Research articles for the 2019-09-04

An analytical perturbative solution to the Merton Garman model using symmetries
Xavier Calmet,Nathaniel Wiesendanger Shaw

In this paper, we introduce an analytical perturbative solution to the Merton Garman model. It is obtained by doing perturbation theory around the exact analytical solution of a model which possesses a two-dimensional Galilean symmetry. We compare our perturbative solution of the Merton Garman model to Monte Carlo simulations and find that our solutions performs surprisingly well for a wide range of parameters. We also show how to use symmetries to build option pricing models. Our results demonstrate that the concept of symmetry is important in mathematical finance.

Characterizing Cryptocurrency market with Levy's stable distributions
Shinji Kakinaka,Ken Umeno

Recent emergence of cryptocurrencies such as Bitcoin and Ethereum has posed possible alternatives to global payments as well as financial assets around the globe, making investors and financial regulators aware of the importance to modeling them properly. The Levy's stable distribution is one of the attractive distributions that well describes the fat tails and scaling phenomena in economic systems.In this paper, we show that the behaviors of price fluctuations in emerging cryptocurrency markets can be characterized fairly well by a Levy's stable distribution with alpha = 1.4, which indicates fatter tails compared to a Gaussian distribution. Moreover, we have come to realize the limitations of the stable model especially under certain conditions outside time intervals ranging from 30 minutes to 4 hours. The arguments are developed under the theoretical background of the General Central Limit Theorem (GCLT) and quantitative valuation defined as a distance function using the Parseval's relation. Our approach can generally be extended for further analysis of statistical properties and contribute to develop proper applications for financial modeling.

Clusters of investors around Initial Public Offering
Margarita Baltakienė,Kęstutis Baltakys,Juho Kanniainen,Dino Pedreschi,Fabrizio Lillo

The complex networks approach has been gaining popularity in analysing investor behaviour and stock markets, but within this approach, initial public offerings (IPO) have barely been explored. We fill this gap in the literature by analysing investor clusters in the first two years after the IPO filing in the Helsinki Stock Exchange by using a statistically validated network method to infer investor links based on the co-occurrences of investors' trade timing for 69 IPO stocks. Our findings show that a rather large part of statistically similar network structures form in different securities' and persist in time for mature and IPO companies. We also find evidence of institutional herding.

Conditional survival probabilities under partial information: a recursive quantization approach with applications
Cheikh Mbaye,Abass Sagna,Frédéric Vrins

We consider a structural model where the survival/default state is observed together with a noisy version of the firm value process. This assumption makes the model more realistic than most of the existing alternatives, but triggers important challenges related to the computation of conditional default probabilities. In order to deal with general diffusions as firm value process, we derive a numerical procedure based on the recursive quantization method to approximate it. Then, we investigate the error approximation induced by our procedure. Eventually, numerical tests are performed to evaluate the performance of the method, and an application is proposed to the pricing of CDS options.

Failure of Smooth Pasting Principle and Nonexistence of Equilibrium Stopping Rules under Time-Inconsistency
Ken Seng Tan,Wei Wei,Xun Yu Zhou

This paper considers a time-inconsistent stopping problem in which the inconsistency arises from non-constant time preference rates. We show that the smooth pasting principle, the main approach that has been used to construct explicit solutions for conventional time-consistent optimal stopping problems, may fail under time-inconsistency. Specifically, we prove that the smooth pasting principle solves a time-inconsistent problem within the intra-personal game theoretic framework if and only if a certain inequality on the model primitives is satisfied. We show that the violation of this inequality can happen even for very simple non-exponential discount functions. Moreover, we demonstrate that the stopping problem does not admit any intra-personal equilibrium whenever the smooth pasting principle fails. The "negative" results in this paper caution blindly extending the classical approaches for time-consistent stopping problems to their time-inconsistent counterparts.

Markovian lifts of positive semidefinite affine Volterra type processes
Christa Cuchiero,Josef Teichmann

We consider stochastic partial differential equations appearing as Markovian lifts of matrix valued (affine) Volterra type processes from the point of view of the generalized Feller property (see e.g., \cite{doetei:10}). We introduce in particular Volterra Wishart processes with fractional kernels and values in the cone of positive semidefinite matrices. They are constructed from matrix products of infinite dimensional Ornstein Uhlenbeck processes whose state space are matrix valued measures. Parallel to that we also consider positive definite Volterra pure jump processes, giving rise to multivariate Hawkes type processes. We apply these affine covariance processes for multivariate (rough) volatility modeling and introduce a (rough) multivariate Volterra Heston type model.

Nash Equilibria in Optimal Reinsurance Bargaining
Michail Anthropelos,Tim J. Boonen

We introduce a strategic behavior in reinsurance bilateral transactions, where agents choose the risk preferences they will appear to have in the transaction. Within a wide class of risk measures, we identify agents' strategic choices to a range of risk aversion coefficients. It is shown that at the strictly beneficial Nash equilibria, agents appear homogeneous with respect to their risk preferences. While the game does not cause any loss of total welfare gain, its allocation between agents is heavily affected by the agents' strategic behavior. This allocation is reflected in the reinsurance premium, while the insurance indemnity remains the same in all strictly beneficial Nash equilibria. Furthermore, the effect of agents' bargaining power vanishes through the game procedure and the agent who gets more welfare gain is the one who has an advantage in choosing the common risk aversion at the equilibrium.

Optimal Stopping and Utility in a Simple Model of Unemployment Insurance
Jason S. Anquandah,Leonid V. Bogachev

Managing unemployment is one of the key issues in social policies. Unemployment insurance schemes are designed to cushion the financial and morale blow of loss of job but also to encourage the unemployed to seek new jobs more pro-actively due to the continuous reduction of benefit payments. In the present paper, a simple model of unemployment insurance is proposed with a focus on optimality of the individual's entry to the scheme. The corresponding optimal stopping problem is solved, and its similarity and differences with the perpetual American call option are discussed. Beyond a purely financial point of view, we argue that in the actuarial context the optimal decisions should take into account other possible preferences through a suitable utility function. Some examples in this direction are worked out.

Robust Utility Maximizing Strategies under Model Uncertainty and their Convergence
Jörn Sass,Dorothee Westphal

In this paper we investigate a utility maximization problem with drift uncertainty in a continuous-time Black--Scholes type financial market. We impose a constraint on the admissible strategies that prevents a pure bond investment and we include uncertainty by means of ellipsoidal uncertainty sets for the drift. Our main results consist in finding an explicit representation of the optimal strategy and the worst-case parameter and proving a minimax theorem that connects our robust utility maximization problem with the corresponding dual problem. Moreover, we show that, as the degree of model uncertainty increases, the optimal strategy converges to a generalized uniform diversification strategy.

Stochastic ordering of Gini indexes for multivariate elliptical random variables
Chuancun Yin

In this paper, we establish the stochastic ordering of the Gini indexes for multivariate elliptical risks which generalized the corresponding results for multivariate normal risks. It is shown that several conditions on dispersion matrices and the components of dispersion matrices of multivariate normal risks for the monotonicity of the Gini index in the usual stochastic order proposed by Samanthi, Wei and Brazauskas (2016) and Kim and Kim (2019) are also suitable for multivariate elliptical risks. We also study the tail probability of Gini index for multivariate elliptical risks and revised a large deviation result for the Gini indexes of multivariate normal risks in Kim and Kim (2019).

Stochastic perturbations and fisheries management
Patrice Loisel

As most natural resources, fisheries are affected by random disturbances. The evolution of such resources may be modelled by a succession of deterministic process and random perturbations on biomass and/or growth rate at random times. We analyze the impact of the characteristics of the perturbations on the management of natural resources. We highlight the importance of using a dynamic programming approach in order to completely characterize the optimal solution, we also present the properties of the controlled model and give the behavior of the optimal harvest for specific jump kernels.