On choice of statistics in credibility estimation

Abstract

The question on what statistics to base our credibility estimators, is discussed in a general model. We introduce concepts of sufficiency, completeness, θ-sufficiency, and θ-completeness that are useful in this connection, and use methods of Rao-Blackwell type. Some of the present results are closely related to results by Taylor (1977).     

On the asymptotic normality of some unbiased estimates

Abstract

Let X _m(n) =(X _j , n, ..., X _{j m,n} ) be a subset of observations of a sample X_n = (X_1n X _2n ... , X _nn ). Here the X_jn 'S in Xn are not necessarily independent or identically distributed, and m(n) mayor may not tend to infinity as n tends to infinity. Suppose the joint density function h_n =h_n (x _m (n); θ) of the X _jn 's in Xm(n) is completely specified except the values of the parameters in the parameter vector θ = (θ1 θ2, ... , θ _k ), where θ belongs to a non-degenerate open subset H of the k-dimensional Euclidean space Rk and k?m(n).     

Remark on a theorem by Gurland

Abstract

Let X ₁,X ₂,...,X _n be a random sample of size from a distribution with probability density function p(x|θ), where the unknown parameter θ belongs to a non-degenerate interval I. The unknown true value of θ will be denoted by θ₀.     

On accurate and provably efficient GARCH option pricing algorithms

The GARCH model has been very successful in capturing the serial correlation of asset return volatilities. As a result, applying the model to options pricing attracts a lot of attention. However, previous tree-based GARCH option pricing algorithms suffer from exponential running time, a cut-off maturity, inaccuracy, or some combination thereof. Specifically, this paper proves that the popular trinomial-tree option pricing algorithms of Ritchken and Trevor (Ritchken, P. and Trevor, R., Pricing options under generalized GARCH and stochastic volatility processes. J. Finance, , 54(1), 377–402.) and Cakici and Topyan (Cakici, N. and Topyan, K., The GARCH option pricing model: a lattice approach. J. Comput. Finance, , 3(4), 71–85.) explode exponentially when the number of partitions per day, n, exceeds a threshold determined by the GARCH parameters. Furthermore, when explosion happens, the tree cannot grow beyond a certain maturity date, making it unable to price derivatives with a longer maturity. As a result, the algorithms must be limited to using small n, which may have accuracy problems. The paper presents an alternative trinomial-tree GARCH option pricing algorithm. This algorithm provably does not have the short-maturity problem. Furthermore, the tree-size growth is guaranteed to be quadratic if n is less than a threshold easily determined by the model parameters. This level of efficiency makes the proposed algorithm practical. The surprising finding for the first time places a tree-based GARCH option pricing algorithm in the same complexity class as binomial trees under the Black–Scholes model. Extensive numerical evaluation is conducted to confirm the analytical results and the numerical accuracy of the proposed algorithm. Of independent interest is a simple and efficient technique to calculate the transition probabilities of a multinomial tree using generating functions.     

Econometric analysis of microscopic simulation models

Microscopic simulation models are often evaluated based on visual inspection of the results. This paper presents formal econometric techniques to compare microscopic simulation (MS) models with real-life data. A related result is a methodology to compare different MS models with each other. For this purpose, possible parameters of interest, such as mean returns, or autocorrelation patterns, are classified and characterized. For each class of characteristics, the appropriate techniques are presented. We illustrate the methodology by comparing the MS model developed by He and Li [J. Econ. Dynam. Control, 2007, 31, 3396–3426, Quant. Finance, 2008, 8, 59–79] with actual data.     

Time-varying long-run mean of commodity prices and the modeling of futures term structures

The exploration of the mean-reversion of commodity prices is important for inventory management, inflation forecasting and contingent claim pricing. Bessembinder et al. [J. Finance, 1995, 50, 361–375] document the mean-reversion of commodity spot prices using futures term structure data; however, mean-reversion to a constant level is rejected in nearly all studies using historical spot price time series. This indicates that the spot prices revert to a stochastic long-run mean. Recognizing this, I propose a reduced-form model with the stochastic long-run mean as a separate factor. This model fits the futures dynamics better than do classical models such as the Gibson–Schwartz [J. Finance, 1990, 45, 959–976] model and the Casassus–Collin-Dufresne [J. Finance, 2005, 60, 2283–2331] model with a constant interest rate. An application for option pricing is also presented in this paper.     

Discussion of Barker and Teixeira ([2018]. Gaps in the IFRS Conceptual Framework. Accounting in Europe, 15) and Van Mourik and Katsuo ([2018]. Profit or loss in the IASB Conceptual Framework. Accounting in Europe, 15)

Abstract

I provide comments on two papers, Barker and Teixeira ([2018]. Gaps in the IFRS Conceptual Framework. Accounting in Europe, 15) and Van Mourik and Katsuo ([2018]. Profit or loss in the IASB Conceptual Framework. Accounting in Europe, 15), in this issue, which were presented at the EAA-IASB research forum in Brussels. The paper accepts the shortcomings of the updated IASB conceptual framework and argues that these are in large part due to the origins of the document. It points out that the original US project was an attempt to make standard-setting more consistent and involved creating principles which would explain existing standards. Constituents have subsequently resisted attempts to make the framework theoretically sound because they fear this will encourage too much innovation. Standard-setters prefer incremental change, so continue to work with a model created to resolve a problem of the 1970s. I suggest that since standard-setting has been professionalised, the more significant need to is to define what information investors find useful. This may involve providing more granular information about the entity’s business model.     

A hierarchical credibility regression model

Abstract

The paper develops a hierarchical credibility regression model with random parameters on two levels. The common credibility regression models appear as special cases by assuming one of the parameter levels degenerate. By specializing into another direction we obtain Jewell's (1975c) hierarchical model as a special case.     

Deriving the arbitrage pricing theory when the number of factors is unknown

This paper examines the use of proxies (or reference variables) for the true factors in the arbitrage pricing theory (APT). It generalizes other authors' existing work and shows that, when there are more reference variables than the true factors, the APT still holds. The possibility of fewer reference variables than the true factors is also considered, but the APT is not shown to hold, in the same sense, for this case. This work builds on an earlier paper by Ingersoll (Ingersoll J 1984 J. Finance 39 1021-39), and our propositions can be thought of as specializations of his theorems. Similar to Nawalkha (Nawalkha S 1997 J. Financial Economics 46 357-81), our work does not use the mathematics of Hilbert and Banach spaces and, thus, is open to a much wider audience. The practical implication of our results is that model builders should be generous with the number of factors they use, as excessively parsimonious models suffer from inaccuracy.     

Purged and partial Markov chains

Abstract

In multiple decrement theory, disability theory, nuptiality and fertility theory, and in other connections, actuaries and demographers often study what in effect is simple Markov chain models. In these models, the transition probabilities, are defined on the basis of the forces of transition (infinitesimal transition probabilities). For various reasons, new models are sometimes constructed by replacing some of the original forces of transition by 0. We shall call the transition probabilities of such a new model partial probabilities.     

A Simple Model of Insurance Market Dynamics

Abstract

This paper constructs and studies a simple but realistic model of an insurance market. The model has a minimalist construction in the sense that the number of parameters defining it is strictly limited and the elimination of any one of them would destroy its realism. There are 11 essential parameters. Each of the parameters has a physical interpretation. Some determine competitive effects within the market, some barriers to entry, and so on. The effect of each on various aspects of the market is examined in the presence of simulated loss experience. The aspects of the market considered include stability of premium rates, profitability, and market concentration. Some of the parameters are capable of use as regulatory controls. Two parameters, in addition to the original 11, are explicit price controls. Despite its simplicity, the model displays considerably complex behavior. Some results are intuitive, but some are not. For this reason, regulatory controls need to be applied with great caution lest they induce perverse effects, possibly even the reverse of those intended. The effect of the parameters on market behavior is first studied in the absence of catastrophic events from the loss experience. Subsequently, the effect of a single such event is studied.     

On certain compound frequency distributions

Abstract

Let ?(χ/y, z, …) be an ordinary frequency distribution where χ is the variate and y, z, … are parameters characterising the function. If then χ is a graduated variate fd 32_1 represents the probability of an observation drawn from this universe, falling between χ and χ + dχ. This probability is a function of χ. It is, however, also a function of the parameters. If one or more parameters are changed, this probability also changes. d f is a relative probability dependent on the values of y, z, …     

A Gravity Model of Mortality Rates for Two Related Populations

Abstract

The mortality rate dynamics between two related but different-sized populations are modeled consistently using a new stochastic mortality model that we call the gravity model. The larger population is modeled independently, and the smaller population is modeled in terms of spreads (or deviations) relative to the evolution of the former, but the spreads in the period and cohort effects between the larger and smaller populations depend on gravity or spread reversion parameters for the two effects. The larger the two gravity parameters, the more strongly the smaller population’s mortality rates move in line with those of the larger population in the long run. This is important where it is believed that the mortality rates between related populations should not diverge over time on grounds of biological reasonableness. The model is illustrated using an extension of the Age-Period-Cohort model and mortality rate data for English and Welsh males representing a large population and the Continuous Mortality Investigation assured male lives representing a smaller related population.     

The stochastic collocation Monte Carlo sampler: highly efficient sampling from ‘expensive’ distributions

In this article, we propose an efficient approach for inverting computationally expensive cumulative distribution functions. A collocation method, called the Stochastic Collocation Monte Carlo sampler (SCMC sampler), within a polynomial chaos expansion framework, allows us the generation of any number of Monte Carlo samples based on only a few inversions of the original distribution plus independent samples from a standard normal variable. We will show that with this path-independent collocation approach the exact simulation of the Heston stochastic volatility model, as proposed in Broadie and Kaya [Oper. Res., 2006, 54, 217–231], can be performed efficiently and accurately. We also show how to efficiently generate samples from the squared Bessel process and perform the exact simulation of the SABR model.     

A Note on Gaussian Estimation of the CKLS and CIR Models with Feedback Effects for Japan

In this note we extend the Gaussian estimation of two factor CKLS and CIR models recently considered in Nowman, K. B. (2001, Gaussian estimation and forecasting of multi-factor term structure models with an application to Japan and the United Kingdom, Asia Pacif. Financ. Markets 8, 23–34) to include feedback effects in the conditional mean as was originally formulated in general continuous time models by Bergstrom, A. R. (1966, Non-recursive models as discrete approximations to systems of stochastic differential equations, Econometrica 34, 173–182) with constant volatility. We use the exact discrete model of Bergstrom, A. R. (1966, Non-recursive models as discrete approximations to systems of stochastic differential equations, Econometrica 34, 173–182) to estimate the parameters which was first used by Brennan, M. J. and Schwartz, E. S. (1979, A continuous time approach to the pricing of bonds, J. Bank. Financ. 3, 133–155) to estimate their two factor interest model but incorporating the assumption of Nowman, K. B. (1997, Gaussian estimation of single-factor continuous time models of the term structure of interest rates, J. Financ. 52, 1695–1706; 2001, Gaussian estimation and forecasting of multi-factor term structure models with an application to Japan and the United Kingdom, Asia Pacif. Financ. Markets 8, 23–34). An application to monthly Japanese Euro currency rates indicates some evidence of feedback from the 1-year rate to the 1-month rate in both the CKLS and CIR models. We also find a low level-volatility effect supporting Nowman, K. B. (2001, Gaussian estimation and forecasting of multi-factor term structure models with an application to Japan and the United Kingdom, Asia Pacif. Financ. Markets 8, 23–34).     

The survival probability of the SABR model: asymptotics and application

The stochastic-alpha-beta-rho (SABR) model is widely used by practitioners in interest rate and foreign exchange markets. The probability of hitting zero sheds light on the arbitrage-free small strike implied volatility of the SABR model (see, e.g. De Marco et al. [SIAM J. Financ. Math., 2017, 8(1), 709–737], Gulisashvili [Int. J. Theor. Appl. Financ., 2015, 18, 1550013], Gulisashvili et al. [Mass at zero in the uncorrelated SABR modeland implied volatility asymptotics, 2016b]), and the survival probability is also closely related to binary knock-out options. Besides, the study of the survival probability is mathematically challenging. This paper provides novel asymptotic formulas for the survival probability of the SABR model as well as error estimates. The formulas give the probability that the forward price does not hit a nonnegative lower boundary before a fixed time horizon.     

Minimizing CVaR in global dynamic hedging with transaction costs

This study develops a global derivatives hedging methodology which takes into account the presence of transaction costs. It extends the Hodges and Neuberger [Rev. Futures Markets, 1989, 8, 222–239] framework in two ways. First, to reduce the occurrence of extreme losses, the expected utility is replaced by the conditional Value-at-Risk (CVaR) coherent risk measure as the objective function. Second, the normality assumption for the underlying asset returns is relaxed: general distributions are considered to improve the realism of the model and to be consistent with fat tails observed empirically. Dynamic programming is used to solve the hedging problem. The CVaR minimization objective is shown to be part of a time-consistent framework. Simulations with parameters estimated from the S&P 500 financial time series show the superiority of the proposed hedging method over multiple benchmarks from the literature in terms of tail risk reduction.     

Hedging jump risk,expected returns and risk premia in jump-diffusion economies

This article presents a pure exchange economy that extends Rubinstein [Bell J. Econ. Manage. Sci., 1976, 7, 407–425] to show how the jump-diffusion option pricing model of Black and Scholes [J. Political Econ., 1973, 81, 637–654] and Merton [J. Financ. Econ., 1976, 4, 125–144] evolves in gamma jumping economies. From empirical analysis and theoretical study, both the aggregate consumption and the stock price are unknown in determining jumping times. By using the pricing kernel, we determine both the aggregate consumption jump time and the stock price jump time from the equilibrium interest rate and CCAPM (Consumption Capital Asset Pricing Model). Our general jump-diffusion option pricing model gives an explicit formula for how the jump process and the jump times alter the pricing. This innovation with predictable jump times enhances our analysis of the expected stock return in equilibrium and of hedging jump risks for jump-diffusion economies.