arXiv daily: Mathematical Finance (q-fin.MF)

Authors:Alexander Gairat, Vadim Shcherbakov

Abstract: This paper concerns a local volatility model in which volatility takes two possible values, and the specific value depends on whether the underlying price is above or below a given threshold value. The model is known, and a number of results have been obtained for it. In particular, explicit pricing formulas for European options have been recently obtained and applied to establish a power law behaviour of the implied volatility skew in the case when the threshold is taken at the money. These results have been obtained by techniques based on the Laplace transform. The purpose of the present paper is to demonstrate how to obtain the same results by another method. This alternative approach is based on the natural relationship of the model with Skew Brownian motion and consists in the systematic use of the joint distribution of this stochastic process and some of its functionals.

Authors:Felix Fießinger, Mitja Stadje

Abstract: Focusing on gains instead of terminal wealth, we consider an asset allocation problem to maximize time-consistently a mean-risk reward function with a general risk measure which is i) law-invariant, ii) cash- or shift-invariant, and iii) positively homogeneous, and possibly plugged into a general function. We model the market via a generalized version of the multi-dimensional Black-Scholes model using $\alpha$-stable L\'evy processes and give supplementary results for the classical Black-Scholes model. The optimal solution to this problem is a Nash subgame equilibrium given by the solution of an extended Hamilton-Jacobi-Bellman equation. Moreover, we show that the optimal solution is deterministic and unique under appropriate assumptions.

Authors:Kamil Fortuna, Janusz Szwabiński

Abstract: Systemic risk is a rapidly developing area of research. Classical financial models often do not adequately reflect the phenomena of bubbles, crises, and transitions between them during credit cycles. To study very improbable events, systemic risk methodologies utilise advanced mathematical and computational tools, such as complex systems, chaos theory, and Monte Carlo simulations. In this paper, a relatively simple mathematical formalism is applied to provide a unified framework for modeling credit cycles and systemic risk assessment. The proposed model is analyzed in detail to assess whether it can reflect very different states of the economy. Basing on those results, measures of systemic risk are constructed to provide information regarding the stability of the system. The formalism is then applied to describe the full credit cycle with the explanation of causal relationships between the phases expressed in terms of parameters derived from real-world quantities. The framework can be naturally interpreted and understood with respect to different economic situations and easily incorporated into the analysis and decision-making process based on classical models, significantly enhancing their quality and flexibility.

Authors:Matteo Gardini, Edoardo Santilli

Abstract: In this article we discuss the application of the Heat-Jarrow-Morton framework Heath et al. [26] to energy markets. The goal of the article is to give a detailed overview of the topic, focusing on practical aspects rather than on theory, which has been widely studied in literature. This work aims to be a guide for practitioners and for all those who deal with the practical issues of this approach to energy market. In particular, we focus on the markets' structure, model calibration by dimension reduction with Principal Component Analysis (PCA), Monte Carlo simulations and derivatives pricing. As application, we focus on European power and gas markets: we calibrate the model on historical futures quotations, we perform futures and spot simulations and we analyze the results.

Authors:David García-Lorite, Raul Merino

Abstract: In this paper, we develop a novel method based on Malliavin calculus to find an approximation for the convexity adjustment for various classical interest rate products. Malliavin calculus provides a simple way to get a template for the convexity adjustment. We find the approximation for Futures, OIS Futures, FRAs, and CMSs under a general family of the one-factor Cheyette model. We have also seen the excellent quality of the numerical accuracy of the formulas obtained.

Authors:Alan Guo

Abstract: We develop a formalism for reasoning about trading on decentralized exchanges on blockchains and a formulation of a particular form of maximal extractable value (MEV) that represents the total arbitrage opportunity extractable from on-chain liquidity. We use this formalism to prove that for blockchains with deterministic block times whose liquidity pools satisfy some natural properties that are satisfied by pools in practice, this form of MEV is invariant under changes to the ordering mechanism of the blockchain and distribution of block times. We do this by characterizing the MEV as the profit of a particularly simple arbitrage strategy when left uncontested. These results can inform design of blockchain protocols by ruling out designs aiming to increase trading opportunity by changing the ordering mechanism or shortening block times.

Authors:Zhuo Jin, Zuo Quan Xu, Bin Zou

Abstract: We study an optimal reinsurance problem under a diffusion risk model for an insurer who aims to minimize the probability of lifetime ruin. To rule out moral hazard issues, we only consider moral-hazard-free reinsurance contracts by imposing the incentive compatibility constraint on indemnity functions. The reinsurance premium is calculated under an extended distortion premium principle, in which the distortion function is not necessarily concave. We first show that an optimal reinsurance contract always exists and then derive two sufficient and necessary conditions to characterize it. Due to the presence of the incentive compatibility constraint and the nonconcavity of the distortion, the optimal contract is obtained as a solution to a double obstacle problem. At last, we apply the general result to study three examples and obtain the optimal contract in (semi)closed form.

Authors:Yunhong Li, Zuo Quan Xu, Xun Yu Zhou

Abstract: We study a continuous-time expected utility maximization problem in which the investor at maturity receives the value of a contingent claim in addition to the investment payoff from the financial market. The investor knows nothing about the claim other than its probability distribution, hence an ``intractable claim''. In view of the lack of necessary information about the claim, we consider a robust formulation to maximize her utility in the worst scenario. We apply the quantile formulation to solve the problem, expressing the quantile function of the optimal terminal investment income as the solution of certain variational inequalities of ordinary differential equations. In the case of an exponential utility, the problem reduces to a (non-robust) rank--dependent utility maximization with probability distortion whose solution is available in the literature.

Authors:Masaaki Fujii, Masashi Sekine

Abstract: In this paper, we study a problem of equilibrium price formation among many investors with exponential utility. The investors are heterogeneous in their initial wealth, risk-averseness parameter, as well as stochastic liability at the terminal time. We characterize the equilibrium risk-premium process of the risky stocks in terms of the solution to a novel mean-field backward stochastic differential equation (BSDE), whose driver has quadratic growth both in the stochastic integrands and in their conditional expectations. We prove the existence of a solution to the mean-field BSDE under several conditions and show that the resultant risk-premium process actually clears the market in the large population limit.