Pricing long-dated insurance contracts with stochastic interest rates and stochastic volatility

We consider the pricing of long-dated insurance contracts under stochastic interest rates and stochastic volatility. In particular, we focus on the valuation of insurance options with long-term equity or foreign exchange exposures. Our modeling framework extends the stochastic volatility model of Schöbel and Zhu (1999) by including stochastic interest rates. Moreover, we allow all driving model factors to be instantaneously correlated with each other, i.e. we allow for a general correlation structure between the instantaneous interest rates, the volatilities and the underlying stock returns. As insurance products often incorporate long-term exposures, they are typically more sensitive to changes in the interest rates, volatility and currencies. Therefore, having the flexibility to correlate the underlying asset price with both the stochastic volatility and the stochastic interest rates, yields a realistic model which is of practical importance for the pricing and hedging of such long-term contracts. We show that European options, typically used for the calibration of the model to market prices, and forward starting options can be priced efficiently and in closed-form by means of Fourier inversion techniques. We extensively discuss the numerical implementation of these pricing formulas, allowing for a fast and accurate valuation of European and forward starting options. The model will be especially useful for the pricing and risk management of insurance contracts and other exotic derivatives involving long-term maturities.     

Volatility estimation for stochastic project value models

One of the key parameters in modeling capital budgeting decisions for investments with embedded options is the project volatility. Most often, however, there is no market or historical data available to provide an accurate estimate for this parameter. A common approach to estimating the project volatility in such instances is to use a Monte Carlo simulation where one or more sources of uncertainty are consolidated into a single stochastic process for the project cash flows, from which the volatility parameter can be determined. Nonetheless, the simulation estimation method originally suggested for this purpose systematically overstates the project volatility, which can result in incorrect option values and non-optimal investment decisions. Examples that illustrate this issue numerically have appeared in several recent papers, along with revised estimation methods that address this problem. In this article, we extend that work by showing analytically the source of the overestimation bias and the adjustment necessary to remove it. We then generalize this development for the cases of levered cash flows and non-constant volatility. In each case, we use an example problem to show how a revised estimation methodology can be applied.     

Probing Option Prices for Information

We present a methodology for extracting information from option prices when the market is viewed as knowledgeable. By expanding the information filtration judiciously and determining conditional characteristic functions for the log of the stock price, we obtain option pricing formulae which when fit to market data may reveal this information. In particular, we consider probing option prices for knowledge of the future stock price, instantaneous volatility, and the asymptotic dividend stream. Additionally the bridge laws developed are also useful for simulation based on stratified sampling that conditions on the terminal values of paths.      

Pricing long-dated insurance contracts with stochastic interest rates and stochastic volatility

We consider the pricing of long-dated insurance contracts under stochastic interest rates and stochastic volatility. In particular, we focus on the valuation of insurance options with long-term equity or foreign exchange exposures. Our modeling framework extends the stochastic volatility model of Schöbel and Zhu (1999) by including stochastic interest rates. Moreover, we allow all driving model factors to be instantaneously correlated with each other, i.e. we allow for a general correlation structure between the instantaneous interest rates, the volatilities and the underlying stock returns. As insurance products often incorporate long-term exposures, they are typically more sensitive to changes in the interest rates, volatility and currencies. Therefore, having the flexibility to correlate the underlying asset price with both the stochastic volatility and the stochastic interest rates, yields a realistic model which is of practical importance for the pricing and hedging of such long-term contracts. We show that European options, typically used for the calibration of the model to market prices, and forward starting options can be priced efficiently and in closed-form by means of Fourier inversion techniques. We extensively discuss the numerical implementation of these pricing formulas, allowing for a fast and accurate valuation of European and forward starting options. The model will be especially useful for the pricing and risk management of insurance contracts and other exotic derivatives involving long-term maturities.     

Statistical Analysis of Financial Volatility by Wavelet Shrinkage

We analyze the Nikkei daily stock index and verify how wavelets can help in identifying, estimating and predicting its volatility features. While we study the conditional mean and variance dynamics, by utilizing statistical parametric inference techniques, we also decompose the observed signal with a data de-noising procedure. We thus investigate how wavelets discriminate among information at different resolution levels and we attempt to understand whether the de-noised data lead to a better identification of the underlying volatility process. We find that the wavelet data pre-processing strategy, by reducing the measurement error of the observed data, is useful for improving the volatility prediction power.     

高频视角下中国股市动态VaR预测模型研究

如何充分挖掘交易数据中有价值的信息对金融风险管理极其重要,现有研究中基于低频波动模型的风险测度方法几乎已经做到了极致,而能达到的预测效果却并不稳健,对高频波动模型的研究相对比较匮乏。那么高频模型能否从高频数据中挖掘出更有价值的信息以便用于风险管理之中呢？本研究通过建立12个低频和9个高频波动模型对上证综指进行样本外动态VaR的滚动预测发现,高频模型相对于低频模型具有更好的稳定性,并且在多数情况下高频模型优于低频模型;多头与空头的风险预测效果具有显著差异,多头风险在高风险情况下高频模型表现出色,低风险情况下并不理想,空头风险则在所有情况下都表现较好。     

Asymptotic Solutions for Australian Options with Low Volatility

Abstract

In this paper we derive asymptotic expansions for Australian options in the case of low volatility using the method of matched asymptotics. The expansion is performed on a volatility scaled parameter. We obtain a solution that is of up to the third order. In case that there is no drift in the underlying, the solution provided is in closed form, for a non-zero drift, all except one of the components of the solutions are in closed form. Additionally, we show that in some non-zero drift cases, the solution can be further simplified and in fact written in closed form as well. Numerical experiments show that the asymptotic solutions derived here are quite accurate for low volatility.     

中国股市期权波动率微笑特征的实证分析

波动率微笑现象显示了期权隐含波动率和执行价格之间的关系.在理想的完全符合Black-Scholes期权定价模型假设的情况下,期权隐含波动率关于执行价格应该是一条水平线.然而,在实证分析中,对隐含波动率和执行价格进行拟合并绘制曲线,会产生一个倾斜或微笑形状的曲线,证明Black-Scholes期权定价模型存在一定的缺陷....     

From regulatory life tables to stochastic mortality projections: The exponential decline model

Often in actuarial practice, mortality projections are obtained by letting age-specific death rates decline exponentially at their own rate. Many life tables used for annuity pricing are built in this way. The present paper adopts this point of view and proposes a simple and powerful mortality projection model in line with this elementary approach, based on the recently studied mortality improvement rates. Two main applications are considered. First, as most reference life tables produced by regulators are deterministic by nature, they can be made stochastic by superposing random departures from the assumed age-specific trend, with a volatility calibrated on market or portfolio data. This allows the actuary to account for the systematic longevity risk in solvency calculations. Second, the model can be fitted to historical data and used to produce longevity forecasts. A number of conservative and tractable approximations are derived to provide the actuary with reasonably accurate approximations for various relevant quantities, available at limited computational cost. Besides applications to stochastic mortality projection models, we also derive useful properties involving supermodular, directionally convex and stop-loss orders.     

Non-linear filtering and optimal investment under partial information for stochastic volatility models

This paper studies the question of filtering and maximizing terminal wealth from expected utility in partial information stochastic volatility models. The special feature is that the only information available to the investor is the one generated by the asset prices, and the unobservable processes will be modeled by stochastic differential equations. Using the change of measure techniques, the partial observation context can be transformed into a full information context such that coefficients depend only on past history of observed prices (filter processes). Adapting the stochastic non-linear filtering, we show that under some assumptions on the model coefficients, the estimation of the filters depend on a priori models for the trend and the stochastic volatility. Moreover, these filters satisfy a stochastic partial differential equations named “Kushner–Stratonovich equations”. Using the martingale duality approach in this partially observed incomplete model, we can characterize the value function and the optimal portfolio. The main result here is that, for power and logarithmic utility, the dual value function associated to the martingale approach can be expressed, via the dynamic programming approach, in terms of the solution to a semilinear partial differential equation which depends on the filters estimate and the volatility. We illustrate our results with some examples of stochastic volatility models popular in the financial literature.     

Maximum likelihood estimation for stochastic volatility in mean models with heavy‐tailed distributions

In this article, we introduce a likelihood‐based estimation method for the stochastic volatility in mean (SVM) model with scale mixtures of normal (SMN) distributions. Our estimation method is based on the fact that the powerful hidden Markov model (HMM) machinery can be applied in order to evaluate an arbitrarily accurate approximation of the likelihood of an SVM model with SMN distributions. Likelihood‐based estimation of the parameters of stochastic volatility models, in general, and SVM models with SMN distributions, in particular, is usually regarded as challenging as the likelihood is a high‐dimensional multiple integral. However, the HMM approximation, which is very easy to implement, makes numerical maximum of the likelihood feasible and leads to simple formulae for forecast distributions, for computing appropriately defined residuals, and for decoding, that is, estimating the volatility of the process. Copyright © 2017 John Wiley & Sons, Ltd.     

Bad news and Dow Jones make the Spanish stocks go round

This paper is a data-based attempt to analyse what kind of information basically affects close-to-open returns, open-to-close returns, volatility and volume in actively traded individual securities on the Spanish stock market. Specifically, we are interested in detecting how these variables react to specific pieces of news considered as exogenous information. However, as volume itself could be interpreted as a proxy of the information flow, we first apply the linear and nonlinear Granger causality tests from volume to return and to volatility. We do not find evidence supporting this latter hypothesis. Furthermore, we only find significant evidence of linear causality from volume to volatility. The other major finding is that both bad news and the Dow Jones play a significant informational role in explaining price changes and volatility. As a consequence of these findings, we also test the residual role of volume as a proxy for noise/liquidity trading after filtering for news, although we do not find evidence in favour of this argument.     

厚尾随机波动率模型的贝叶斯参数估计及实证研究

针对现有时间序列模型难以刻画参数渐变性的问题,对厚尾随机波动(SV)模型的参数估计方法进行了推广,采用基于贝叶斯的MCMC方法,选取2013年5月~2016年6月这一经历多轮震荡的上证指数作为实证分析对象,构造了基于Gibbs抽样的MCMC过程进行仿真分析.结果显示,以卡方分布作为厚尾参数的先验分布能够有效地描述数据波动的厚尾特征,并且能得到较高精度的参数估计结果.结果表明,厚尾SV模型能有效反映出我国股市尖峰厚尾和波动长期记忆性的特征.     

金融系统的非线性分析:交易量对股价波动的非线性影响

如何研究股价波动和成交量之间的关系一直是金融系统研究中感兴趣的话题.Lamoureux 和 Lastrapes 认为选择日交易量度量每天流入市场的信息量是合理的,但他们假定交易量对波动率的影响是线性的.提出部分非线性GARCH模型分析交易量对股票市场波动率的影响,基于GARCH模型局部线性化非参数似然估计方法,对中国证券市场股票价格和交易量数据进行实证研究.结果表明,交易量对股价波动的影响具有显著的非线性性.     

价格波动幅度变动率——一个新的市场风险度量指标

在度量风险时,一个常用的指标就是收益率的方差.然而,用该指标来度量风险会丢失掉一些非常有用的信息,如一日之内的开盘价,最高价,最低价等.鉴于传统指标的不足,提出用另一种指标---价格波动幅度变动率来度量市场风险.为了说明该指标的合理性及优良性,用标普500指数考察了该指标的主要经验统计特性和动态结构.结果表明新的指标具有近似正态性,建模的简易性及可预测性,因此可以作为一个新的风险度量指标.     

Implied Filtering Densities on the Hidden State of Stochastic Volatility

Abstract

We formulate and analyse an inverse problem using derivative prices to obtain an implied filtering density on volatility’s hidden state. Stochastic volatility is the unobserved state in a hidden Markov model (HMM) and can be tracked using Bayesian filtering. However, derivative data can be considered as conditional expectations that are already observed in the market, and which can be used as input to an inverse problem whose solution is an implied conditional density on volatility. Our analysis relies on a specification of the martingale change of measure, which we refer to as separability. This specification has a multiplicative component that behaves like a risk premium on volatility uncertainty in the market. When applied to SPX options data, the estimated model and implied densities produce variance-swap rates that are consistent with the VIX volatility index. The implied densities are relatively stable over time and pick up some of the monthly effects that occur due to the options’ expiration, indicating that the volatility-uncertainty premium could experience cyclic effects due to the maturity date of the options.     

高阶矩风险溢酬:信息含量及影响因素

本文基于台湾股市和期权市场数据,研究波动率、偏度和峰度等高阶矩风险溢酬的信息含量及影响因素,并结合边际贡献度分析了各常见影响因素对高阶矩风险溢酬的解释力。研究结论表明:波动率间的相关性高于偏度和峰度,波动率也更易于预测;期权市场的隐含矩信息对未来实际矩有较好解释力,且引入多个市场信息的预测效果要优于单一市场。市场情绪和异质信念是各高阶矩风险溢酬的主要影响因素;流动性信息次之;市场因子的解释相对较弱。对高阶矩风险溢酬影响因素的分析还应引入市场极端风险、风险厌恶信息和宏观经济变量等因素的考察。     

Support Vector Machine as an Efficient Framework for Stock Market Volatility Forecasting

Advantages and limitations of the existing models for practical forecasting of stock market volatility have been identified. Support vector machine (SVM) have been proposed as a complimentary volatility model that is capable to extract information from multiscale and high-dimensional market data. Presented results for SP500 index suggest that SVM can efficiently work with high-dimensional inputs to account for volatility long-memory and multiscale effects and is often superior to the main-stream volatility models. SVM-based framework for volatility forecasting is expected to be important in the development of the novel strategies for volatility trading, advanced risk management systems, and other applications dealing with multi-scale and high-dimensional market data.     

波动率风险溢价:基于香港权证市场的实证

波动率风险溢价包含了关于投资者风险厌恶的重要信息,它的估计是金融计量学文献关注的一个核心问题。本文基于香港权证市场数据和GARCH扩散随机波动率(SV)模型,对香港证券市场的波动率风险溢价进行了估计研究。采用香港恒生指数和指数权证数据,通过建立基于有效重要性抽样的极大似然(EIS-ML)方法联合估计了GARCH扩散模型的客观与风险中性测度,进而得到了香港证券市场的波动率风险溢价。研究结果发现,在香港证券市场上,市场投资者对波动率风险进行了定价,即存在波动率风险溢价,且波动率风险溢价在绝大多数情形下为正,说明市场投资者总体表现为风险爱好。     

光滑树图期权定价模型的叉熵分析法

为了克服CRR模型收敛的波动性,以及强调历史信息的预测作用的情况,提出了一个新奇的光滑收敛的树图模型.新模型基于历史信息,运用最小叉熵原理
来推导树图的关键参数p,u,d, 然后使用倒推法推断期权的价格.显然,新模型所得的期权的价格隐含着历史信息.由于最小叉熵原理是一个凸规划问题,能求得唯一的最优解,所以,新模型也适用于不完全金融市场期权定价.最后,数值算例表明,相比于CRR模型,新模型收敛光滑平稳且有更高的计算精度;对上涨(下跌)的二元期权、欧式期权,新模型都能光滑收敛于B-S公式.