regime-switching stochastic volatility jump-diffusion model with stochastic intensity, stochastic interest and dependent jump

This paper investigates the performance of bivariate options in the hypothesis of association between two underlying assets. Instead of the classical jump-diffusion process, the volatility of assets and the intensity of Poisson co-jump are both subject to the regime-switching square root process in this price dynamics. The endogenous and exogenous interest rate processes are introduced to examine the effect of interest rate on bivariate options pricing, respectively. An analytic pricing expression of bivariate options are deduced by joint discounted conditional characteristic function. Furthermore, the Fourier cosine expansion method is applied to obtain the approximated solutions of bivariate options price. Simulation and numerical examples are realized to examine the effect of the proposed model, the Fourier cosine expansion method, and the sensitivity of key arguments. The results indicate that embedding stochastic intensity, dependent structure of co-jump, and Markov regime-switching into the pricing dynamics have a significant influence on option pricing, and options prices are robust with respect to the choice of interest rate process.

Exact perturbation approximations for the conditional moments of a multifactor CIR term structure model with a weak mean-reversion influence

In this paper, we derive exact series expressions for the conditional moments of a multivariate Cox-Ingersoll-Ross term structure model with a weak mean-reversion effect assumption. First, we construct the perturbation solution for the system of Riccati equations, which illustrates the conditional characteristic function of this term structure model. Second, we derive the power series expression for the perturbation solution. Via the power series solutions, we establish the exact series expressions for conditional moments. Furthermore, we demonstrate the error bounds for these perturbation approximations of the conditional moments. Numerical analysis indicates that our method is more efficient, accurate and superior to the Monte Carlo approach with the Euler scheme for computing the conditional moments.

A Slicing‐Free Perspective to Sufficient Dimension Reduction: Selective Review and Recent Developments

SummarySince the pioneering work of sliced inverse regression, sufficient dimension reduction has been growing into a mature field in statistics and it has broad applications to regression diagnostics, data visualisation, image processing and machine learning. In this paper, we provide a review of several popular inverse regression methods, including sliced inverse regression (SIR) method and principal hessian directions (PHD) method. In addition, we adopt a conditional characteristic function approach and develop a new class of slicing‐free methods, which are parallel to the classical SIR and PHD, and are named weighted inverse regression ensemble (WIRE) and weighted PHD (WPHD), respectively. Relationship with recently developed martingale difference divergence matrix is also revealed. Numerical studies and a real data example show that the proposed slicing‐free alternatives have superior performance than SIR and PHD.

Forward starting options pricing under a regime-switching jump-diffusion model with Wishart stochastic volatility and stochastic interest rate

Vanilla options become effective immediately after they are entered, while some exotic options will only come to effective some time after they are bought or sold. Forward starting options are one kind of such exotic options started actually at some pre-specified future date. This paper presents an extension of regime-switching jump diffusion model, in which the parameters are driven by a continuous time and stationary Markov chain on a finite state space, by introducing the Wishart process into the instantaneous variance-covariance matrix of the risky asset price and stochastic interest rate. We derive the discounted conditional joint characteristic function and the forward characteristic function of the log-asset price and its the instantaneous variance-covariance process, and thereby the price of forward starting options are well evaluated by the probabilistic approach combined with the Fourier-cosine (COS) method. We also provide efficient Monte Carlo simulation of this proposed model, and simulated solutions to forward starting options pricing within a two-state regime switching framework. Numerical results show that the COS method is accurate and efficient for pricing forward starting options. Finally, we analyse impacts of some main parameters (especially, parameters in the Wishart stochastic volatility) in this proposed model on option prices and Δ values. Also, we consider the forward implied volatility. Furthermore, the forward starting options under the regime-switching jump diffusion model with Wishart stochastic volatility and stochastic interest rate which we derived are more generalized than those recently appeared in the derivatives pricing literature, and thus have wider application.

Pricing Variance Swaps under MRG Model with Regime-Switching: Discrete Observations Case

In this paper, we creatively price the discretely sampled variance swaps under the mean-reverting Gaussian model (MRG model in short) with regime-switching asymmetric double exponential jump diffusion. We extend the traditional MRG model by further considering the trend of the financial market as well as a sudden and unexpected event of the market. This new model is meaningful because it uses observable Markov chains that represent market states to adjust its parameters, which helps capture the movement of the market and fluctuations in asset prices. By utilizing the characteristic function and the conditional transition characteristic function, we obtain analytical solutions for pricing formulae. Note that this is our first effort to provide the analytical solution for the ordinary differential equations satisfied by the Feynman–Kac theorem. To achieve this, we have developed a new methodology in Proposition 2 that involves dividing the sampling interval into more detailed switching and non-switching intervals. One significant advantage of our closed-form solution is its high computational accuracy and efficiency. Subsequent semi-Monte Carlo simulations will provide specific validation results.

Evaluation of integrals with fractional Brownian motion for different Hurst indices

In this paper, we will evaluate integrals that define the conditional expectation, variance and characteristic function of stochastic processes with respect to fractional Brownian motion (fBm) for all relevant Hurst indices, i.e. . Particularly, the fractional Ornstein–Uhlenbeck (fOU) process gives rise to highly nontrivial integration formulas that need careful analysis when considering the whole range of Hurst indices. We will show that the classical technique of analytic continuation, from complex analysis, provides a way of extending the domain of validity of an integral from to the larger domain . Numerical experiments for different Hurst indices confirm the robustness and efficiency of the integral formulations presented. Moreover, we provide accurate and highly efficient financial option pricing results for processes that are related to the fOU process, with the help of Fourier cosine expansions.

Determination of the characteristic function of discrete-time conditional linear random process and its application

The discrete-time conditional linear random process is defined, and its properties in the context of application for mathematical modelling of information signals in energy and medicine are analyzed. The relation to the continuous-time counterpart is considered on the basis of time sampling and aggregation. One-dimensional and multidimensional characteristic functions of discrete-time conditional linear random process are obtained using conditional characteristic function approach. The conditions for the investigated model to be strict sense stationary are justified.

A numerical inversion of the bivariate characteristic function

We propose a numerical algorithm for the inversion of the bivariate characteristic function. This will allow the complex probability distribution specified by the characteristic function to be used in practise. Subsequently, it will be possible to create numerical algorithms for a copula function. We will also propose an algorithm for generating random numbers for the case where the bivariate distribution is specified by its characteristic function. This algorithm will be based on the conditional characteristic function. The concept and application of the algorithms will be illustrated using a version of the bivariate logistic distribution specified by its characteristic function.

Cost-effective optimization strategies and sampling plan for Weibull quantiles under type-II censoring

This article introduces a conditional reliability sampling plan for the Weibull distribution. The plan is applicable when the interested quality characteristic is a lower quantile and the life data are observed according to a progressive type-II censoring scheme. It is called “conditional”, since its operating characteristic function is derived by conditioning on a vector of ancillary statistics. We approximate the conditional operating characteristic function by a fixed vector that does not change by the sample. The plan is optimally designed by concentrating on economic considerations while the producer’s and consumer’s confidences are satisfied. In this regard, three cost models are proposed. The first cost function is constructed based on the inspection cost, while the second model accounts for the inaccurate estimation cost. The third cost function is a weighted-relative combination of the other two costs. Based on some combinations of requirements, the optimal plan parameters are prepared in some tables. We further conduct a series of simulations to show that the approximated conditional operating characteristic function can ensure the inspection’s risks well. Finally, an example based on the well-known data set on the strength of carbon fibers is discussed to demonstrate the applicability of the proposed plan.

Embedding non-linear filtering in autonomous underwater vehicle dynamics via the Kolmogorov backward equation

This paper revisits the state vector of an autonomous underwater vehicle (AUV) dynamics coupled with the underwater Markovian stochasticity in the ‘non-linear filtering’ context. The underwater stochasticity is attributed to atmospheric turbulence, planetary interactions, sea surface conditions and astronomical phenomena. In this paper, we adopt the Itô process, a homogeneous Markov process, to describe the AUV state vector evolution equation. This paper accounts for the process noise as well as observation noise correction terms by considering the underwater filtering model. The non-linear filtering of the paper is achieved using the Kolmogorov backward equation and the evolution of the conditional characteristic function. The non-linear filtering equation is the cornerstone formalism of stochastic optimal control systems. Most notably, this paper introduces the non-linear filtering theory into an underwater vehicle stochastic system by constructing a lemma and a theorem for the underwater vehicle stochastic differential equation that were not available in the literature.

Key Technique of Almost Exact Simulation for Non-affine Heston Model

In order to sample asset price more accurately under the non-affine Heston model in the situation where the Feller condition was unsatisfied, we proposed the key technique of almost exact simulation for non-affine Heston model, by fusing the approximate analytic solution of conditional variance characteristic function, the second-order Newton’s method and interpolation method. Numerical results on four representative benchmarks show that our newly proposed simulation scheme has both good convergence and accuracy, which are much better than those of the Itô-Taylor schemes. Especially, our scheme performs well in the case where the Feller condition is extremely unsatisfied, while the alternatives don’t work.

Nonparametric spot volatility from options

We propose a nonparametric estimator of spot volatility from noisy short-dated option data. The estimator is based on forming portfolios of options with different strikes that replicate the (risk-neutral) conditional characteristic function of the underlying price in a model-free way. The separation of volatility from jumps is done by making use of the dominant role of the volatility in the conditional characteristic function over short time intervals and for large values of the characteristic exponent. The latter is chosen in an adaptive way in order to account for the time-varying volatility. We show that the volatility estimator is near rate-optimal in minimax sense. We further derive a feasible joint central limit theorem for the proposed option-based volatility estimator and existing high-frequency return-based volatility estimators. The limit distribution is mixed Gaussian reflecting the time-varying precision in the volatility recovery.

Affine processes beyond stochastic continuity

In this paper, we study time-inhomogeneous affine processes beyond the common assumption of stochastic continuity. In this setting, times of jumps can be both inaccessible and predictable. To this end, we develop a general theory of finite dimensional affine semimartingales under very weak assumptions. We show that the corresponding semimartingale characteristics have affine form and that the conditional characteristic function can be represented with solutions to measure differential equations of Riccati type. We prove existence of affine Markov processes and affine semimartingales under mild conditions and elaborate on examples and applications including affine processes in discrete time.

Error bounds for the perturbation solution of the transition density under a multi-factor CIR term structure model with weak mean-reversion effect

We consider a multi-factor Cox-Ingersoll-Ross (CIR) model of the term structure of interest rates with weak mean-reversion effect. We use perturbation theory to analyze its conditional characteristic function illustrated by a system of Riccati equations and derive the error bounds for the perturbation approximations. Using the Fourier inversion theorem, we clarify that the perturbation approximation of the conditional characteristic function can be applied to estimate the transition density and likelihood function. We provide their error bounds and accuracy orders. Finally, we discuss the performance of the perturbation approximation in estimating the transition density via simulation.

Nonparametric implied Lévy densities

This paper develops a nonparametric estimator for the Lévy density of an asset price, following an Itô semimartingale, implied by short-maturity options. The asymptotic setup is one in which the time to maturity of the available options decreases, the mesh of the available strike grid shrinks and the strike range expands. The estimation is based on aggregating the observed option data into nonparametric estimates of the conditional characteristic function of the return distribution, the derivatives of which allow to infer the Fourier transform of a known transform of the Lévy density in a way which is robust to the level of the unknown diffusive volatility of the asset price. The Lévy density estimate is then constructed via Fourier inversion. We derive an asymptotic bound for the integrated squared error of the estimator in the general case as well as its probability limit in the special Lévy case. We further show rate optimality of our Lévy density estimator in a minimax sense. An empirical application to market index options reveals relative stability of the left tail decay during high and low volatility periods.

Skew CIR process, conditional characteristic function, moments and bond pricing

This paper is concerned with one general Feller’s Branching Diffusion, called skew CIR process. We derive the conditional characteristic function and moment of this general diffusion process first. Then with the same computing idea, we handle with its application in bond pricing. All the results we adopt are closed forms.

Bond valuation for generalized Langevin processes with integrated Lévy noise

PurposeRecently, Stein et al. (2016) studied theoretical properties and parameter estimation of continuous time processes derived as solutions of a generalized Langevin equation (GLE). In this paper, the authors extend the model to a wider class of memory kernels and then propose a bond and bond option valuation model based on the extension of the generalized Langevin process of Stein et al. (2016).Design/methodology/approachBond and bond option pricing based on the proposed interest rate models presents new difficulties as the standard partial differential equation method of stochastic calculus for bond pricing cannot be used directly. The authors obtain bond and bond option prices by finding the closed form expression of the conditional characteristic function of the integrated short rate process driven by a general Lévy noise.FindingsThe authors obtain zero-coupon default-free bond and bond option prices for short rate models driven by a variety of Lévy processes, which include Vasicek model and the short rate model obtained by solving a second-order Langevin stochastic differential equation (SDE) as special cases.Originality/valueBond and bond option pricing plays an important role in capital markets and risk management. In this paper, the authors derive closed form expressions for bond and bond option prices for a wider class of interest rate models including second-order SDE models. Closed form expressions may be especially instrumental in facilitating parameter estimation in these models.

Exact Simulation of the Wishart Multidimensional Stochastic Volatility Model

In this article, we propose an exact simulation method of the Wishart multidimensional stochastic volatility (WMSV) model—a single asset model with a multidimensional Wishart variance process. Our method is based on analysis of the conditional characteristic function of the log-price given a terminal volatility level. In particular, we found an explicit expression for the conditional characteristic function for the Heston model. Numerical experiments demonstrate that our new method is much faster and reliable than the Euler discretization method.

Option Pricing Under a Stochastic Interest Rate and Volatility Model with Hidden Markovian Regime-Switching

In this paper we discuss an option pricing problem in a hidden Markovian regime-switching model with a stochastic interest rate and volatility. Regime switches are attributed to structural changes in an hidden economic environment and are described by a continuous-time, finite-state, unobservable Markov chain. The model is then applied to the valuation of a standard European option. By means of the standard separation principle, filtering and option valuation problems are separated. Robust filters for the hidden states of the economy and their robust filtered estimates of unknown parameters from the expectation maximization algorithm are presented based on standard techniques in filtering theory. Then an explicit expression of a conditional characteristic function relevant to option pricing is presented and the valuation of the option is discussed using the inverse Fourier transformation approach. Using the limiting behavior of the conditional characteristic function, an efficient implementation of the transform inversion integral is considered. Numerical experiments are given to illustrate the flexibility of filtering algorithms and the significance of regime-switching in option pricing.

A kernel estimate method for characteristic function-based uncertainty importance measure

In this paper, we propose a fast computation method based on a kernel function for the characteristic function-based moment-independent uncertainty importance measure θi. We first point out that the possible computational complexity problems that exist in the estimation of θi. Since the convergence rate of a double-loop Monte Carlo (MC) simulation is O(N−1/4), the first possible problem is the use of double-loop MC simulation. And because the norm of the difference between the unconditional and conditional characteristic function of model output in θi is a Lebesgue integral over the infinite interval, another possible problem is the computation of this norm. Then a kernel function is introduced to avoid the use of double-loop MC simulation, and a longer enough bounded interval is selected to instead of the infinite interval to calculate the norm. According to these improvements, a kind of fast computational methods is introduced for θi, and during the whole process, all θi can be obtained by using a single quasi-MC sequence. From the comparison of numerical error analysis, it can be shown that the proposed method is an effective and helpful approach for computing the characteristic function-based moment-independent importance index θi.