First passage time for multivariate jump‐diffusion processes in finance and other areas of applications

Abstract

The first passage time (FPT) problem is an important problem with a wide range of applications in science, engineering, economics, and industry. Mathematically, such a problem can be reduced to estimating the probability of a stochastic process first to reach a boundary level. In most important applications in the financial industry, the FPT problem does not have an analytical solution and the development of efficient numerical methods becomes the only practical avenue for its solution. Most of our examples in this contribution are centered around the evaluation of default correlations in credit risk analysis, where we are concerned with the joint defaults of several correlated firms, the task that is reducible to a FPT problem. This task represents a great challenge for jump-diffusion processes (JDP). In this contribution, we develop further our previous fast Monte Carlo method in the case of multivariate (and correlated) JDP. This generalization allows us, among other things, to evaluate the default events of several correlated assets based on a set of empirical data. The developed technique is an efficient tool for a number of financial, economic, and business applications, such as credit analysis, barrier option pricing, macroeconomic dynamics, and the evaluation of risk, as well as for a number of other areas of applications in science and engineering, where the FPT problem arises.Many applications in science, engineering, finance, economics, and industry require the solution of the first passage problem. Examples range from population genetics and other applications in biology, to neurobiology and neurophysiology, to molecule kinetics and problems in chemistry and physics, to rare events modelling and quantum information, and to aerospace engineering applications (e.g. [1][2][3][4]). This problem also arises in financial applications. Although methodologies developed in this paper are applicable to a number of other areas, we exemplify our discussion here with problems from finance.In the financial world, individual companies are usually linked together via economic conditions, so default correlation, defined as the risk of multiple companies' default together, has been an important area of research in credit analysis with applications to joint defaults, credit derivatives, asset pricing, and risk management.Currently, there are two dominant groups of theoretical models used in default correlation. One is a reduced form model, such as in [5] that uses a Copula function to parameterize the default correlation. In this context, it is worthwhile noting that Chen and Sopranzetti [6] have translated the joint default probability into a bivariate normal probability function, making the analysis and applications of such models potentially more convenient.The second group of models for default correlation is a structural form model. Zhou [7] and Hull and White [8] were the first to incorporate default correlation into the Black-Cox first passage structural model. They obtained similar closed-form solutions for two ...

A phase transition in the first passage of a Brownian process through a fluctuating boundary with implications for neural coding

“…Finding the very first such time, τ ¼ infft j W ðtÞ ¼ 1g; known as the "first passage" of the process through the boundary B = 1, is easier said than done, one of those classical problems whose concise statements conceal their difficulty (1-4). For general fluctuating random processes, the first-passage time problem is both extremely difficult (5-9) and highly relevant, due to its manifold practical applications: it models phenomena as diverse as the onset of chemical reactions (10)(11)(12)(13)(14), transitions of macromolecular assemblies (15)(16)(17)(18)(19), time-to-failure of a device (20)(21)(22), accumulation of evidence in neural decision-making circuits (23), the "gambler's ruin" problem in game theory (24), species extinction probabilities in ecology (25), survival probabilities of patients and disease progression (26)(27)(28), triggering of orders in the stock market (29)(30)(31), and firing of neural action potentials (32)(33)(34)(35)(36)(37).…”

Weak convergence of marked point processes generated by crossings of multivariate jump processes. Applications to neural network modeling

“…This convergence also holds if we reset the firing components: assume η 1 = j and then η 1;n = j for n large enough. Then X * j;n ( τ 1;n ) = r 0; η1;n = Y * j ( τ 1 ), (14) and thus X * n ( τ 1;n ) → Y * ( τ 1 ), implying (13).…”

“…From step m = 1, X * ( τ 1;n ) → Y * ( τ 1 ), and since Y * ( τ 1 ) < B and Y * ∈ H k , we can apply Lemma 3. Then, (13) follows noting that (10) also holds if we reset the firing components η 2;n and η 2 , as done in (14).…”

Probability of Default and Default Correlations

“…The evaluations of probability of default and default correlation from the previous literature are not only computationally intense but also inconsistent. Zhang and Melnik [17] discuss the first passage time for multivariate jump-diffusion processes without any explicit default correlation or default probability. Li and Krehbiel [16] indicate some numerical results to the inconsistency of their default rate and default correlation.…”