Compound trend renewal process with discounted claims: a unified approach

We derive recursive formulas for the moments of compound trend renewal sums with discounted claims. An integral expression for the moment generating function of this risk process is then obtained, from which particular distribution functions are found. We extend the compound (deterministic) trend renewal process by assuming a stochastic trend, a stochastic force of net interest and a stochastic dependence between the inter-occurrence times and the severities of the claims. Finally, stochastic dominance ordering is also observed between the compound trend renewal process and an associated non-homogeneous Poisson process.     

Negative real interest rates

Standard textbook general equilibrium term structure models such as that developed by Cox, Ingersoll, and Ross [1985b. “A Theory of the Term Structure of Interest Rates.” Econometrica 53 (2): 385–407], do not accommodate negative real interest rates. Given this, the Cox, Ingersoll, and Ross [1985b. “A Theory of the Term Structure of Interest Rates.” Econometrica 53 (2): 385–407] ‘technological uncertainty variable’ is formulated in terms of the Pearson Type IV probability density. The Pearson Type IV encompasses mean-reverting sample paths, time-varying volatility and also allows for negative real interest rates. The Fokker–Planck (i.e. the Chapman–Kolmogorov) equation is then used to determine the conditional moments of the instantaneous real rate of interest. These enable one to determine the mean and variance of the accumulated (i.e. integrated) real rate of interest on a bank (or loan) account when interest accumulates at the instantaneous real rate of interest defined by the Pearson Type IV probability density. A pricing formula for pure discount bonds is also developed. Our empirical analysis of short-dated Treasury bills shows that real interest rates in the UK and the USA are strongly compatible with a general equilibrium term structure model based on the Pearson Type IV probability density.     

Moment generating functions of compound renewal sums with discounted claims

Léveillé & Garrido (2001a, 2001b) have obtained recursive formulas for the moments of compound renewal sums with discounted claims, which incorporate both, Andersen's (1957) generalization of the classical risk model, where the claim number process is an ordinary renewal process, and Taylor's (1979), where the joint effect of the claims cost inflation and investment income on a compound Poisson risk process is considered.

In this paper, assuming certain regularity conditions, we improve the preceding results by examining more deeply the asymptotic and finite time moment generating functions of the discounted aggregate claims process. Examples are given for claim inter-arrival times and claim severity following phase-type distributions, such as the Erlang case.     

On the discounted aggregate claim costs until ruin in dependent Sparre Andersen risk processes

In this paper, a dependent Sparre Andersen risk process in which the joint density of the interclaim time and the resulting claim severity satisfies the factorization as in Willmot and Woo is considered. We study a generalization of the Gerber–Shiu function (i) whose penalty function further depends on the surplus level immediately after the second last claim before ruin; and (ii) which involves the moments of the discounted aggregate claim costs until ruin. The generalized discounted density with a moment-based component proposed in Cheung plays a key role in deriving recursive defective renewal equations. We pay special attention to the case where the marginal distribution of the interclaim times is Coxian, and the required components in the recursion are obtained. A reverse type of dependency structure, where the claim severities follow a combination of exponentials, is also briefly discussed, and this leads to a nice explicit expression for the expected discounted aggregate claims until ruin. Our results are applied to generate some numerical examples involving (i) the covariance of the time of ruin and the discounted aggregate claims until ruin; and (ii) the expectation, variance and third central moment of the discounted aggregate claims until ruin.     

Stochastic volatility: A tale of co-jumps,non-normality,GMM and high frequency data

In this article we introduce a linear–quadratic volatility model with co-jumps and show how to calibrate this model to a rich dataset. We apply GMM and more specifically match the moments of realized power and multi-power variations, which are obtained from high-frequency stock market data. Our model incorporates two salient features: the setting of simultaneous jumps in both return process and volatility process and the superposition structure of a continuous linear–quadratic volatility process and a Lévy-driven Ornstein–Uhlenbeck process. We compare the quality of fit for several models, and show that our model outperforms the conventional jump diffusion or Bates model. Besides that, we find evidence that the jump sizes are not normally distributed and that our model performs best when the distribution of jump-sizes is only specified through certain (co-) moment conditions. Monte Carlo experiments are employed to confirm this.     

Recursive Moments of Compound Renewal Sums with Discounted Claims

Under regularity conditions, Le´veille´& Garrido [6] gives a derivation of the first two moments (resp. asymptotic) of a Compound Renewal Present Value Risk (CRPVR) process using renewal theory arguments. In this paper, with the same procedure and assuming that all the moments of the claim severity and the claims number process exist, we get recursive formulas for all the moments (resp. asymptotic) of the CRPVR process.     

The Valuation of American Options with Stochastic Interest Rates: A Generalization of the Geske—Johnson Technique

The Geske–Johnson approach provides an efficient and intuitively appealing technique for the valuation and hedging of American-style contingent claims. Here, we generalize their approach to a stochastic interest rate economy. The method is implemented using options exercisable on one of a finite number of dates. We illustrate how the value of an American-style option increases with interest rate volatility. The magnitude of this effect depends on the extent to which the option is in the money, the volatilities of the underlying asset and the interest rates, as well as the correlation between them.     

Ruin probabilities in classical risk models with gamma claims

In this paper, we provide three equivalent expressions for ruin probabilities in a Cramér–Lundberg model with gamma distributed claims. The results are solutions of integro-differential equations, derived by means of (inverse) Laplace transforms. All the three formulas have infinite series forms, two involving Mittag–Leffler functions and the third one involving moments of the claims distribution. This last result applies to any other claim size distributions that exhibits finite moments.     

Moments of renewal shot-noise processes and their applications

In this paper, we study the family of renewal shot-noise processes. The Feynmann–Kac formula is obtained based on the piecewise deterministic Markov process theory and the martingale methodology. We then derive the Laplace transforms of the conditional moments and asymptotic moments of the processes. In general, by inverting the Laplace transforms, the asymptotic moments and the first conditional moments can be derived explicitly; however, other conditional moments may need to be estimated numerically. As an example, we develop a very efficient and general algorithm of Monte Carlo exact simulation for estimating the second conditional moments. The results can be then easily transformed to the counterparts of discounted aggregate claims for insurance applications, and we apply the first two conditional moments for the actuarial net premium calculation. Similarly, they can also be applied to credit risk and reliability modelling. Numerical examples with four distribution choices for interarrival times are provided to illustrate how the models can be implemented.     

Realised higher moments: theory and practice

This paper examines the incorporation of higher moments in portfolio selection problems utilising high-frequency data. Our approach combines innovations from the realised volatility literature with a portfolio selection methodology utilising higher moments. We provide an empirical study of the measurement of higher moments from tick by tick data and implement the model for a selection of stocks from the DOW 30 over the time period 2005–2011. We demonstrate a novel estimator for moments and co-moments in the presence of microstructure noise.     

m-Double Poisson Lévy markets

We develop novel mispricing of markets under asymmetric information and jumps for informed and uninformed investors, called m-Double Poisson markets, driven by independent Double Poisson processes. In the special case m?=?1, called the Double Poisson pure-jump Lévy market, both types of investors hold the same optimal portfolio and expected utility, and hence, the informed investor has no utility advantage over the uninformed. For the general market, instantaneous centralized moments of returns are used to compute optimal portfolios and utilities. The mean, variance, skewness and kurtosis of instantaneous returns are reported using jump amplitudes and frequencies.     

On alternative interest rate processes

In this paper alternative interest rate processes are estimated for Denmark, Germany, Sweden, and the UK, using the generalized method of moments (GMM). In line with the study by Chan, Karolyi, Longstaff, and Sanders (1992) on US data, there seems to be a positive relation between interest rate level and volatility for some countries. In contrast to their study, it is found that mean-reversion plays an important role for the specification of the interest rate dynamics. The results seem to be robust to the use of different moment conditions, and simulations of the estimated models reveal that they are fairly able to capture non-fitted moments as well. In addition, there is evidence of a structural change in the Danish interest rate process in August 1985, which may be due to a change in monetary policy. The small sample properties of the GMM estimators are also studied through simulations.     

Optimal portfolio allocation with higher moments

We model the risky asset as driven by a pure jump process, with non-trivial and tractable higher moments. We compute the optimal portfolio strategy of an investor with CRRA utility and study the sensitivity of the investment in the risky asset to the higher moments, as well as the resulting wealth loss from ignoring higher moments. We find that ignoring higher moments can lead to significant overinvestment in risky securities, especially when volatility is high.      

A new approach for option pricing under stochastic volatility

We develop a new approach for pricing European-style contingent claims written on the time T spot price of an underlying asset whose volatility is stochastic. Like most of the stochastic volatility literature, we assume continuous dynamics for the price of the underlying asset. In contrast to most of the stochastic volatility literature, we do not directly model the dynamics of the instantaneous volatility. Instead, taking advantage of the recent rise of the variance swap market, we directly assume continuous dynamics for the time T variance swap rate. The initial value of this variance swap rate can either be directly observed, or inferred from option prices. We make no assumption concerning the real world drift of this process. We assume that the ratio of the volatility of the variance swap rate to the instantaneous volatility of the underlying asset just depends on the variance swap rate and on the variance swap maturity. Since this ratio is assumed to be independent of calendar time, we term this key assumption the stationary volatility ratio hypothesis (SVRH). The instantaneous volatility of the futures follows an unspecified stochastic process, so both the underlying futures price and the variance swap rate have unspecified stochastic volatility. Despite this, we show that the payoff to a path-independent contingent claim can be perfectly replicated by dynamic trading in futures contracts and variance swaps of the same maturity. As a result, the contingent claim is uniquely valued relative to its underlying’s futures price and the assumed observable variance swap rate. In contrast to standard models of stochastic volatility, our approach does not require specifying the market price of volatility risk or observing the initial level of instantaneous volatility. As a consequence of our SVRH, the partial differential equation (PDE) governing the arbitrage-free value of the contingent claim just depends on two state variables rather than the usual three. We then focus on the consistency of our SVRH with the standard assumption that the risk-neutral process for the instantaneous variance is a diffusion whose coefficients are independent of the variance swap maturity. We show that the combination of this maturity independent diffusion hypothesis (MIDH) and our SVRH implies a very special form of the risk-neutral diffusion process for the instantaneous variance. Fortunately, this process is tractable, well-behaved, and enjoys empirical support. Finally, we show that our model can also be used to robustly price and hedge volatility derivatives.     

Insurance Considering a New Stochastic Model for the Discount Factor

In many empirical situations (e.g.: Libor), the rate of interest will remain fixed at a certain level (random instantaneous rate i i ) for a random period of time ( t i ) until a new random rate should be considered, i i + 1 , that will remain for t i + 1 , waiting time until the next change in the rate of interest. Three models were developed using the approach cited above for random rate of interest and random waiting times between changes in the rate of interest. Using easy integral transforms (Laplace & Fourier) we will be able to calculate the moments of the probability function of the discount factor, V ( t ), and even its c.d.f. The approach will also be extended to the calculation of the expected value (net premium) and variance of a term insurance and we will get its c.d.f., something not very common in actuarial literature due to its complexity, but very useful when the law of large numbers cannot be applied and consequently use normal approximations.     

The maximum entropy mortality model: forecasting mortality using statistical moments

ABSTRACT

The age-at-death distribution is a representation of the mortality experience in a population. Although it proves to be highly informative, it is often neglected when it comes to the practice of past or future mortality assessment. We propose an innovative method to mortality modeling and forecasting by making use of the location and shape measures of a density function, i.e. statistical moments. Time series methods for extrapolating a limited number of moments are used and then the reconstruction of the future age-at-death distribution is performed. The predictive power of the method seems to be net superior when compared to the results obtained using classical approaches to extrapolating age-specific-death rates, and the accuracy of the point forecast (MASE) is improved on average by 33% respective to the state-of-the-art, the Lee–Carter model. The method is tested using data from the Human Mortality Database and implemented in a publicly available R package.     

A model of discontinuous interest rate behavior,yield curves,and volatility

This paper develops an equilibrium model in which interest rates follow a discontinuous (generalized) gamma process. The gamma process has finite variation, takes an infinite number of “small” jumps in every interval, and includes the Wiener process as a limiting case. The gamma interest rate model produces yield curves that closely resemble those of diffusion models. But in contrast to diffusion models, the curvature of the yield curve does not directly depend on the true volatility of the interest rate process, but instead depends on a different risk-neutral volatility. The gamma model appears to fit the distribution of interest rates changes and the jump characteristics of interest rate paths. Empirical tests reject a diffusion model of interest rates in favor of the more general gamma model because daily interest rate innovations are highly leptokurtic. The author appreciates comments from George Constantinides, Jon Ingersoll, Herbert Johnson, Ray Rishel, and an anonymous referee, computational assistance from Kerry Back and Saikat Nandi, and support from Atlantic Asset Management. Any errors are the responsibility of the author.     

A binomial approximation for two-state Markovian HJM models

This article develops a lattice algorithm for pricing interest rate derivatives under the Heath et al. (Econometrica 60:77–105, 1992) paradigm when the volatility structure of forward rates obeys the Ritchken and Sankarasubramanian (Math Financ 5:55–72) condition. In such a framework, the entire term structure of the interest rate may be represented using a two-dimensional Markov process, where one state variable is the spot rate and the other is an accrued variance statistic. Unlike in the usual approach based on the Nelson-Ramaswamy (Rev Financ Stud 3:393–430) transformation, we directly discretize the heteroskedastic spot rate process by a recombining binomial tree. Further, we reduce the computational cost of the pricing problem by associating with each node of the lattice a fixed number of accrued variance values computed on a subset of paths reaching that node. A backward induction scheme coupled with linear interpolation is used to evaluate interest rate contingent claims.     

Two-Sided Bounds for the Finite Time Probability of Ruin

Explicit, two-sided bounds are derived for the probability of ruin of an insurance company, whose premium income is represented by an arbitrary, increasing real function, the claims are dependent, integer valued r.v.s and their inter-occurrence times are exponentially, non-identically distributed. It is shown, that the two bounds coincide when the moments of the claims form a Poisson point process. An expression for the survival probability is further derived in this special case, assuming that the claims are integer valued, i.i.d. r.v.s. This expression is compared with a different formula, obtained recently by Picard & Lefevre (1997) in terms of generalized Appell polynomials. The particular case of constant rate premium income and non-zero initial capital is considered. A connection of the survival probability to multivariate B -splines is also established.     

General equilibrium pricing of currency and currency options

This paper presents a consumption-based general equilibrium model for valuing foreign exchange contingent claims. The model identifies a novel economic mechanism by exploiting highly but imperfectly shared consumption disaster with variable intensities which are the concerns to the representative investor under recursive utility. When applied to the data, the model simultaneously replicates (i) the moderate option-implied volatilities; (ii) substantial variations in the risk-neutral skewness of currency returns; (iii) the uncovered interest rate parity puzzle; and (iv) the first two moments of carry trade returns. Furthermore, the model rationalizes salient features of the aggregate stock, government bonds, and equity index options.