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.     

马尔可夫交换Lévy过程的最小熵鞅测度

研究了由马尔可夫交换Lévy过程的随机指数所驱动的风险资产的期权定价问题,即市场的利率、风险资产的波动率以及N个状态的补偿子都依赖于不可观的经济状态,而这些经济状态服从于一个连续时间的隐马氏链模型.一般地,由马尔可夫交换Levy过程的随机指数所描述的市场是不完备的,因此,鞅测度不是唯一的.通过采用状态转换Esscher变换来确定等价鞅测度,并且证明了所得到的定价测度就是最小熵鞅测度.     

马尔可夫交换Levy过程的最小熵鞅测度

研究了由马尔可夫交换Levy过程的随机指数所驱动的风险资产的期权定价问题,即市场的利率、风险资产的波动率以及N个状态的补偿子都依赖于不可观的经济状态,而这些经济状态服从于一个连续时间的隐马氏链模型.一般地,由马尔可夫交换Levy过程的随机指数所描述的市场是不完备的,因此,鞅测度不是唯一的.通过采用状态转换Esscher变换来确定等价鞅测度,并且证明了所得到的定价测度就是最小熵鞅测度.     

跳扩散模型下股价服从几何布朗运动的Esscher变换定价

考虑跳扩散模型下期权的Esscher变换定价,给出了Esscher变换下带跳的B-S矩生成函数和复合泊松过程下的矩生成函数,推导出跳扩散模型下期权的Esscher变换定价公式.     

体制转换模型下巨灾权益卖权的定价研究

本文探讨体制转换跳扩散模型下巨灾权益卖权的定价问题.模型参数,包括无风险利率、保险公司股价的平均回报率和波动率均随着经济状态的变化而改变.文中假设经济环境采用一个连续时间、有限状态、可观测的马尔可夫链来刻画,从而可以将经济条件的变化考虑到产品定价中.通过体制转换Esscher变换选取一个等价鞅测度,然后通过快速傅立叶变换对巨灾权益卖权进行定价.     

标的资产服从NIG-Levy过程的亚式期权数值定价分析

考虑到金融时间序列的厚尾性即呈现尖峰厚尾分布,波动率具有聚集性和持续性等特点,也即标的资产的价格可能会出现间断的跳跃,我们展示了在标的资产价格对数收益服从NIG-Levy过程的条件下,如何构建和计算等价鞅测度,我们考虑通过Esscher转换得到Q等价鞅测度,并以此为基础寻找风险中性概率的条件,最后利用这些条件探讨亚式期权的数值定价问题,利用低差异序列中的Halton、Sobol、Faure序列对亚式期权进行了数值定价分析.     

多峰映射族中非双曲奇异吸引子的丰富性

本文考虑多峰映射族中非双曲奇异吸引子的丰富性,证明多维参数空间中存在正测度的参数集合,对应系统具有绝对连续的不变测度.     

基于非对称漂移双gamma跳-扩散过程的创新幂型期权定价模型

针对假设股价的对数收益率布朗运动在期权定价时产生的无法解释股价对数收益率的尖峰厚尾和序列相关性的缺陷,采用了Zhang提出的非对称漂移双gamma跳-扩散过程来描述资产(股价)的对数收益率运动形态(该过程是kou提出的双指数跳-扩散过程的推广),并利用Esscher风险中性变换,研究了幂型期权的定价公式.还设计了两种创新的幂型期权,在非对称漂移双gamma跳-扩散过程下给出了相应的定价公式.     

Esscher变换下扩展的欧式交换期权定价

文章研究Esscher变换下标的资产价格服从几何布朗运动的扩展的几种欧式交换期权(包括广义交换期权,复合交换期权,障碍交换期权,红绿灯期权)定价问题.首先,给出了带漂移布朗运动的反射原理和性质;其次,借助Gerber和Shiu (1994)给出了多维独立平稳增量过程和二维带漂移布朗运动的Esscher变换定义及其性质;最后,应用Esscher变换的相关理论给出了标的资产价格服从几何布朗运动的扩展的多种欧式交换期权定价公式.特别,本文所得到的期权定价公式与以往文献中给出的结果是一致的.     

聚合风险模型下Esscher风险度量的估计及其大样本性质

Esscher度量是一种重要的风险度量,在金融风险管理、保险精算中有广泛的应用,然而大部分关于Esscher风险度量的文献均在个体风险模型下考虑的.本文建立了聚合风险模型下Esscher度量的估计模型,得到了相应的非参数估计,并证明了估计的强相合性和渐近正态性,最后,通过数值模拟的方法验证了估计的大样本性质.     

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.     

The Risk Premium and the Esscher Transform in Power Markets

In power markets one frequently encounters a risk premium being positive in the short end of the forward curve and negative in the long end. Economically it has been argued that the positive premium is reflecting retailers aversion for spike risk, wheras in the long end of the forward curve, the hedging pressure kicks in as in other commodity markets. Mathematically, forward prices are expressed as risk-neutral expectations of the spot at delivery. We apply the Esscher transform on power spot models based on mean-reverting processes driven by independent increment (time-inhomogeneous Lévy) processes. It is shown that the Esscher transform is yielding a change of mean-reversion level. Moreover, we show that an Esscher transform together with jumps occuring seasonally may explain the occurence of a positive risk premium in the short end. This is demonstrated both mathematically and by a numerical example for a two-factor spot model being relevant for electricity markets.     

Esscher transforms and consumption-based models

The Esscher transform is an important tool in actuarial science. Since the pioneering work of Gerber and Shiu (1994), the use of the Esscher transform for option valuation has also been investigated extensively. However, the relationships between the asset pricing model based on the Esscher transform and some fundamental equilibrium-based asset pricing models, such as consumption-based models, have so far not been well-explored. In this paper, we attempt to bridge the gap between consumption-based models and asset pricing models based on Esscher-type transformations in a discrete-time setting. Based on certain assumptions for the distributions of asset returns, changes in aggregate consumptions and returns on the market portfolio, we construct pricing measures that are consistent with those arising from Esscher-type transformations. Explicit relationships between the market price of risk, and the risk preference parameters are derived for some particular cases.     

Actuarial risk measures for financial derivative pricing

We present an axiomatic characterization of price measures that are superadditive and comonotonic additive for normally distributed random variables. The price representation derived involves a probability measure transform that is closely related to the Esscher transform, and we call it the Esscher-Girsanov transform. In a financial market in which the primary asset price is represented by a stochastic differential equation with respect to Brownian motion, the price mechanism based on the Esscher-Girsanov transform can generate approximate-arbitrage-free financial derivative prices.     

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.     

Analytical pricing of vulnerable options under a generalized jump–diffusion model

In this paper we propose a model to price European vulnerable options. We formulate their credit risk in a reduced form model and the dynamics of the spot price in a completely random generalized jump–diffusion model, which nests a number of important models in finance. We obtain a closed-form price for the vulnerable option by (1) determining an equivalent martingale measure, using the Esscher transform and (2) manipulating the pay-off structure of the option four further times, by using the Esscher–Girsanov transform.     

Exponential change of measure applied to term structures of interest rates and exchange rates

In this paper, we study the term structures of interest rates and foreign exchange rates through establishing a state-price deflator. The state-price deflator considered here can be viewed as an extension to the potential representation of the state-price density in [Rogers, L.C.G., 1997. The potential approach to the term structure of interest rates and foreign exchange rates. Mathematical Finance 7(2), 157-164]. We identify a risk-neutral probability measure from the state-price deflator by a technique of exponential change of measure for Markov processes proposed by [Palmowski, Z., Rolski, T., 2002. A technique for exponential change of measure for Markov processes. Bernoulli 8(6), 767-785] and present examples of several classes of diffusion processes (jump-diffusions and diffusions with regime-switching) to illustrate the proposed theory. A comparison between the exponential change of measure and the Esscher transform for identifying risk-neutral measures is also presented. Finally, we consider the exchange rate dynamics by virtue of the ratio of the current state-price deflators between two economies and then discuss the pricing of currency options.