A hidden Markov regime-switching model for option valuation

We investigate two approaches, namely, the Esscher transform and the extended Girsanov’s principle, for option valuation in a discrete-time hidden Markov regime-switching Gaussian model. The model’s parameters including the interest rate, the appreciation rate and the volatility of a risky asset are governed by a discrete-time, finite-state, hidden Markov chain whose states represent the hidden states of an economy. We give a recursive filter for the hidden Markov chain and estimates of model parameters using a filter-based EM algorithm. We also derive predictors for the hidden Markov chain and some related quantities. These quantities are used to estimate the price of a standard European call option. Numerical examples based on real financial data are provided to illustrate the implementation of the proposed method.     

The estimation of a nonlinear moving average model

We consider discrete-parameter stochastic processes that are the output of a nonlinear filter driven by white noise. For a simple model, we derive estimates of the unknown coefficients in the transfer function and the noise variance, and investigate their asymptotic properties. We prove some lemmas that can also be used to obtain rates of convergence in the weak and strong laws of large numbers, and central limit theorems, for estimates of more general nonlinear models.     

Recursive estimation for continuous time stochastic volatility models

Volatility plays an important role in portfolio management and option pricing. Recently, there has been a growing interest in modeling volatility of the observed process by nonlinear stochastic process [S.J. Taylor, Asset Price Dynamics, Volatility, and Prediction, Princeton University Press, 2005; H. Kawakatsu, Specification and estimation of discrete time quadratic stochastic volatility models, Journal of Empirical Finance 14 (2007) 424–442]. In [H. Gong, A. Thavaneswaran, J. Singh, Filtering for some time series models by using transformation, Math Scientist 33 (2008) 141–147], we have studied the recursive estimates for discrete time stochastic volatility models driven by normal errors. In this paper, we study the recursive estimates for various classes of continuous time nonlinear non-Gaussian stochastic volatility models used for option pricing in finance.     

随机波动模型的马尔可夫链—蒙特卡洛模拟方法——在沪市收益率序列上的应用

针对具有Markov区制转移的、波动均值状态相依的随机波动模型,基于贝叶斯分析,我们推导并给出了对区制转移随机波动模型的MCMC估计方法,其中对参数估计采用Gibbs抽样方法,对潜在对数波动和区制的状态变量估计采用"向前滤波、向后抽样"的多步移动方法;利用该模型,对我国上证综指周收益率进行了实证分析,发现对沪市波动性有较好的描述,捕捉了波动的时变性、聚类性和非线性特征,同时刻画了沪市的高低波动状态转换过程。     

Nonlinear Filter Estimation of Volatility

A discrete time nonlinear filter is used to estimate the volatility in a financial model. New filters are derived for sums of unobserved quantities and the EM algorithm applied to determine the parameters of the model.     

The robust Merton problem of an ambiguity averse investor

We derive a closed form portfolio optimization rule for an investor who is diffident about mean return and volatility estimates, and has a CRRA utility. Confidence is here represented using ellipsoidal uncertainty sets for the drift, given a (compact valued) volatility realization. This specification affords a simple and concise analysis, as the agent becomes observationally equivalent to one with constant, worst case parameters. The result is based on a max–min Hamilton–Jacobi–Bellman–Isaacs PDE, which extends the classical Merton problem and reverts to it for an ambiguity-neutral investor.     

A note on the derivation of filter regularization operators for nonlinear evolution equations

Despite the strong focus of regularization on ill-posed problems, the general construction of such methods has not been fully explored. Moreover, many previous studies cannot be clearly adapted to handle more complex scenarios, albeit the greatly increasing concerns on the improvement of wider classes. In this note, we rigorously study a general theory for filter regularized operators in a Hilbert space for nonlinear evolution equations which have occurred naturally in different areas of science. The starting point lies in problems that are in principle ill-posed with respect to the initial/final data – these basically include the Cauchy problem for nonlinear elliptic equations and the backward-in-time nonlinear parabolic equations. We derive general filters that can be used to stabilize those problems. Essentially, we establish the corresponding well-posed problem whose solution converges to the solution of the ill-posed problem. The approximation can be confirmed by the error estimates in the Hilbert space. This work improves very much many papers in the same field of research.     

Gradient estimates for nonlinear diffusion semigroups by coupling methods

In this paper, we obtain gradient estimates for certain nonlinear partial differential equations by coupling methods. First, we derive uniform gradient estimates for certain semi-linear PDEs based on the coupling method introduced by Wang in 2011 and the theory of backward SDEs. Then we generalize Wang's coupling to the G-expectation space and obtain gradient estimates for nonlinear diffusion semigroups, which correspond to the solutions of certain fully nonlinear PDEs.     

Medium‐term horizon volatility forecasting: A comparative study

In this paper, volatility is estimated and then forecast using unobserved components‐realized volatility (UC‐RV) models as well as constant volatility and GARCH models. With the objective of forecasting medium‐term horizon volatility, various prediction methods are employed: multi‐period prediction, variable sampling intervals and scaling. The optimality of these methods is compared in terms of their forecasting performance. To this end, several UC‐RV models are presented and then calibrated using the Kalman filter. Validation is based on the standard errors on the parameter estimates and a comparison with other models employed in the literature such as constant volatility and GARCH models. Although we have volatility forecasting for the computation of Value‐at‐Risk in mind the methodology presented has wider applications. This investigation into practical volatility forecasting complements the substantial body of work on realized volatility‐based modelling in business. Copyright © 2007 John Wiley & Sons, Ltd.     

Filltering of a partially observed process in the case of a high signal –to–noise ratio for correlated systems

This paper concerns a nonlinear filtering problem with correlated noises in the case of a high signal–to–noise ratio, when only one component of the signal is observed. We compute an approximate filter for the unnormalized filter associated to the system and derive both a Zakai and a Kushner-Stratonovitch type equation for the approximate filter     

Nonlinear continuous-discrete filtering using kernel density estimatesand functional integrals

We develop filter algorithms for nonlinear stochastic differential equations with discrete time measurements (continuous-discrete state space model). The apriori density (time update) is computed by Monte Carlo simulations of the Fokker-Planck equation using kernel density estimators and measurement updates are obtained by using the extended Kalman filter (EKF) updates. For small sampling intervals, a discretized continuous sampling approach (DCS) is used. A third algorithm utilizes a functional (path) integral representation of the transition density (functional integral filter FIF). The kernel density filter (KDF), DCS, and FIF are compared with the EKF and the Gaussian sum filter by using a Ginzburg-Landau-equation and a stochastic volatility model.     

Discrete–time nonlinear filtering with marked point process observations: ii. risk–sensitive filters

We discuss in this article the risk–sensitive filtering problem of estimating a nonlinear signal process, with nonadditive non–Gaussian noise, via a marked point process observation. This extends to the risk sensitive case all the risk–neutral results studied in Dufour and Kannan [2].By going into a change of measure, we derive the unnormalized conditional density of the signal conditioned on the observation history. We also derive the unnormalized prediction density. Using these, we present two separate expressions for the optimal estimate of the signal. A similar analysis of the smoothing density of the signal is also studied under both the risk–sensitive and risk–neutral cases. We specialize the above optimal estimation to the linear signal dynamics and marked point process observation under some Gaussian assumptions. We obtain a Kalman type risk-sensitive filter. Due to the special nature of the observation process, the conditional mean and covariance estimates directly depend now on the point process     

Forgetting the initial distribution for Hidden Markov Models

The forgetting of the initial distribution for discrete Hidden Markov Models (HMM) is addressed: a new set of conditions is proposed, to establish the forgetting property of the filter, at a polynomial and geometric rate. Both a pathwise-type convergence of the total variation distance of the filter started from two different initial distributions, and a convergence in expectation are considered. The results are illustrated using different HMM of interest: the dynamic tobit model, the nonlinear state space model and the stochastic volatility model.     

Numerical Solutions for Option Pricing Models Including Transaction Costs and Stochastic Volatility

The option pricing problem when the asset is driven by a stochastic volatility process and in the presence of transaction costs leads to solving a nonlinear partial differential equation. The nonlinear term in the PDE reflects the presence of transaction costs. Under a particular market completion assumption we derive the nonlinear PDE whose solution may be used to find the price of options. In this paper under suitable conditions, we give an algorithmic scheme to obtain the solution of the problem by an iterative method and provide numerical solutions using the finite difference method.     

Regime-switching stochastic volatility model: estimation and calibration to VIX options

We develop and implement a method for maximum likelihood estimation of a regime-switching stochastic volatility model. Our model uses a continuous time stochastic process for the stock dynamics with the instantaneous variance driven by a Cox–Ingersoll–Ross process and each parameter modulated by a hidden Markov chain. We propose an extension of the EM algorithm through the Baum–Welch implementation to estimate our model and filter the hidden state of the Markov chain while using the VIX index to invert the latent volatility state. Using Monte Carlo simulations, we test the convergence of our algorithm and compare it with an approximate likelihood procedure where the volatility state is replaced by the VIX index. We found that our method is more accurate than the approximate procedure. Then, we apply Fourier methods to derive a semi-analytical expression of S&P500 and VIX option prices, which we calibrate to market data. We show that the model is sufficiently rich to encapsulate important features of the joint dynamics of the stock and the volatility and to consistently fit option market prices.     

Individual homogenization of nonlinear parabolic operators

In this article, we prove an individual homogenization result for a class of almost periodic nonlinear parabolic operators. The spatial and temporal heterogeneities are almost periodic functions in the sense of Besicovitch. The latter allows discontinuities and is suitable for many applications. First, we derive stability and comparison estimates for sequences of G-convergent nonlinear parabolic operators. Furthermore, using these estimates, the individual homogenization result is shown.     

A uniform asymptotic expansion for stochastic volatility model in pricing multi‐asset European options

In this paper, we consider a stochastic volatility model for pricing multi‐asset European options that are widely used in the real world, under the assumption that the volatilities are driven by different OU processes. Using the singular perturbation method for multi‐parameter and the boundary layer theory, we derive a uniform asymptotic expansion for the option prices, as well as the uniform error estimates. Copyright © 2011 John Wiley & Sons, Ltd.     

Liouville theorems,a priori estimates,and blow-up rates for solutions of indefinite superlinear parabolic problems

In this paper we establish new nonlinear Liouville theorems for parabolic problems on half spaces. Based on the Liouville theorems, we derive estimates for the blow-up of positive solutions of indefinite parabolic problems and investigate the complete blow-up of these solutions. We also discuss a priori estimates for indefinite elliptic problems.     

A posteriori error estimates of finite element methods for nonlinear quadratic boundary optimal control problem

This paper is aimed at studying finite element discretization for a class of quadratic boundary optimal control problems governed by nonlinear elliptic equations. We derive a posteriori error estimates for the coupled state and control approximation. Such estimates can be used to construct a reliable adaptive finite element approximation for the boundary optimal control problem. Finally, we present a numerical example to confirm our theoretical results.     

Bochner–Weitzenböck formula and Li–Yau estimates on Finsler manifolds

We prove the Bochner–Weitzenböck formula for the (nonlinear) Laplacian on general Finsler manifolds and derive Li–Yau type gradient estimates as well as parabolic Harnack inequalities. Moreover, we deduce Bakry–Émery gradient estimates. All these estimates depend on lower bounds for the weighted flag Ricci tensor.