A Markov-modulated model for stocks paying discrete dividends

We extend the model in [Korn, R., Rogers, L.C.G., 2005. Stock paying discrete dividends: modelling and option pricing. Journal of Derivatives 13, 44–49] for (discrete) dividend processes to incorporate the dependence of assets on the market mode or the state of the economy, where the latter is modeled by a hidden finite-state Markov chain. We then derive the resulting dynamics of the stock price and various option-pricing formulae. It turns out that the stock price jumps not only at the time of the dividend payment, but also when the underlying Markov chain jumps.     

Explicit solutions to European options in a regime-switching economy

We provide closed-form solutions for European option values when the dynamics of both the short rate and volatility of the underlying price process are modulated by a continuous-time Markov chain with a finite number of “economic states”. Extensions involving dividends, currencies and cost of carry are further explored.     

On the Esscher transforms and other equivalent martingale measures for Barndorff-Nielsen and Shephard stochastic volatility models with jumps

We compute and then discuss the Esscher martingale transform for exponential processes, the Esscher martingale transform for linear processes, the minimal martingale measure, the class of structure preserving martingale measures, and the minimum entropy martingale measure for stochastic volatility models of the Ornstein–Uhlenbeck type as introduced by Barndorff-Nielsen and Shephard. We show that in the model with leverage, with jumps both in the volatility and in the returns, all those measures are different, whereas in the model without leverage, with jumps in the volatility only and a continuous return process, several measures coincide, some simplifications can be made and the results are more explicit. We illustrate our results with parametric examples used in the literature.     

复合泊松过程和Meixner过程驱动下的期权定价

本文讨论了股票价格对数过程由复合泊松过程、Meixner过程驱动下的欧式看涨期权的定价问题.利用Esscher变换和风险中性Esscher测度得到了两类过程驱动下的期权定价公式,为实践者提供了理论上的参考价格.     

GARCH option pricing: A semiparametric approach

Option pricing based on GARCH models is typically obtained under the assumption that the random innovations are standard normal (normal GARCH models). However, these models fail to capture the skewness and the leptokurtosis in financial data. We propose a new method to compute option prices using a nonparametric density estimator for the distribution of the driving noise. We investigate the pricing performances of this approach using two different risk neutral measures: the Esscher transform pioneered by Gerber and Shiu [Gerber, H.U., Shiu, E.S.W., 1994a. Option pricing by Esscher transforms (with discussions). Trans. Soc. Actuar. 46, 99–91], and the extended Girsanov principle introduced by Elliot and Madan [Elliot, R.J., Madan, D.G., 1998. A discrete time equivalent martingale 9 measure. Math. Finance 8, 127–152]. Both measures are justified by economic arguments and are consistent with Duan’s [Duan, J.-C., 1995. The GARCH option pricing model. Math. Finance 5, 13–32] local risk neutral valuation relationship (LRNVR) for normal GARCH models. The main advantage of the two measures is that one can price derivatives using skewed or heavier tailed innovations distributions to model the returns. An empirical study regarding the European Call option valuation on S&P500 Index shows: (i) under both risk neutral measures our semiparametric algorithm performs better than the existing normal GARCH models if we allow for a leverage effect and (ii) the pricing errors when using the Esscher transform are quite small even though our estimation procedure is based only on historical return data.     

On option pricing under a completely random measure via a generalized Esscher transform

In this paper, we develop an option valuation model when the price dynamics of the underlying risky asset is governed by the exponential of a pure jump process specified by a shifted kernel-biased completely random measure. The class of kernel-biased completely random measures is a rich class of jump-type processes introduced in [James, L.F., 2005. Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages. Ann. Statist. 33, 1771–1799; James, L.F., 2006. Poisson calculus for spatial neutral to the right processes. Ann. Statist. 34, 416–440] and it provides a great deal of flexibility to incorporate both finite and infinite jump activities. It includes a general class of processes, namely, the generalized Gamma process, which in its turn includes the stable process, the Gamma process and the inverse Gaussian process as particular cases. The kernel-biased representation is a nice representation form and can describe different types of finite and infinite jump activities by choosing different mixing kernel functions. We employ a dynamic version of the Esscher transform, which resembles an exponential change of measures or a disintegration formula based on the Laplace functional used by James, to determine an equivalent martingale measure in the incomplete market. Closed-form option pricing formulae are obtained in some parametric cases, which provide practitioners with a convenient way to evaluate option prices.     

Option pricing when the regime-switching risk is priced

We study the pricing of an option when the price dynamic of the underlying risky asset is governed by a Markov-modulated geometric Brownian motion. We suppose that the drift and volatility of the underlying risky asset are modulated by an observable continuous-time, finite-state Markov chain. We develop a two- stage pricing model which can price both the diffusion risk and the regime-switching risk based on the Esscher transform and the minimization of the maximum entropy between an equivalent martingale measure and the real-world probability measure over different states. Numerical experiments are conducted and their results reveal that the impact of pricing regime-switching risk on the option prices is significant.     

双参数Esscher变换及其应用

通过把Lévy过程分解为两个独立过程之和,将Esscher变换由单参数推广到双参数,并给出了双参数Esscher变换测度为等价鞅测度的充要条件．     

On approximation of smoothing probabilities for hidden Markov models

We consider the smoothing probabilities of hidden Markov model (HMM). We show that under fairly general conditions for HMM, the exponential forgetting still holds, and the smoothing probabilities can be well approximated with the ones of double-sided HMM. This makes it possible to use ergodic theorems. As an application we consider the pointwise maximum a posteriori segmentation, and show that the corresponding risks converge.     

A correction on approximation of smoothing probabilities for hidden Markov models

In this note, we correct a mistake concerning Theorem 2.1 in Lember (2011a).     

An extension of the Euler Laplace transform inversion algorithm with applications in option pricing

We show that the Euler algorithm for Laplace transform inversion can be extended to functions defined on the entire real line, if they have specific decay features. Our objective is to apply the method to option pricing problems, specifically when inverting Laplace transforms of the option price in the logarithm of the strike.     

Small-time expansions for state-dependent local jump–diffusion models with infinite jump activity

In this article, we consider a Markov process ${\left\{X_t\right\}}_{t?0}$, which solves a stochastic differential equation driven by a Brownian motion and an independent pure jump component exhibiting both state-dependent jump intensity and infinite jump activity. A second order expansion is derived for the tail probability $\mathbb{P}[X_t?x+y]$ in small time $t$, where $x$ is the initial value of the process and $y>0$. As an application of this expansion and a suitable change of the underlying probability measure, a second order expansion, near expiration, for out-of-the-money European call option prices is obtained when the underlying stock price is modeled as the exponential of the jump–diffusion process ${\left\{X_t\right\}}_{t?0}$ under the risk-neutral probability measure.     

受控AR信号模型下基于Kullback-Leibler信息量HMM参数的估计算法

本是在对现实世界中常见的信号模型一受控AR模型的处理中引进HMM的，并且基于Kullback-Leibler(简记为K-L)信息量在此特定信号模型下蛤出了HMM参数的估计算法。     

Smoothly truncated stable distributions,GARCH-models,and option pricing

Although asset return distributions are known to be conditionally leptokurtic, this fact has rarely been addressed in the recent GARCH model literature. For this reason, we introduce the class of smoothly truncated stable distributions (STS distributions) and derive a generalized GARCH option pricing framework based on non-Gaussian innovations. Our empirical results show that (1) the model’s performance in the objective as well as the risk-neutral world is substantially improved by allowing for non-Gaussian innovations and (2) the model’s best option pricing performance is achieved with a new estimation approach where all model parameters are obtained from time-series information whereas the market price of risk and the spot variance are inverted from market prices of options. The paper subsumes a previous one under the title “A New Class of Probability Distributions and Its Application to Finance”. The authors gratefully acknowledge comments made by seminar participants at University of California, Santa Barbara, University of Washington, Seattle, Hochschule für Banken, Frankfurt, Cornell University, Princeton University, American University, Washington DC, and the Risk Management and Financial Engineering Conference held in Gainesville, FL in April 2005. All views and opinions expressed in this article are strictly those of the author and do not necessarily represent the views of Sal. Oppenheim.     

非齐次隐马尔可夫模型随机变换的若干强极限定理

利用鞅方法讨论了非齐次隐马尔可夫模型变换的强极限定理,作为特殊情形,将随机选择的概念拓展到非齐次隐马尔可夫模型中,得到了关于有限非齐次隐马尔可夫模型随机选择与随机公平比的若干极限定理.