Joint Survival Probability Research Articles

Dependent censoring with simultaneous death times based on the Generalized Marshall–Olkin model

In this paper, we consider the problem of dependent censoring models with a positive probability that the times of failure are equal. In this context, we propose to consider the Marshall–Olkin type model and studied some properties of the associated survival copula in its application to censored data. We also introduce estimators for the marginal distributions and the joint survival probabilities under different schemes and show their asymptotic normality under appropriate conditions. Finally, we evaluate the finite-sample performance of our approach relying on a small simulation study with synthetic data real data applications.

Credit default swap pricing with counterparty risk in a reduced form model with Hawkes process

Credit default swap pricing with counterparty risk in a reduced form model with Hawkes process

On the joint survival probability of two collaborating firms

AbstractWe consider the problem of controlling the drift and diffusion rate of the endowment processes of two firms such that the joint survival probability is maximized. We assume that the endowment processes are continuous diffusions, driven by independent Brownian motions, and that the aggregate endowment is a Brownian motion with constant drift and diffusion rate. Our results reveal that the maximal joint survival probability depends only on the aggregate risk-adjusted return and on the maximal risk-adjusted return that can be implemented in each firm. Here the risk-adjusted return is understood as the drift rate divided by the squared diffusion rate.

Modeling joint survival probabilities of runs scored and balls faced in limited overs cricket using copulas

In limited overs cricket, the goal of a batsman is to score a maximum number of runs within a limited number of balls. Therefore, the number of runs scored and the number of balls faced are the two key statistics used to evaluate the performance of a batsman. In cricket, as the batsmen play as pairs, having longer partnerships is also key to building strong innings. Moreover, having a steady opening partnership is extremely important as a team aims to build such a stronger innings. In this study, we have shown a way to evaluate the performance of opening partnerships in Twenty20 (T20) cricket and the performance of individual batsmen in One Day International Cricket (ODI) by modeling the joint distribution of runs scored and balls faced using copula functions. The joint survival probabilities derived from this approach are then used to evaluate the batting performance of opening partnerships and individual batsmen for different stages of the innings. Results of the study have shown that cricket managers and team officials can use the proposed method in selecting appropriate partnership pairs and individual batsmen in an efficient manner for specific situations in the match.

Credit default swap pricing with counterparty risk in a reduced form model with a common jump process

AbstractIn this paper, we study the credit default swap (CDS) pricing with counterparty risk in a reduced form model. The default jump intensities of the reference firm and counterparty are both assumed to follow the mean-reverting CIR processes with independent jumps respectively and a common jump. The approximate closed-form solutions of the joint survival probability density and the probability density of the first default can be obtained by using the PDE method. Then with the expressions of the probability densities, we can get the formula for the CDS price with counterparty risk in a reduced form model with a common jump. In the numerical analysis part, we find that the default of the reference asset has a greater impact on the CDS price than that of the default of counterparty after introducing the common jump process.

Basket Credit Default Swap Pricing with Two Defaultable Counterparties

In this paper, we study the basket CDS pricing with two defaultable counterparties based on the reduced‐form model. The default jump intensities of the reference firms and counterparties are all assumed to follow the mean‐reverting constant elasticity of variance (CEV) processes. Taking the Vasicek process which is a special case of CEV process as an example, the approximate analytic solutions of the joint survival probability density, the probability densities of the first default and the first two defaults can be solved by using PDE method. In addition, we also extend the Vasciek process to the Vasciek process with cojumps and obtain the approximate closed‐form solutions of the relevant default probability densities. Then with the expressions of the probability densities, we can get the formula of the basket CDS price with two defaultable counterparties. In the numerical analysis, we do sensitivity analysis and compare the basket CDS prices under our model with that with only one defaultable counterparty. The numerical results show that our model can be applied into practice.

Finite-time ruin probability for correlated Brownian motions

Let be a two-dimensional Gaussian process with standard Brownian motion marginals and constant correlation . Define the joint survival probability of both supremum functionals by where and u, v are given positive constants. Approximation of is of interest for the analysis of ruin probability in bivariate Brownian risk model, as well as in the study of the power of bivariate test statistics. In this contribution, we derive tight bounds for in the case and obtain precise approximations for all by letting and taking v = au for some fixed positive constant a.

Stochastic decompositions in bivariate risk and queueing models with mutual assistance

We consider two bivariate models with two-way interactions in context of risk and queueing theory. The two entities interact with each other by providing assistance but otherwise evolve independently. We focus on certain random quantities underlying the joint survival probability and the joint stationary workload, and show that these admit stochastic decomposition. Each one can be seen as an independent sum of respective quantities for the two models with one-way interaction. Additionally, we discuss a rather general method of establishing decompositions from a given kernel equation by identifying two independent random variables from their difference, which may be useful for other models. Finally, we point out that the same decomposition is true for uncorrelated Brownian motion reflected to stay in a quadrant, and it concerns the face measures appearing in the basic adjoint relationship.

The survival probability of critical and subcritical branching processes in finite state space Markovian environment

Let (Zn)n≥0 be a branching process in a random environment defined by a Markov chain (Xn)n≥0 with values in a finite state space X. Let Pi be the probability law generated by the trajectories of Xnn≥0 starting at X0=i∈X. We study the asymptotic behaviour of the joint survival probability PiZn>0,Xn=j, j∈X as n→+∞ in the critical and strongly, intermediate and weakly subcritical cases.

Numerical analysis of an extended structural default model with mutual liabilities and jump risk

We consider a structural default model in an interconnected banking network as in [1], with mutual obligations between each pair of banks. We analyse the model numerically for two banks with jumps in their asset value processes. Specifically, we develop a finite difference method for the resulting two-dimensional partial integro-differential equation, and study its stability and consistency. We then compute joint and marginal survival probabilities, as well as prices of credit default swaps (CDS), first-to-default swaps (FTD), Credit and Debt Value Adjustments (CVA and DVA). Finally, we calibrate the model to market data and assess the impact of jump risk.

Structural default model with mutual obligations

In this paper we consider mutual obligations in an interconnected bank system and analyze their impact on the joint and marginal survival probabilities for individual banks. We also calculate prices of the corresponding credit default swaps and first-to-default swaps. To make the role of mutual obligations more transparent, we develop a simple structural default model with banks’ assets driven by correlated multidimensional Brownian motion with drift. We calculate closed form expressions for many quantities of interest and use them for the efficient model calibration. We demonstrate that mutual obligations have noticeable impact on the system behavior.

Joint survival probability via truncated invariant copula

Given an intensity-based credit risk model, this paper studies dependence structure between default intensities. To model this structure, we use a multivariate shot noise intensity process, where jumps occur simultaneously and their sizes are correlated. Through very lengthy algebra, we obtain explicitly the joint survival probability of the integrated intensities by using the truncated invariant Farlie–Gumbel–Morgenstern copula with exponential marginal distributions. We also apply our theoretical result to pricing basket default swap spreads. This result can provide a useful guide for credit risk management.

Estimating the joint survival probabilities of married individuals

We estimate the joint survival probability of spouses using a large random sample drawn from a Dutch census. As benchmarks we use two bivariate Weibull models. We consider more flexible models, using a semi-nonparametric approach, by extending the independent Weibull distribution using squared polynomials. Also based on a nonparametric comparison, we find that extending the independent Weibull distribution by a squared third order polynomial shows the best performance. We illustrate our model by calculating remaining life expectancies and annuity values. We find that the husbands life expectancy at birth is generally increasing with his wifes age of death and the wifes life expectancy at birth is generally increasing with her husbands age of death. Ignoring the dependence between the remaining lifetimes of spouses may lead to an underestimation of the value of a joint annuity and an overestimation of the value of a single-life annuity, but less than suggested on the basis of the previous literature.

Efficient solution of structural default models with correlated jumps and mutual obligations

The structural default model of Lipton and Sepp [Credit value adjustment for credit default swaps via the structural default model, J. Credit Risk 5(2) (2009), pp. 123–146] is generalized for a set of banks with mutual interbank liabilities whose assets are driven by correlated Lévy processes with idiosyncratic and common components. The multi-dimensional problem is made tractable via a novel computational method, which generalizes the one-dimensional fractional partial differential equation method of Itkin [Efficient solution of backward jump-diffusion PIDEs with splitting and matrix exponentials, J. Comput. Financ. (2014), forthcoming. Available at http://arxiv.org/abs/1304.3159] to the two- and three-dimensional cases. This method is unconditionally stable and of the second order of approximation in space and time; in addition, for many popular Lévy models it has linear complexity in each dimension. Marginal and joint survival probabilities for two and three banks with mutual liabilities are computed. The effects of mutual liabilities are discussed, and numerical examples are given to illustrate these effects.

Modeling the Dependency Structure of Integrated Intensity Processes.

This paper studies an important issue of dependence structure. To model this structure, the intensities within the Cox processes are driven by dependent shot noise processes, where jumps occur simultaneously and their sizes are correlated. The joint survival probability of the integrated intensities is explicitly obtained from the copula with exponential marginal distributions. Subsequently, this result can provide a very useful guide for credit risk management.

Optimal stop-loss reinsurance: a dependence analysis

The stop-loss reinsurance is one of the most important reinsurance contracts in the insurance market. From the insurer point of view, it presents an interesting property: it is optimal if the criterion of minimizing the variance of the cost of the insurer is used. The aim of the paper is to contribute to the analysis of the stop-loss contract in one period from the point of view of the insurer and the reinsurer. Firstly, the influence of the parameters of the reinsurance contract on the correlation coefficient between the cost of the insurer and the cost of the reinsurer is studied. Secondly, the optimal stop-loss contract is obtained if the criterion used is the maximization of the joint survival probability of the insurer and the reinsurer in one period.

Optimal Stop-Loss Reinsurance: A Dependence Analysis

The stop-loss reinsurance is one of the most important reinsurance contracts in the insurance market. From the insurer point of view, it presents an interesting property: it is optimal if the criterion of minimizing the variance of the cost of the insurer is used. The aim of the paper is to contribute to the analysis of the stop-loss contract in one period from the point of view of the insurer and the reinsurer. Firstly, the influence of the parameters of the reinsurance contract on the correlation coefficient between the cost of the insurer and the cost of the reinsurer is studied. Secondly, the optimal stop-loss contract is obtained if the criterion used is the maximization of the joint survival probability of the insurer and the reinsurer in one period.

Sensitivity of the joint survival probability for reinsurance schemes

We model the joint risk process for an insurer and a reinsurer using a diffusion approximation and obtain expressions for the sensitivity of the joint survival probability with respect to parameters of the reinsurance scheme. The approach used leads, more generally, to explicit expressions for the sensitivity of functionals of diffusions in with constant coefficients, whose drift vector and covariance matrix are differentiable functions of a parameter, in a form suitable for efficient Monte Carlo simulation. The functionals examined depend on the values of the diffusion at a finite number of time epochs, and the sensitivities are calculated using the likelihood ratio method. An extension to dynamic reinsurance schemes is also briefly described, and sensitivity estimators are provided using the integration by parts formula of the Malliavin calculus. Copyright © 2013 John Wiley & Sons, Ltd.

Optimal combination of quota-share and stop-loss reinsurance treaties under the joint survival probability

Optimal combination of quota-share and stop-loss reinsurance treaties under the joint survival probability

Modelling Default Correlations in a Two-Firm Model with Dynamic Leverage Ratios

This article provides a generalized two-firm model of default correlation, based on the structural approach that incorporates interest rate risk. In most structural models default is driven by the firms' asset dynamics. In this article, a two-firm model of default is instead driven by the dynamic leverage ratios, which combines the measure of risks of the firms' total liabilities and assets. This article investigates analytical methods and numerical tools to solve the two-dimensional first passage time problem with time-dependent parameters. We carry out a comparative analysis of the impact of model parameters and provide some insights of their effects on joint survival probabilities and default correlations.