Donatien Hainaut

A model for interest rates with clustering effects

We propose a model for short-term rates driven by a self-exciting jump process to reproduce the clustering of shocks on the Euro overnight index average (EONIA). The key element of the model is the feedback effect between the absolute value of jumps and the intensity of their arrival process. In this setting, we obtain a closed-form solution for the characteristic function for interest rates and their integral. We introduce a class of equivalent measures under which the features of the process are preserved. We infer the prices of bonds and their dynamics under a risk-neutral measure. The question of derivatives pricing is developed under a forward measure, and a numerical algorithm is proposed to evaluate caplets and floorlets. The model is fitted to EONIA rates from 2004 to 2014 using a peaks-over-threshold procedure. From observation of swap curves over the same period, we filter the evolution of risk premiums for Brownian and jump components. Finally, we analyse the sensitivity of implied caplet volatility to parameters defining the level of self-excitation.

A structural model for credit risk with switching processes and synchronous jumps

This paper studies a switching regime version of Mertons structural model for the pricing of default risk. The default event depends on the total value of the firms asset modeled by a switching Lévy process. The novelty of this approach is to consider that firms asset jumps synchronously with a change in the regime. After a discussion of dynamics under the risk neutral measure, two models are presented. In the first one, the default happens at bond maturity, when the firms value falls below a predetermined barrier. In the second version, the firm can enter bankruptcy at multiple predetermined discrete times. The use of a Markov chain to model switches in hidden external factors makes it possible to capture the effects of changes in trends and volatilities exhibited by default probabilities. With synchronous jumps, the firms asset and state processes are no longer uncorrelated. Finally, some econometric evidence that switching Lévy processes, with synchronous jumps, fit well historical time series is provided.

Life annuitization : why and how much

This paper addresses some of the problems a majority of retired individuals face: Why and in what proportion should they invest in a life annuity to maximize the utility of their future consumption or a bequest? The market considered in this work is made up of three assets: a life annuity, a risky asset and a cash account. As this problem doesn’t accept any suitable explicit solution, it is numerically solved by the Markov Chain approximation developed by Kushner and Dupuis. Without a bequest motive, we observe that the optimal planning of consumption is divided into two periods and that optimal asset allocation should include the risky asset. Next, the influence of a bequest on consumption and investment pattern is developed. We demonstrate that even with a bequest motive, pensioners should allocate a part of their wealth to the purchase of life annuities.

Dynamic Asset Allocation Under VAR Constraint with Stochastic Interest Rates

This paper addresses the problem of dynamic asset allocation under a bounded shortfall risk in a market composed of three assets: cash, stocks and a zero coupon bond. The dynamics of the instantaneous short rates is driven by a Hull and White model. In this setting, we determine and compare optimal investment strategies maximizing the CRRA utility of terminal wealth with and without value at risk constraint.

Impulse control of pension fund contributions, in a regime switching economy

In defined benefit pension plans, allowances are independent from the financial performance of the fund. And the sponsoring firm pays regularly contributions to limit deviations of fund assets from the mathematical reserve, necessary for covering the promised liabilities. This research paper proposes a method to optimize the timing and size of contributions, in a regime switching economy. The model takes into consideration important market frictions, like transactions costs, late payments and illiquidity. The problem is solved numerically using dynamic programming and impulse control techniques. Our approach is based on parallel grids, with trinomial links, discretizing the asset return in each economic regime.

Optimal funding of defined benefit pension plans

In this paper, we address the issue of determining the optimal contribution rate of a defined benefit pension fund. The affiliates mortality is modelled by a jump process and the benefits paid at retirement are function of the evolution of future salaries. Assets of the fund are invested in cash, stocks, and a rolling bond. Interest rates are driven by a Vasicek model. The objective is to minimize both the quadratic spread between the contribution rate and the normal cost, and the quadratic spread between the terminal wealth and the mathematical reserve required to cover benefits. The optimization is done under a budget constraint that guarantees the actuarial equilibrium between the current asset and future contributions and benefits. The method of resolution is based on the Cox–Huang approach and on dynamic programming.

An intensity model for credit risk with switching Lévy processes

We develop a switching regime version of the intensity model for credit risk pricing. The default event is specified by a Poisson process whose intensity is modeled by a switching Levy process. This model presents several interesting features. First, as Levy processes encompass numerous jump processes, our model can duplicate the sudden jumps observed in credit spreads. Also, due to the presence of jumps, probabilities do not vanish at very short maturities, contrary to models based on Brownian dynamics. Furthermore, as the parameters of the Levy process are modulated by a hidden Markov chain, our approach is well suited to model changes of volatility trends in credit spreads, related to modifications of unobservable economic factors.

Default probabilities of a holding company, with complete and partial information

Abstract This paper studies the valuation of credit risk for firms that own several subsidiaries or business lines. We provide simple analytical approximating expressions for probabilities of default, and for equity–debt market values, both in the case when the information is available in continuous time as well as in the case that it is not instantaneously available. The total firm’s asset value being modeled as a sum of lognormal random variables, we use convex upper and lower approximations to infer these analytical approximating expressions. We extend the model to firms financed by multiple stochastic liabilities and conclude by numerical illustrations.

A Fractal Version of the Hull-White Interest Rate Model

This paper develops a new version of the Hull–Whites model of interest rates, in which the volatility of the short term rate is driven by a Markov switching multifractal model. The interest rate dynamics is still mean reverting but the constant volatility of the Brownian motion is replaced by a multifractal process so as to capture persistent volatility shocks. In this setting, we infer properties of the short term rate distribution, a semi-closed form expression for bond prices and their dynamics under a forward measure. Finally, our work is illustrated by a numerical application in which we assess the exposure of a bonds portfolio to the interest risk.

Frequency and Severity Modelling Using Multifractal Processes: An Application to Tornado Occurrence in the USA and CAT Bonds

This paper proposes a statistical model for insurance claims arising from climatic events, such as tornadoes in the USA, that exhibit a large variability both in frequency and intensity. To represent this variability and seasonality, the claims process modelled by a Poisson process of intensity equal to the product of a periodic function, and a multifractal process is proposed. The size of claims is modelled in a similar way, with gamma random variables. This method is shown to enable simulation of the peak times of damage. A two-dimensional multifractal model is also investigated. The work concludes with an analysis of the impact of the model on the yield of weather bonds linked to damage caused by tornadoes.