Finance and Stochastics

We investigate the problem of optimal dividend distribution for a company in the presence of regime shifts. We consider a company whose cumulative net revenues evolve as a Brownian motion with positive drift that is modulated by a finite state Markov chain, and model the discount rate as a deterministic function of the current state of the chain. In this setting, the objective of the company is to maximize the expected cumulative discounted dividend payments until the moment of bankruptcy, which is taken to be the first time that the cash reserves (the cumulative net revenues minus cumulative dividend payments) are zero. We show that if the drift is positive in each state, it is optimal to adopt a barrier strategy at certain positive regime-dependent levels, and provide an explicit characterization of the value function as the fixed point of a contraction. In the case that the drift is small and negative in one state, the optimal strategy takes a different form, which we explicitly identify if there are two regimes. We also provide a numerical illustration of the sensitivities of the optimal barriers and the influence of regime switching.

This paper quantifies the interplay between the no-arbitrage notion of no unbounded profit with bounded risk (NUPBR) and additional progressive information generated by a random time. This study complements the one of Aksamit et al. (Finance Stoch. 21:1103–1139, 2017) in which the authors have studied similar topics for the model stopped at the random time, while here we deal with the question of what happens after the random time. Given that the existing literature proves that NUPBR is always violated after honest times that avoid stopping times in a continuous filtration, we propose here a new class of honest times for which NUPBR can be preserved for some models. For these honest times, we obtain two principal results. The first result characterizes the pairs of initial market and honest time for which the resulting model preserves NUPBR, while the second result characterizes honest times that do not affect NUPBR of any quasi-left-continuous model (i.e., in which the asset price process has no predictable jump times). Furthermore, we construct explicitly local martingale deflators for a large class of models.

Under the assumption that the asset value follows a phase-type jump-diffusion, we show that the expected discounted penalty satisfies an ODE and obtain a general form for the expected discounted penalty. In particular, if only downward jumps are allowed, we get an explicit formula in terms of the penalty function and jump distribution. On the other hand, if the downward jump distribution is a mixture of exponential distributions (and upward jumps are determined by a general Lévy measure), we obtain closed-form solutions for the expected discounted penalty. As an application, we work out an example in Leland’s structural model with jumps. For earlier and related results, see Gerber and Landry [Insur. Math. Econ. 22:263–276, 1998], Hilberink and Rogers [Finance Stoch. 6:237–263, 2002], Asmussen et al. [Stoch. Proc. Appl. 109:79–111, 2004], and Kyprianou and Surya [Finance Stoch. 11:131–152, 2007].

We consider a continuous-time market with proportional transaction costs. Under appropriate assumptions, we prove the existence of optimal strategies for investors who maximise their worst-case utility over a class of possible models. We consider utility functions defined on either the positive axis or the whole real line.

We consider the problem of identifying the worst case dependence structure of a portfolio X 1,…,X n of d-dimensional risks, which yields the largest risk of the joint portfolio. Based on a recent characterization result of law invariant convex risk measures, the worst case portfolio structure is identified as a μ-comonotone risk vector for some worst case scenario measure μ. It turns out that typically there will be a diversification effect even in worst case situations. The only exceptions arise when risks are measured by translated max correlation risk measures. We determine the worst case portfolio structure and the worst case diversification effect in several classes of examples as, e.g. in elliptical, Euclidean spherical, and Archimedean type distribution classes.

We consider a nonstandard ruin problem where: (i) the increments of the process are heavy-tailed and Markov-dependent, modulated by a general Harris recurrent Markov chain; (ii) ruin occurs when a positive boundary is attained within a sufficiently small time. Our main result provides sharp asymptotics for the small-time probability of ruin, viz., P(sup  n≤δ u S n ≥u), where {S n } denotes the discrete partial sums of the process and δ∈(0,1/μ), where μ is the mean drift. We apply our results to obtain risk estimates which quantify, e.g., repetitive operational risk losses or the extremal behavior for a GARCH(1,1) process.

The implied volatility skew has received relatively little attention in the literature on short-term asymptotics for financial models with jumps, despite its importance in model selection and calibration. We rectify this by providing high order asymptotic expansions for the at-the-money implied volatility skew, under a rich class of stochastic volatility models with independent stable-like jumps of infinite variation. The case of a pure-jump stable-like Lévy model is also considered under the minimal possible conditions for the resulting expansion to be well defined. Unlike recent results for “near-the-money” option prices and implied volatility, the results herein aid in understanding how the implied volatility smile near expiry is affected by important features of the continuous component, such as the leverage and vol-of-vol parameters. As intermediary results, we obtain high order expansions for at-the-money digital call option prices, which furthermore allow us to infer analogous results for the delta of at-the-money options. Simulation results indicate that our asymptotic expansions give good fits for options with maturities up to one month, underpinning their relevance in practical applications, and an analysis of the implied volatility skew in recent S&P 500 options data shows it to be consistent with the infinite variation jump component of our models.

The aim of this paper is to prove the fundamental theorem of asset pricing (FTAP) in finite discrete time with proportional transaction costs by utility maximization. The idea goes back to L.C.G. Rogers’ proof of the classical FTAP for a model without transaction costs. We consider one risky asset and show that under the robust no-arbitrage condition, the investor can maximize his expected utility. Using the optimal portfolio, a consistent price system is derived.