arXiv subjects

Masahiko Egami

Publications and source records attributed to Masahiko Egami.

At least 19 recordsLinked to original sources

arXiv2026

A potential-theoretic approach to optimal stopping in a spectrally Lévy Model

We establish a systematic solution method for optimal stopping problems of spectrally negative Lévy processes. Our approach relies essentially on potential theory, that is, on the analysis of superharmonic and subharmonic functions and their analytic properties, including the maximum principle. Using these mathematical results, we not only derive necessary and sufficient conditions of optimality for a broad class of reward functions, but also develop a method to tackle general problems in a direct and constructive way (without pre-specifying the solution form). To reinforce the latter point, we also present several examples with complex solution structures that illustrate the effectiveness of our approach, including continuation regions with multiple connected components.

math.OC↗

arXiv2025

Loss-Given-Default Modeling by Post-Last Passage Time Process

This study proposes a stochastic model for loss-given-default (LGD) which provides the LGD distribution based on credit market and company-specific financial conditions. The model utilizes last passage time of a linear diffusion (representing firm value) to a certain threshold point, after which default occurs as a surprising event. By treating the post-last passage time process in a continuum of the original process, we are able to use firm-value approach before and intensity-based approach after the last passage time, leading to a hybrid model. Under minimal and standard assumptions, we obtain the distributions of default time and LGD explicitly. We provide a computationally simple estimation procedure and real-world examples of estimated LGD distribution implied in CDS market.

q-fin.RM↗

arXiv2024

On Decomposition of the Last Passage Time of Diffusions

For a regular transient diffusion, we provide a decomposition of its last passage time to a certain state $α$. This is accomplished by transforming the original diffusion into two diffusions using the occupation time of the area above and below $α$. Based on these two processes, both having a reflecting boundary at $α$, we derive the decomposition formula of the Laplace transform of the last passage time explicitly in a simple form in terms of Green functions. This equation also leads to the Green function's decomposition formula. We demonstrate an application of these formulas to a diffusion with two-valued parameters.

math.PR↗

arXiv2019

Time Reversal and Last Passage Time of Diffusions with Applications to Credit Risk Management

We study time reversal, last passage time, and $h$-transform of linear diffusions. For general diffusions with killing, we obtain the probability density of the last passage time to an arbitrary level and analyze the distribution of the time left until killing after the last passage time. With these tools, we develop a new risk management framework for companies based on the leverage process (the ratio of a company asset process over its debt) and its corresponding alarming level. We also suggest how a company can determine the alarming level for the leverage process by constructing a relevant optimization problem.

q-fin.MF↗

arXiv2018

A Direct Solution Method for Pricing Options in Regime-switching Models

Pricing financial or real options with arbitrary payoffs in regime-switching models is an important problem in finance. Mathematically, it is to solve, under certain standard assumptions, a general form of optimal stopping problems in regime-switching models. In this article, we reduce an optimal stopping problem with an arbitrary value function in a two-regime environment to a pair of optimal stopping problems without regime switching. We then propose a method for finding optimal stopping rules using the techniques available for non-switching problems. In contrast to other methods, our systematic solution procedure is more direct since we first obtain the explicit form of the value functions. In the end, we discuss an option pricing problem which may not be dealt with by the conventional methods, demonstrating the simplicity of our approach.

q-fin.MF↗

arXiv2016

Explicit Solutions for Optimal Stopping of Linear Diffusion and its Maximum

We provide, in a general setting, explicit solutions for optimal stopping problems that involve diffusion process and its running maximum. Our approach is to use the excursion theory for Levy processes. Since general diffusions are, in particular, not of independent increments, we use an appropriate measure change to make the process have that property. Then we rewrite the original two-dimensional problem as an infinite number of one-dimensional ones and complete the solution. We show general solution methods with explicit value functions and corresponding optimal strategies, illustrating them by some examples.

math.OC↗

arXiv2016

Explicit Solutions for Optimal Stopping of Maximum Process with Absorbing Boundary that Varies with It

We provide, in a general setting, explicit solutions for optimal stopping problems that involve a diffusion process and its running maximum. Besides, a new feature includes absorbing boundaries that vary with the value of the running maximum. The existence of the absorbing boundary of this type makes the problem harder but more practical and flexible. Our approach is to use the excursion theory for Levy processes. Since general diffusions are, in particular, not of independent increments, we use an appropriate measure change to make the process have that property. Then we rewrite the original two-dimensional problem as an infinite number of one-dimensional ones and complete the solution. We show general solution methods with explicit value functions and corresponding optimal strategies, illustrating them by some examples.

math.OC↗

arXiv2015

Optimal Stopping When the Absorbing Boundary is Following After

We consider a new type of optimal stopping problems where the absorbing boundary moves as the state process X attains new maxima S. More specifically, we set the absorbing boundary as S-b where b is a certain constant. This problem is naturally connected with excursions from zero of the reflected process S-X. We examine this constrained optimization with the state variable X as a spectrally negative Levy process. The problem is in nature a two-dimensional one. The threshold strategy given by the path of X is not in fact optimal. It turns out, however, that we can reduce the original problem to an infinite number of one-dimensional optimal stopping problems, and we find explicit solutions. This work is motivated by the bank's profit maximization with the constraint that it maintain a certain level of leverage ratio. When the bank's asset value severely deteriorates, the bank's required capital requirement shall be violated. This situation corresponds to X<S-b in our setting. This model may well describe a real-life situation where even a big bank can fail because the absorbing boundary is keeping up with the size of the bank.

math.PR↗

arXiv2014

Phase-type fitting of scale functions for spectrally negative Levy processes

We study the scale function of the spectrally negative phase-type Levy process. Its scale function admits an analytical expression and so do a number of its fluctuation identities. Motivated by the fact that the class of phase-type distributions is dense in the class of all positive-valued distributions, we propose a new approach to approximating the scale function and the associated fluctuation identities for a general spectrally negative Levy process. Numerical examples are provided to illustrate the effectiveness of the approximation method.

math.PR↗

arXiv2013

An Excursion-Theoretic Approach to Regulator's Bank Reorganization Problem

The importance of the global financial system cannot be exaggerated. When a large financial institution becomes problematic and is bailed out, that bank is often claimed as "too big to fail". On the other hand, to prevent bank's failure, regulatory authorities adopt the Prompt Corrective Action (PCA) against a bank that violates certain criteria, often measured by its leverage ratio. In this article, we provide a framework where one can analyze the cost and effect of PCA's. We model a large bank with deteriorating asset and regulatory actions attempting to prevent a failure. The model uses the excursion theory of Levy processes and finds an optimal leverage ratio that triggers a PCA. A nice feature includes it incorporates the fact that social cost associated with PCA's are be greatly affected by the size of banks subject to PCA's, so that one can see the cost of rescuing a bank "too big to fail".

q-fin.PR↗

arXiv2013

On the continuous and smooth fit principle for optimal stopping problems in spectrally negative Levy models

We consider a class of infinite-time horizon optimal stopping problems for spectrally negative Levy processes. Focusing on strategies of threshold type, we write explicit expressions for the corresponding expected payoff via the scale function, and further pursue optimal candidate threshold levels. We obtain and show the equivalence of the continuous/smooth fit condition and the first-order condition for maximization over threshold levels. As examples of its applications, we give a short proof of the McKean optimal stopping problem (perpetual American put option) and solve an extension to Egami and Yamazaki (2013).

math.OC↗

arXiv2012

Default Swap Games Driven by Spectrally Negative Levy Processes

This paper studies game-type credit default swaps that allow the protection buyer and seller to raise or reduce their respective positions once prior to default. This leads to the study of an optimal stopping game subject to early default termination. Under a structural credit risk model based on spectrally negative Levy processes, we apply the principles of smooth and continuous fit to identify the equilibrium exercise strategies for the buyer and the seller. We then rigorously prove the existence of the Nash equilibrium and compute the contract value at equilibrium. Numerical examples are provided to illustrate the impacts of default risk and other contractual features on the players' exercise timing at equilibrium.

q-fin.PR↗

arXiv2011

Precautionary Measures for Credit Risk Management in Jump Models

Sustaining efficiency and stability by properly controlling the equity to asset ratio is one of the most important and difficult challenges in bank management. Due to unexpected and abrupt decline of asset values, a bank must closely monitor its net worth as well as market conditions, and one of its important concerns is when to raise more capital so as not to violate capital adequacy requirements. In this paper, we model the tradeoff between avoiding costs of delay and premature capital raising, and solve the corresponding optimal stopping problem. In order to model defaults in a bank's loan/credit business portfolios, we represent its net worth by Levy processes, and solve explicitly for the double exponential jump diffusion process and for a general spectrally negative Levy process.

q-fin.RM↗

arXiv2011

An optimal life insurance policy in the investment-consumption problem in an incomplete market

This paper considers an optimal life insurance for a householder subject to mortality risk. The household receives a wage income continuously, which is terminated by unexpected (premature) loss of earning power or (planned and intended) retirement, whichever happens first. In order to hedge the risk of losing income stream by householder's unpredictable event, the household enters a life insurance contract by paying a premium to an insurance company. The household may also invest their wealth into a financial market. The problem is to determine an optimal insurance/investment/consumption strategy in order to maximize the expected total, discounted utility from consumption and terminal wealth. To reflect a real-life situation better, we consider an incomplete market where the householder cannot trade insurance contracts continuously. To our best knowledge, such a model is new in the insurance and finance literature. The case of exponential utilities is considered in detail to derive an explicit solution. We also provide numerical experiments for that particular case to illustrate our results.

q-fin.PM↗

arXiv2010

Solving Optimal Dividend Problems via Phase-type Fitting Approximation of Scale Functions

The optimal dividend problem by De Finetti (1957) has been recently generalized to the spectrally negative Lévy model where the implementation of optimal strategies draws upon the computation of scale functions and their derivatives. This paper proposes a phase-type fitting approximation of the optimal strategy. We consider spectrally negative Lévy processes with phase-type jumps as well as meromorphic Lévy processes (Kuznetsov et al., 2010a), and use their scale functions to approximate the scale function for a general spectrally negative Lévy process. We obtain analytically the convergence results and illustrate numerically the effectiveness of the approximation methods using examples with the spectrally negative Lévy process with i.i.d. Weibull-distributed jumps, the β-family and CGMY process.

q-fin.CP↗

arXiv2009

On the One-Dimensional Optimal Switching Problem

We explicitly solve the optimal switching problem for one-dimensional diffusions by directly employing the dynamic programming principle and the excessive characterization of the value function. The shape of the value function and the smooth fit principle then can be proved using the properties of concave functions.

math.OC↗

arXiv2009

A Unified Treatment of Dividend Payment Problems under Fixed Cost and Implementation Delays

In this paper we solve the dividend optimization problem for a corporation or a financial institution when the managers of the corporation are facing (regulatory) implementation delays. We consider several cash reservoir models for the firm including two mean-reverting processes, Ornstein-Uhlenbeck and square-root processes. We provide our solution via a new characterization of the value function for one-dimensional diffusions and provide easily implementable algorithms to find the optimal control and the value function.

math.OC↗

arXiv2007

Optimizing Venture Capital Investments in a Jump Diffusion Model

We study a practical optimization problems for venture capital investments and/or Research and Development (R&D) investments. The first problem is that, given the amount of the initial investment and the reward function at the initial public offering (IPO) market, the venture capitalist wants to maximize overall discounted cash flows after subtracting subsequent (if needed) investments. We describe this problem as a mixture of singular stochastic control and optimal stopping problems and give an explicit solution. The former corresponds to finding an optimal subsequent investment policy for the purpose that the value of the investee company stays away from zero. The latter corresponds to finding an optimal stopping rule in order to maximize the harvest of their investments. The second kind problem is concerned about optimal dividend policy. Rather than selling the holding stock, the investor may extract dividends when it is appropriate. We will find a quasi-explicit optimal solution to this problem and prove the existence and uniqueness of the solution and the optimality of the proposed strategy.

math.OC↗