Bayesian comparison of several continuous time models of the Australian short rate

This paper provides an empirical analysis of a range of alternative single‐factor continuous time models for the Australian short‐term interest rate. The models are nested in a general single‐factor diffusion process for the short rate, with each alternative model indexed by the level effect parameter for the volatility. The inferential approach adopted is Bayesian, with estimation of the models proceeding through a Markov chain Monte Carlo simulation scheme. Discrimination between the alternative models is based on Bayes factors. A data augmentation approach is used to improve the accuracy of the discrete time approximation of the continuous time models. An empirical investigation is conducted using weekly observations on the Australian 90 day interest rate from January 1990 to July 2000. The Bayes factors indicate that the square root diffusion model has the highest posterior probability of all models considered.     

Expected Log-Utility Maximization Under Incomplete Information and with Cox-Process Observations

We consider the portfolio optimization problem for the criterion of maximization of expected terminal log-utility. The underlying market model is a regime-switching diffusion model where the regime is determined by an unobservable factor process forming a finite state Markov process. The main novelty is due to the fact that prices are observed and the portfolio is rebalanced only at random times corresponding to a Cox process where the intensity is driven by the unobserved Markovian factor process as well. This leads to a more realistic modeling for many practical situations, like in markets with liquidity restrictions; on the other hand it considerably complicates the problem to the point that traditional methodologies cannot be directly applied. The approach presented here is specific to the log-utility. For power utilities a different approach is presented in the companion paper (Fujimoto et al. in Appl Math Optim 67(1):33–72, 2013).     

Moments of the Dividend Payments and Related Problems in a Markov-Modulated Risk Model

Abstract

In this paper we derive some results on the dividend payments prior to ruin in a Markovmodulated risk process in which the rate for the Poisson claim arrival process and the distribution of the claim sizes vary in time depending on the state of an underlying (external) Markov jump process {J(t); t ≥ 0}. The main feature of the model is the flexibility in modeling the arrival process in the sense that periods with very frequent arrivals and periods with very few arrivals may alternate, and that the states of {J(t); t ≥ 0} could describe, for example, epidemic types in health insurance or weather conditions in car insurance. A system of integro-differential equations with boundary conditions satisfied by the nth moment of the present value of the total dividends prior to ruin, given the initial environment state, is derived and solved. We show that the probabilities that the surplus process attains a dividend barrier from the initial surplus without first falling below zero and the Laplace transforms of the time that the surplus process first hits a barrier without ruin occurring can be expressed in terms of the solution of the above-mentioned system of integro-differential equations. In the two-state model, explicit results are obtained when both claim amounts are exponentially distributed.     

Counterparty Risk for Credit Default Swaps: Markov Chain Interacting Intensities Model with Stochastic Intensity

We analyze the counterparty risk for credit default swaps using the Markov chain model of portfolio credit risk of multiple obligors with interacting default intensity processes. The default correlation between the protection seller and underlying entity is modeled by an increment in default intensity upon the occurrence of an external shock event. The arrival of the shock event is a Cox process whose stochastic intensity is assumed to follow an affine diffusion process with jumps. We examine how the correlated default risks between the protection seller and the underlying entity may affect the credit default premium in a credit default swap.     

Risk theory in a Markovian environment

Abstract

We consider risk processes _{t t?0} with the property that the rate β of the Poisson arrival process and the distribution of B of the claim sizes are not fixed in time but depend on the state of an underlying Markov jump process {Z_t } _t?0 such that β=β _i and B=B_i when Z_t=i . A variety of methods, including approximations, simulation and numerical methods, for assessing the values of the ruin probabilities are studied and in particular we look at the Cramér-Lundberg approximation and diffusion approximations with correction terms. The mathematical framework is Markov-modulated random walks in discrete and continuous time, and in particular Wiener-Hopf factorisation problems and conjugate distributions (Esscher transforms) are involved.     

Markov Aging Process and Phase-Type Law of Mortality

Abstract

In this article, we propose a finite-state Markov process with one absorbing state to model human mortality. A health index called physiological age is introduced and modeled by the Markov process. Under this model the time of death follows a phase-type distribution. The model possesses many desirable analytical properties useful for mortality analysis. Closed-form expressions are available for many quantities of interest including the conditional survival probabilities of the time of death and the actuarial present values of the whole life insurance and annuity. The heterogeneity or frailty effect of a cohort can be expressed explicitly. The model is also able to explain some stylized facts of observed mortality data. We fit the model to some Swedish population cohort data and life tables compiled by the U.S. Social Security Administration. The fitting results are very satisfactory.     

A general characterization of one factor affine term structure models

We give a complete characterization of affine term structure models based on a general nonnegative Markov short rate process. This applies to the classical CIR model but includes as well short rate processes with jumps. We provide a link to the theory of branching processes and show how CBI-processes naturally enter the field of term structure modelling. Using Markov semigroup theory we exploit the full structure behind an affine term structure model and provide a deeper understanding of some well-known properties of the CIR model. Based on these fundamental results we construct a new short rate model with jumps, which extends the CIR model and still gives closed form expressions for bond options. Manusript received: June 2000, final version received: October 2000     

“Social Security—Adequacy,Equity, and Progressiveness: A Review of Criteria Based on Experience In Canada and the United States”, Robert L. Brown,January 2000

Abstract

We consider the valuation of credit default swaps (CDSs) under an extended version of Merton’s structural model for a firm’s corporate liabilities. In particular, the interest rate process of a money market account, the appreciation rate, and the volatility of the firm’s value have switching dynamics governed by a finite-state Markov chain in continuous time. The states of the Markov chain are deemed to represent the states of an economy. The shift from one economic state to another may be attributed to certain factors that affect the profits or earnings of a firm; examples of such factors include changes in business conditions, corporate decisions, company operations, management strategies, macroeconomic conditions, and business cycles. In this article, the Esscher transform, which is a well-known tool in actuarial science, is employed to determine an equivalent martingale measure for the valuation problem in the incomplete market setting. Systems of coupled partial differential equations (PDEs) satisfied by the real-world and risk-neutral default probabilities are derived. The consequences for the swap rate of a CDS brought about by the regimeswitching effect of the firm’s value are investigated via a numerical example for the case of a two-state Markov chain. We perform sensitivity analyses for the real-world default probability and the swap rate when different model parameters vary. We also investigate the accuracy and efficiency of the PDE approach by comparing the numerical results from the PDE approach to those from the Monte Carlo simulation.     

Convexity theory for the term structure equation

We study the convexity and model parameter monotonicity properties for prices of bonds and bond options when the short rate is modeled by a diffusion process. We provide sharp conditions on the model parameters under which the convexity of the price in the short rate is guaranteed. Under these conditions, the price is decreasing in the drift and increasing in the volatility of the short rate. We also study the convexity properties of the logarithm of the price and find simple conditions on the coefficients that guarantee that the price is log-convex or log-concave.      

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     

Sensitivity analysis of long-term cash flows

This paper conducts a sensitivity analysis of long-term cash flows. The price of the cash flow at time zero is given by the pricing operator of a Markov diffusion acting on the cash flow function. We study the extent to which the price of the cash flow is affected by small perturbations of the underlying Markov diffusion. The main tool is the Hansen–Scheinkman decomposition, which is a method to express the cash flow in terms of eigenvalues and eigenfunctions of the pricing operator. By incorporating techniques developed by Fournié et al. (Finance Stoch. 3:391–412, 1999), the sensitivities of long-term cash flows can be represented via simple expressions in terms of eigenvalues and eigenfunctions.     

A stochastic differential game for optimal investment of an insurer with regime switching

We introduce a model to discuss an optimal investment problem of an insurance company using a game theoretic approach. The model is general enough to include economic risk, financial risk, insurance risk, and model risk. The insurance company invests its surplus in a bond and a stock index. The interest rate of the bond is stochastic and depends on the state of an economy described by a continuous-time, finite-state, Markov chain. The stock index dynamics are governed by a Markov, regime-switching, geometric Brownian motion modulated by the chain. The company receives premiums and pays aggregate claims. Here the aggregate insurance claims process is modeled by either a Markov, regime-switching, random measure or a Markov, regime-switching, diffusion process modulated by the chain. We adopt a robust approach to model risk, or uncertainty, and generate a family of probability measures using a general approach for a measure change to incorporate model risk. In particular, we adopt a Girsanov transform for the regime-switching Markov chain to incorporate model risk in modeling economic risk by the Markov chain. The goal of the insurance company is to select an optimal investment strategy so as to maximize either the expected exponential utility of terminal wealth or the survival probability of the company in the ‘worst-case’ scenario. We formulate the optimal investment problems as two-player, zero-sum, stochastic differential games between the insurance company and the market. Verification theorems for the HJB solutions to the optimal investment problems are provided and explicit solutions for optimal strategies are obtained in some particular cases.     

Modelling and forecasting short-term interest rate volatility: A semiparametric approach

This paper employs a semiparametric procedure to estimate the diffusion process of short-term interest rates. The Monte Carlo study shows that the semiparametric approach produces more accurate volatility estimates than models that accommodate asymmetry, level effect and serial dependence in the conditional variance. Moreover, the semiparametric approach yields robust volatility estimates even if the short rate drift function and the underlying innovation distribution are misspecified. Empirical investigation with the U.S. three-month Treasury bill rates suggests that the semiparametric procedure produces superior in-sample and out-of-sample forecast of short rate changes volatility compared with the widely used single-factor diffusion models. This forecast improvement has implications for pricing interest rate derivatives.     

A multivariate Markov chain stock model

ABSTRACT

We propose a dividend stock valuation model where multiple dividend growth series and their dependencies are modelled using a multivariate Markov chain. Our model advances existing Markov chain stock models. First, we determine assumptions that guarantee the finiteness of the price and risk as well as the fulfilment of transversality conditions. Then, we compute the first- and second-order price-dividend ratios by solving corresponding linear systems of equations and show that a different price-dividend ratio is attached to each combination of states of the dividend growth process of each stock. Subsequently, we provide a formula for the computation of the variances and covariances between stocks in a portfolio. Finally, we apply the theoretical model to the dividend series of three US stocks and perform comparisons with existing models. The results could also be applied for actuarial purposes as a general stochastic investment model and for calculating the initial endowment to fund a portfolio of dependent perpetuities.     

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.     

A PDE approach for risk measures for derivatives with regime switching

This paper considers a partial differential equation (PDE) approach to evaluate coherent risk measures for derivative instruments when the dynamics of the risky underlying asset are governed by a Markov-modulated geometric Brownian motion (GBM); that is, the appreciation rate and the volatility of the underlying risky asset switch over time according to the state of a continuous-time hidden Markov chain model which describes the state of an economy. The PDE approach provides market practitioners with a flexible and effective way to evaluate risk measures in the Markov-modulated Black–Scholes model. We shall derive the PDEs satisfied by the risk measures for European-style options, barrier options and American-style options.      

Regime-Switching Periodic Models For Claim Counts

Abstract

We study a Cox risk model that accounts for both seasonal variations and random fluctuations in the claims intensity. This occurs with natural phenomena that evolve in a seasonal environment and affect insurance claims, such as hurricanes.

More precisely, we define an intensity process governed by a periodic function with a random peak level. The periodic intensity function follows a deterministic pattern in each short-term period and is illustrated by a beta-type function. A Markov chain with m states, corresponding to different risk levels, is chosen for the level process, yielding a so-called regime-switching process.

The properties of the corresponding claim-counting process are discussed in detail. By properly defining a Lundberg-type coefficient, we derive upper bounds for finite time ruin probabilities in a two-state case.     

Optimal portfolio choice in the bond market

We consider the Merton problem of optimal portfolio choice when the traded instruments are the set of zero-coupon bonds. Working within a Markovian Heath–Jarrow–Morton model of the interest rate term structure driven by an infinite-dimensional Wiener process, we give sufficient conditions for the existence and uniqueness of an optimal trading strategy. When there is uniqueness, we provide a characterization of the optimal portfolio as a sum of mutual funds. Furthermore, we show that a Gauss–Markov random field model proposed by Kennedy [Math. Financ. 4, 247–258(1994)] can be treated in this framework, and explicitly calculate the optimal portfolio. We show that the optimal portfolio in this case can be identified with the discontinuities of a certain function of the market parameters.