Bounds for Ratios of Posterior Expectations: Applications in the Collective Risk Model

This paper considers the collective risk model for the insurance claims process. We will adopt a Bayesian point of view, where uncertainty concerning the specification of the prior distribution is a common question. The robust Bayesian approach uses a class of prior distributions which model uncertainty about the prior, instead of a single distribution. Relatively little research has dealt with robustness with respect to ratios of posterior expectations as occurs with the Esscher and Variance premium principles. Appropriate techniques are developed in this paper to solve this problem using the k -contamination class in the collective risk model.     

Pricing Annuity Guarantees Under a Regime-Switching Model

Abstract

We consider the pricing problem of equity-linked annuities and variable annuities under a regimeswitching model when the dynamic of the market value of a reference asset is driven by a generalized geometric Brownian motion model with regime switching. In particular, we assume that regime switching over time according to a continuous-time Markov chain with a finite number state space representing economy states. We use the Esscher transform to determine an equivalent martingale measure for fair valuation in the incomplete market setting. The paper is complemented with some numerical examples to highlight the implications of our model on pricing these guarantees.     

State Price Density,Esscher Transforms,and Pricing Options on Stocks,Bonds, and Foreign Exchange Rates

Abstract

The state price density is modeled as an exponential function of the underlying state variables, and the Esscher transform is used to specify the forward-risk-adjusted measure. With the aid of state price densities, Esscher transforms, and characteristic functions, this paper provides a consistent framework for pricing options on stocks, interest rates, and foreign exchange rates. The framework discussed is quite general and is related to many popular models.     

Option Pricing Under Autoregressive Random Variance Models

Abstract

The autoregressive random variance (ARV) model introduced by Taylor (1980, 1982, 1986) is a popular version of stochastic volatility (SV) models and a discrete-time simplification of the continuous-time diffusion SV models. This paper introduces a valuation model for options under a discrete-time ARV model with general stock and volatility innovations. It employs the discretetime version of the Esscher transform to determine an equivalent martingale measure under an incomplete market. Various parametric cases of the ARV models, are considered, namely, the log-normal ARV models, the jump-type Poisson ARV models, and the gamma ARV models, and more explicit pricing formulas of a European call option under these parametric cases are provided. A Monte Carlo experiment for some parametric cases is also conducted.     

Option pricing and Esscher transform under regime switching

Summary We consider the option pricing problem when the risky underlying assets are driven by Markov-modulated Geometric Brownian Motion (GBM). That is, the market parameters, for instance, the market interest rate, the appreciation rate and the volatility of the underlying risky asset, depend on unobservable states of the economy which are modelled by a continuous-time Hidden Markov process. The market described by the Markov-modulated GBM model is incomplete in general and, hence, the martingale measure is not unique. We adopt a regime switching random Esscher transform to determine an equivalent martingale pricing measure. As in Miyahara [33], we can justify our pricing result by the minimal entropy martingale measure (MEMM).We would like to thank the referees for many helpful and insightful comments and suggestions.Correspondence to: R. J. Elliott     

Option pricing and hedging under a stochastic volatility Lévy process model

In this paper, we discuss a stochastic volatility model with a Lévy driving process and then apply the model to option pricing and hedging. The stochastic volatility in our model is defined by the continuous Markov chain. The risk-neutral measure is obtained by applying the Esscher transform. The option price using this model is computed by the Fourier transform method. We obtain the closed-form solution for the hedge ratio by applying locally risk-minimizing hedging.     

Valuing Equity-Indexed Annuities

Abstract

Equity-indexed annuities have generated a great deal of interest and excitement among both insurers and their customers since they were first introduced to the marketplace in early 1995. Because of the embedded options in these products, the insurers are presented with some challenging mathematical problems when it comes to the pricing and management of equity indexed annuities. This paper explores the pricing aspect of three of the most common product designs: the point-to-point, the cliquet, and the lookback. Based on certain assumptions, we are able to present the pricing formulas in closed form for the three product designs. The method of Esscher transforms is the fundamental tool for pricing such deferred annuities.     

Skewness and Stock Option Prices

Abstract

In the classical Black-Scholes model, the logarithm of the stock price has a normal distribution, which excludes skewness. In this paper we consider models that allow for skewness. We propose an option-pricing formula that contains a linear adjustment to the Black-Scholes formula. This approximation is derived in the shifted Poisson model, which is a complete market model in which the exact option price has some undesirable features. The same formula is obtained in some incomplete market models in which it is assumed that the price of an option is defined by the Esscher method. For a European call option, the adjustment for skewness can be positive or negative, depending on the strike price.     

The systematic pricing of market sentiment shock

ABSTRACT

We show that market sentiment shocks create demand shocks for risky assets and a systematic risk for assets. We measure a market sentiment shock as the unexpected portion of the University of Michigan Consumer Sentiment Index’s growth. This shock prices stock returns in arbitrage pricing theory framework at 1% after controlling for market, size, value, momentum, and liquidity risk factors. Its premium lowered the implied risk aversion by 97.9% to 11.46 between 1978 and 2009 in our sentiment consumption-based capital-asset-pricing model. Merton’s [1973. “An Intertemporal Capital Asset Pricing Model.” Econometrica 41: 867–887]. intertemporal capital-asset-pricing model reconfirms our finding that this market sentiment shock is a systematic risk factor that provides investment opportunities.     

Modeling Operational Risk With Bayesian Networks

Bayesian networks is an emerging tool for a wide range of risk management applications, one of which is the modeling of operational risk. This comes at a time when changes in the supervision of financial institutions have resulted in increased scrutiny on the risk management of banks and insurance companies, thus giving the industry an impetus to measure and manage operational risk. The more established methods for risk quantification are linear models such as time series models, econometric models, empirical actuarial models, and extreme value theory. Due to data limitations and complex interaction between operational risk variables, various nonlinear methods have been proposed, one of which is the focus of this article: Bayesian networks. Using an idealized example of a fictitious on line business, we construct a Bayesian network that models various risk factors and their combination into an overall loss distribution. Using this model, we show how established Bayesian network methodology can be applied to: (1) form posterior marginal distributions of variables based on evidence, (2) simulate scenarios, (3) update the parameters of the model using data, and (4) quantify in real‐time how well the model predictions compare to actual data. A specific example of Bayesian networks application to operational risk in an insurance setting is then suggested.     

Forward-looking portfolio selection with multivariate non-Gaussian models

In this study, we suggest a portfolio selection framework based on time series of stock log-returns, option-implied information, and multivariate non-Gaussian processes. We empirically assess a multivariate extension of the normal tempered stable (NTS) model and of the generalized hyperbolic (GH) one by implementing an estimation method that simultaneously calibrates the multivariate time series of log-returns and, for each margin, the univariate observed one-month implied volatility smile. To extract option-implied information, the connection between the historical measure P and the risk-neutral measure Q, needed to price options, is provided by the multivariate Esscher transform. The method is applied to fit a 50-dimensional series of stock returns, to evaluate widely known portfolio risk measures and to perform a forward-looking portfolio selection analysis. The proposed models are able to produce asymmetries, heavy tails, both linear and non-linear dependence and, to calibrate them, there is no need for liquid multivariate derivative quotes.     

Principal Applications of Bayesian Methods in Actuarial Science

Abstract

Bayesian ideas were introduced into actuarial science in the late 1960s in the form of empirical credibility methods for premium setting. The advance of the Bayesian methodology was slow due to its subjective nature and to the computational difficulties associated with the full Bayesian analysis. This paper offers a brief survey of Bayesian solutions to some actuarial problems and discusses the current state of research.     

Bayesian Premium Rating with Latent Structure

We propose a fully Bayesian approach to non-life risk premium rating, based on hierarchical models with latent variables for both claim frequency and claim size. Inference is based on the joint posterior distribution and is performed by Markov Chain Monte Carlo. Rather than plug-in point estimates of all unknown parameters, we take into account all sources of uncertainty simultaneously when the model is used to predict claims and estimate risk premiums. Several models are fitted to both a simulated dataset and a small portfolio regarding theft from cars. We show that interaction among latent variables can improve predictions significantly. We also investigate when interaction is not necessary. We compare our results with those obtained under a standard generalized linear model and show through numerical simulation that geographically located and spatially interacting latent variables can successfully compensate for missing covariates. However, when applied to the real portfolio data, the proposed models are not better than standard models due to the lack of spatial structure in the data.     

Comparison of Some Bayesian Analyses of Heterogeneity in Group Life Insurance

Norberg (1989) analyses the heterogeneity in a portfolio of group life insurances using a parametric empirical Bayesian approach. In the present paper the model of Norberg is compared to a parametric fully Bayesian model and to a non-parametric fully Bayesian model.     

Risk-neutral and actual default probabilities with an endogenous bankruptcy jump-diffusion model

This paper focuses on historical and risk-neutral default probabilities in a structural model, when the firm assets dynamics are modeled by a double exponential jump diffusion process. Relying on the Leland [(1994a) Journal of Finance, 49, 1213–1252; (1994b) Bond prices, yield spreads, and optimal capital structure with default risk. Working paper no. 240, IBER, University of California, Berkeley] or Leland and Toft [(1996) Journal of Finance, 51(3), 987–1019] endogenous structural approaches, as formalized by Hilberink and Rogers [(2002) Finance and Stochastics, 6(2), 237–263], this article gives a coherent construction of historical default probabilities. The risk-neutral world where evolve the firm assets, modeled by a class of geometric Lévy processes, is constructed based on the Esscher measure, yielding useful and new analytical relations between historical and risk-neutral probabilities. We do a complete numerical analysis of the predictions of our framework, and compare these predictions with actual data. In particular, this new framework displays an enhanced predictive power w.r.t. current Gaussian endogenous structural models.      

Interval Estimation of Actuarial Risk Measures

Abstract

This article investigates performance of interval estimators of various actuarial risk measures. We consider the following risk measures: proportional hazards transform (PHT), Wang transform (WT), value-at-risk (VaR), and conditional tail expectation (CTE). Confidence intervals for these measures are constructed by applying nonparametric approaches (empirical and bootstrap), the strict parametric approach (based on the maximum likelihood estimators), and robust parametric procedures (based on trimmed means).

Using Monte Carlo simulations, we compare the average lengths and proportions of coverage (of the true measure) of the intervals under two data-generating scenarios: “clean” data and “contaminated” data. In the “clean” case, data sets are generated by the following (similar shape) parametric families: exponential, Pareto, and lognormal. Parameters of these distributions are selected so that all three families are equally risky with respect to a fixed risk measure. In the “contaminated” case, the “clean” data sets from these distributions are mixed with a small fraction of unusual observations (outliers). It is found that approximate knowledge of the underlying distribution combined with a sufficiently robust estimator (designed for that distribution) yields intervals with satisfactory performance under both scenarios.     

Model misspecification,Bayesian versus credibility estimation,and Gibbs posteriors

ABSTRACT

In the context of predicting future claims, a fully Bayesian analysis – one that specifies a statistical model, prior distribution, and updates using Bayes's formula – is often viewed as the gold-standard, while Bühlmann's credibility estimator serves as a simple approximation. But those desirable properties that give the Bayesian solution its elevated status depend critically on the posited model being correctly specified. Here we investigate the asymptotic behavior of Bayesian posterior distributions under a misspecified model, and our conclusion is that misspecification bias generally has damaging effects that can lead to inaccurate inference and prediction. The credibility estimator, on the other hand, is not sensitive at all to model misspecification, giving it an advantage over the Bayesian solution in those practically relevant cases where the model is uncertain. This begs the question: does robustness to model misspecification require that we abandon uncertainty quantification based on a posterior distribution? Our answer to this question is No, and we offer an alternative Gibbs posterior construction. Furthermore, we argue that this Gibbs perspective provides a new characterization of Bühlmann's credibility estimator.     

Modeling and Pricing Longevity Derivatives Using Stochastic Mortality Rates and the Esscher Transform

The Lee-Carter mortality model provides a structure for stochastically modeling mortality rates incorporating both time (year) and age mortality dynamics. Their model is constructed by modeling the mortality rate as a function of both an age and a year effect. Recently the MBMM model (Mitchell et al. 2013) showed the Lee Carter model can be improved by fitting with the growth rates of mortality rates over time and age rather than the mortality rates themselves. The MBMM modification of the Lee-Carter model performs better than the original and many of the subsequent variants. In order to model the mortality rate under the martingale measure and to apply it for pricing the longevity derivatives, we adapt the MBMM structure and introduce a Lévy stochastic process with a normal inverse Gaussian (NIG) distribution in our model. The model has two advantages in addition to better fit: first, it can mimic the jumps in the mortality rates since the NIG distribution is fat-tailed with high kurtosis, and, second, this mortality model lends itself to pricing of longevity derivatives based on the assumed mortality model. Using the Esscher transformation we show how to find a related martingale measure, allowing martingale pricing for mortality/longevity risk–related derivatives. Finally, we apply our model to pricing a q-forward longevity derivative utilizing the structure proposed by Life and Longevity Markets Association.     

Pricing derivatives with modeling CO2 emission allowance using a regime-switching jump diffusion model: with regime-switching risk premium

Carbon markets trade the spot European Union Allowance (EUA), with one EUA providing the right to emit one tone of carbon dioxide (CO₂). We examine the spot EUA returns in BlueNext that exhibit jumps and a volatility clustering feature. We propose a regime-switching jump diffusion model (RSJM) with a hidden Markov chain to capture not only a volatility clustering feature, but also the dynamics of the spot EUA returns that are influenced by change in the CO₂ emission economic conditions. In addition, the switching jump intensities of the RSJM are shown to be affected by change in the carbon-market macroeconomic environment. We further derive the theoretical futures-option prices with a constant convenience yield under the RSJM via the generalized Esscher transform where regime-switching risk is priced with a risk premium. The empirical study shows that the derived futures-option pricing model under the RSJM with regime-switching risk is a more complete model than a jump diffusion model for pricing CO₂ options.