Mogens Bladt

An actuarial approach to option pricing under the physical measure and without market assumptions

Abstract As the title may indicate, this paper uses merely probabilistic and actuarial considerations for pricing options. There are no economical considerations involved, and our approach is valid even when an equilibrium price measure does not exist (arbitrage, non-equilibrium) or is not unique (incompleteness). We only make use of the physical measure that generates the pay-out distributions. The approach does not in general carry over to general derivative securities, since we use an interpretation of the securities under consideration as being potential losses or claims from the issuers point of view. Under this interpretation we calculate the price of the security as the fair premium needed to insure the potential loss. As a special case of our formula we derive the Black and Scholes formula.

A Review on Phase-type Distributions and their Use in Risk Theory

Phase-type distributions, defined as the distributions of absorption times of certain Markov jump processes, constitute a class of distributions on the positive real axis which seems to strike a balance between generality and tractability. Indeed, any positive distribution may be approximated arbitrarily closely by phase-type distributions whereas exact solutions to many complex problems in stochastic modeling can be obtained either explicitly or numerically. In this paper we introduce phase-type distributions and retrieve some of their basic properties through appealing probabilistic arguments which, indeed, constitute their main feature of being mathematically tractable. This is illustrated in an example where we calculate the ruin probability for a rather general class of surplus processes where the premium rate is allowed to depend on the current reserve and where claims sizes are assumed to be of phase-type. Finally we discuss issues concerning statistical inference for phase-type distributions and related functionals such as e.g. a ruin probability.

Matrix‐Exponential Distributions: Calculus and Interpretations via Flows

By considering randomly stopped deterministic flow models, we develop an intuitively appealing way to generate probability distributions with rational Laplace–Stieltjes transforms on [0,∞). That approach includes and generalizes the formalism of PH-distributions. That generalization results in the class of matrix-exponential probability distributions. To illustrate the novel way of thinking that is required to use these in stochastic models, we retrace the derivations of some results from matrix-exponential renewal theory and prove a new extension of a result from risk theory. Essentially the flow models allows for keeping track of the dynamics of a mechanism that generates matrix-exponential distributions in a similar way to the probabilistic arguments used for phase-type distributions involving transition rates. We also sketch a generalization of the Markovian arrival process (MAP) to the setting of matrix-exponential distribution. That process is known as the Rational arrival process (RAP).

The estimation of phase-type related functionals using Markov chain Monte Carlo methods

In this paper we present a method for estimation of functionals depending on one or several phase-type distributions. This could for example be the ruin probability in a risk reserve process where claims are assumed to be of phase-type. The proposed method uses a Markov chain Monte Carlo simulation to reconstruct the Markov jump processes underlying the phase-type variables in combination with Gibbs sampling to obtain a stationary sequence of phase-type probability measures from the posterior distribution of these given the observations. This enables us to find quantiles of posterior distributions of functionals of interest, thereby representing estimation uncertainty in a flexible way. We compare our estimates to those obtained by the method of maximum likelihood and find a good agreement. We illustrate the statistical potential of the method by estimating ruin probabilities in simulated examples.

Simple simulation of diffusion bridges with application to likelihood inference for diffusions

With a view to statistical inference for discretely observed diffusion models, we propose simple methods of simulating diffusion bridges, approximately and exactly. Diffusion bridge simulation plays a fundamental role in likelihood and Bayesian inference for diffusion processes. First a simple method of simulating approximate diffusion bridges is proposed and studied. Then these approximate bridges are used as proposal for an easily implemented Metropolis-Hastings algorithm that produces exact diffusion bridges. The new method utilizes time-reversibility properties of one-dimensional diffusions and is applicable to all one-dimensional diffusion processes with finite speed-measure. One advantage of the new approach is that simple simulation methods like the Milstein scheme can be applied to bridge simulation. Another advantage over previous bridge simulation methods is that the proposed method works well for diffusion bridges in long intervals because the computational complexity of the method is linear in the length of the interval. For

Efficient estimation of transition rates between credit ratings from observations at discrete time points

\rho

Phase-type distributions and risk processes with state-dependent premiums

-mixing diffusions the approximate method is shown to be particularly accurate for long time intervals. In a simulation study, we investigate the accuracy and efficiency of the approximate method and compare it to exact simulation methods. In the study, our method provides a very good approximation to the distribution of a diffusion bridge for bridges that are likely to occur in applications to statistical inference. To illustrate the usefulness of the new method, we present an EM-algorithm for a discretely observed diffusion process.

DISTRIBUTIONS OF REWARD FUNCTIONS ON CONTINUOUS-TIME MARKOV CHAINS

The paper demonstrates how discrete time credit rating data (e.g. annual observations) can be analysed by means of a continuous-time Markov model. Two methods for estimating the transition intensities are given: the EM algorithm and an MCMC approach. The estimated transition intensities can be used to estimate the matrix of probabilities of transitions between all credit ratings, including default probabilities, over any time horizon. Thus the advantages of a continuous-time model can be obtained without continuous-time data. Estimates of the variance of estimators as well as confidence and credibility intervals are presented, and a test for equality of two intensity matrices is proposed. The methods are demonstrated by analysis of a large data set drawn from Moodys Corporate Bond Default Database, where reasonable estimates are obtained from annual observations.

On the Construction of Bivariate Exponential Distributions with an Arbitrary Correlation Coefficient

Abstract Consider a risk reserve process with initial reserve u, Poisson arrivals, premium rule p(r) depending on the current reserve r and claim size distribution which is phase-type in the sense of Neuts. It is shown that the ruin probabilities ψ(u) can be expressed as the solution of a finite set of differential equations, and similar results are obtained for the case where the process evolves in a Markovian environment (e.g., a numerical example of a stochastic interest rate is presented). Further, an explicit formula for ψ(u) is presented for the case where p(r) is a two-step function. By duality, the results apply also to the stationary distribution of storage processes with the same input and release rate p(r) at content r.

Poisson's equation for queues driven by a Markovian marked point process

We develop algorithms for the computation of the distribution of the total reward accrued during [0, t) in a finite continuous-parameter Markov chain. During sojourns, the reward grows linearly at a rate depending on the state visited. At transitions, there can be instantaneous rewards whose values depend on the states involved in the transition. For moderate values of t, the reward distribution is obtained by implementing a series representation, due to Sericola, that is based on the uniformization method. As an alternative, that distribution can also be computed by the numerical inversion of its Laplace-Stieltjes transform. For larger values of t, we implement a matrix convolution product to compute a related semi-Markov matrix efficiently and accurately.