# Topics in Lévy and Jump Processes

Erik Baurdoux (LSE), Nan Chen (Chinese University of Hong Kong), Masahiko Egami (Kyoto), Arturo Kohatsu-Higa (Ritsumeikan), Andreas Kyprianou (Bath), Budhi Arta Surya (Bandung Institute of Technology),

One Day Seminar
Topics in Lévy and Jump Processes

Erik Baurdoux (LSE), Nan Chen (Chinese University of Hong Kong), Masahiko Egami (Kyoto), Arturo Kohatsu-Higa (Ritsumeikan), Andreas Kyprianou (Bath), Budhi Arta Surya (Bandung Institute of Technology),

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10:30-11:15 Andreas Kyprianou (University of Bath)
Title: "Censored Stable Processes"
Abstract: We look at a general two-sided jumping strictly alpha-stable process where alpha is in (0,2). By censoring its path each time it enters the negative half line we show that the resulting process is a positive self-similar Markov Process. Using Lamperti's transformation we uncover an underlying driving Lévy process and, moreover, we are able to describe in surprisingly explicit detail the Wiener-Hopf factorization of the latter. Using this Wiener-Hopf factorization together with a series of spatial path transformations, it is now possible to produce an explicit formula for the law of the original stable processes as it first *enters* a finite interval, thereby generalizing a result of Blumenthal, Getoor and Ray for symmetric stable processes from 1961.

11:20-12:05 Budhi Arta Surya (Bandung Institute of Technology)
Title: "Finite Maturity Corporate Debt Valuation with Sharing Rule Upon Default Under Lévy Models"
Abstract: This paper discusses the valuation of finite maturity corporate debt under the Lévy models of underlying firm's assets. The debt specifies that both equity and debt holders will share the total value of the firm upon default due to a certain rule. As proposed by Fan and Sundaresan (2000), the sharing rule is determined by Nash bargaining game. Due to conflict of interest between the equity and debt holders, the firm faces two-stage optimization problem. The firm will perform the first-stage optimization to maximize the total firm value, as the debt holders will act in the anticipation of what the equity holders may do later when the debt is issued. Once the total firm value is calculated, the equity holders will find the optimal default level that maximizes the firm equity value. Since the nature of the optimization problem is of finite maturity, there is no closed form valuation formula available. Therefore, we resolve to numerical method proposed in Surya (2012). The method is based on the associated partial-integro differential equation formulations of the optimization problem. The numerical results verify the recent result of Surya (2012) and Kyprianou and Surya (2006) that the smooth pasting condition may not hold for general Lévy processes.

14:00-14:45 Arturo Kohatsu-Higa (Ritsumeikan University)
Title: "Approximation schemes for Lévy driven stochastic differential equations"
Abstract: Stochastic differential equations (sde's) driven by Lévy processes have various applications. Between them notably in Finance, these are infinite activity Lévy processes which are mainly characterized by the fact that their Lévy measure has infinite mass and therefore an infinite amount of jumps in any interval. It is clear (in general) that the path of such Lévy process can not be simulated exactly and therefore usually approximating Lévy process are used. The problem of efficiently simulating the solution of sde's driven by such processes is a challenging problem and we will present some of our recent results in the area. From the mathematical point of view, our approach profits from the splitting method written in a probabilistic framework. We will show that the use of such frame allows a tractable study of the error and therefore providing the study of the optimality of the approximating Lévy process.

14:50-15:35 Nan Chen (Chinese University of Hong Kong)
Title: "Cocos, bail-in, and tail risks"
Abstract: We develop a capital structure model to analyze the incentives created by contingent convertibles (CoCos) and bail-in debt, two variants of debt that converts to equity as a bank nears or reaches financial distress. Our formulation includes firm-specific and market-wide tail risk in the form of two types of jumps and leads to a tractable jump-diffusion model of the firm's income and asset value. The firm's liabilities include insured deposits and senior and subordinated debt, as well as convertible debt. Our model combines endogenous default, debt rollover, and jumps; these features are essential in examining how changes in capital structure to include CoCos or bail-in debt change incentives for equity holders. We derive closed-form expressions to value the firm and its liabilities, and we use these to investigate how CoCos affect debt overhang, asset substitution, the firm's ability to absorb losses, the sensitivity of equity holders to various types of risk, and how these properties interact with the firm's debt maturity profile, the tax treatment of CoCo coupons, and the pricing of deposit insurance. We examine the effects of varying the two main design features of CoCos, the conversion trigger and the conversion ratio, and we compare the effects of CoCos with the effects of reduced bankruptcy costs through orderly resolution. Across a wide set of considerations, we find that CoCos generally have positive incentive effects when the conversion trigger is not set too low. The need to roll over debt, the debt tax shield, and tail risk in the firm's income and asset value have particular impact on the effects of CoCos. We also identify a phenomenon of debt-induced collapse that occurs when a firm issues CoCos and then takes on excessive additional debt: the added debt burden can induce equity holders to raise their default barrier above the conversion trigger, effectively changing CoCos to junior straight debt; equity value experiences a sudden drop at the point at which this occurs. Finally, we calibrate the model to past data on the largest U.S. bank holding companies to see what impact CoCos might have had on the financial crisis. We use the calibration to gauge the increase in loss absorbing capacity and the reduction in debt overhang costs resulting from CoCos. We also time approximate conversion dates for high and low conversion triggers. A joint work with Paul Glasserman, Behzad Nouri, Markus Pelger.

15:35-16:00 coffee break

16:00-16:45 Masahiko Egami (Kyoto University)
Title: "Optimal Stopping When the Absorbing Boundary is Chasing After"
Abstract: 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 problem with the state variable X as a spectrally negative Lévy process. 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 deteriorates, the required capital requirement is endangered. This situation corresponds to X<S-b in our setting. This model may well describes a real-world problem where even a big bank can fail because the absorbing boundary is keeping up with the size of the bank. Moreover, the problem is in nature a two-dimensional one in which, unlike one-dimensional cases, we show that the "threshold strategy" is not in fact optimal. This is a joint work with T. Oryu.

16:50-17:35 Erik Baurdoux (London School of Economics)
Title: "Future Drawdowns of Lévy processes"
Abstract: For a Lévy process $X$ we study the future draw-down defined as $D_{t,s} = \inf_{t\leq u<t+s} (X_u-X_t)$ which is the minimum increment of $X$ starting at $t$ and length at most $s$. As an application, consider the case where the value of a stock is modeled by the exponential of a Lévy process. Then $D_{t,s}$ can be thought of as the lowest future $\log$ return in the time window $[t,t+s]$. We study $\overline{D}_{T,S}=\sup_{0\leq t\leq T}D_{t,S}$ which is the largest such future return with $t$ ranging over $[0,T]$. When $S=\infty$ we find the exact asymptotics of $\mathbb{P}(\overline{D}_{T,S}>x)$ (as $x\rightarrow\infty$) in the case when the Lévy measure is light tailed. Furthermore, in the case when $X$ only has one-sided jumps we express $\mathbb{P}(\overline{D}_{\mathbf{e}_1.\mathbf{e}_2}>x)$ in terms of scale functions when $\mathbf{e}_1$ and $\mathbf{e}_2$ are independent, exponentially distributed random variables.

Erik Baurdoux (LSE), Nan Chen (Chinese University of Hong Kong), Masahiko Egami (Kyoto), Arturo Kohatsu-Higa (Ritsumeikan), Andreas Kyprianou (Bath), Budhi Arta Surya (Bandung Institute of Technology),
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Topics in Lévy and Jump Processes

2012年09月07日（金）10:30-17:35

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