Research articles for the 2020-12-17

Decentralized Finance, Centralized Ownership? An Iterative Mapping Process to Measure Protocol Token Distribution
Matthias Nadler,Fabian Schär

In this paper, we analyze various Decentralized Finance (DeFi) protocols in terms of their token distributions. We propose an iterative mapping process that allows us to split aggregate token holdings from custodial and escrow contracts and assign them to their economic beneficiaries. This method accounts for liquidity-, lending-, and staking-pools, as well as token wrappers, and can be used to break down token holdings, even for high nesting levels. We compute individual address balances for several snapshots and analyze intertemporal distribution changes. In addition, we study reallocation and protocol usage data, and propose a proxy for measuring token dependencies and ecosystem integration. The paper offers new insights on DeFi interoperability as well as token ownership distribution and may serve as a foundation for further research.

Dynamic Reinsurance in Discrete Time Minimizing the Insurer's Cost of Capital
Alexander Glauner

In the classical static optimal reinsurance problem, the cost of capital for the insurer's risk exposure determined by a monetary risk measure is minimized over the class of reinsurance treaties represented by increasing Lipschitz retained loss functions. In this paper, we consider a dynamic extension of this reinsurance problem in discrete time which can be viewed as a risk-sensitive Markov Decision Process. The model allows for both insurance claims and premium income to be stochastic and operates with general risk measures and premium principles. We derive the Bellman equation and show the existence of a Markovian optimal reinsurance policy. Under an infinite planning horizon, the model is shown to be contractive and the optimal reinsurance policy to be stationary. The results are illustrated with examples where the optimal policy can be determined explicitly.

Explicit RKF-Compact Scheme for Pricing Regime Switching American Options with Varying Time Step
Chinonso Nwankwo,Weizhong Dai

In this research work, an explicit Runge-Kutta-Fehlberg time integration with a fourth-order compact finite difference scheme in space is employed for solving the regime-switching pricing model. First, we recast the free boundary problem into a system of nonlinear partial differential equations with a multi-fixed domain. We further introduce a transformation based on the square root function with a fixed free boundary from which a high order analytical approximation is obtained for computing the derivative of the optimal exercise boundary in each regime. The high order analytical approximation is achieved by the method of extrapolation. As such, it enables us to employ fourth-order spatial discretization and an adaptive time integration with Dirichlet boundary conditions for obtaining the numerical solution of the asset option, option Greeks, and the optimal exercise boundary for each regime. In the set of equations, Hermite interpolation with Newton basis is used to estimate the coupled assets options and option Greeks. A numerical experiment is carried out with two- and four-regimes examples and results are compared with the existing methods. The results obtained from the numerical experiment show that the present method provides better performance in terms of computational speed and more accurate solutions with a large step size.

Geometric Brownian motion with affine drift and its time-integral
Runhuan Feng,Pingping Jiang,Hans Volkmer

The joint distribution of a geometric Brownian motion and its time-integral was derived in a seminal paper by Yor (1992) using Lamperti's transformation, leading to explicit solutions in terms of modified Bessel functions. In this paper, we revisit this classic result using the simple Laplace transform approach in connection to the Heun differential equation. We extend the methodology to the geometric Brownian motion with affine drift and show that the joint distribution of this process and its time-integral can be determined by a doubly-confluent Heun equation. Furthermore, the joint Laplace transform of the process and its time-integral is derived from the asymptotics of the solutions. In addition, we provide an application by using the results for the asymptotics of the double-confluent Heun equation in pricing Asian options. Numerical results show the accuracy and efficiency of this new method.

Levelling Down and the COVID-19 Lockdowns: Uneven Regional Recovery in UK Consumer Spending
John Gathergood,Fabian Gunzinger,Benedict Guttman-Kenney,Edika Quispe-Torreblanca,Neil Stewart

We show the recovery in consumer spending in the United Kingdom through the second half of 2020 is unevenly distributed across regions. We utilise Fable Data: a real-time source of consumption data that is a highly correlated, leading indicator of Bank of England and Office for National Statistics data. The UK's recovery is heavily weighted towards the "home counties" around outer London and the South. We observe a stark contrast between strong online spending growth while offline spending contracts. The strongest recovery in spending is seen in online spending in the "commuter belt" areas in outer London and the surrounding localities and also in areas of high second home ownership, where working from home (including working from second homes) has significantly displaced the location of spending. Year-on-year spending growth in November 2020 in localities facing the UK's new tighter "Tier 3" restrictions (mostly the midlands and northern areas) was 38.4% lower compared with areas facing the less restrictive "Tier 2" (mostly London and the South). These patterns have been further exacerbated during November 2020 when a second national lockdown was imposed. To prevent such COVID-19-driven regional inequalities from becoming persistent we propose governments introduce temporary, regionally-targeted interventions in 2021. The availability of real-time, regional data enables policymakers to efficiently decide when, where and how to implement such regional interventions and to be able to rapidly evaluate their effectiveness to consider whether to expand, modify or remove them.

Optimal switch from a fossil-fueled to an electric vehicle
Paolo Falbo,Giorgio Ferrari,Giorgio Rizzini,Maren Diane Schmeck

In this paper we propose and solve a real options model for the optimal adoption of an electric vehicle. A policymaker promotes the abeyance of fossil-fueled vehicles through an incentive, and the representative fossil-fueled vehicle's owner decides the time at which buying an electric vehicle, while minimizing a certain expected cost. This involves a combination of various types of costs: the stochastic opportunity cost of driving one unit distance with a traditional fossil-fueled vehicle instead of an electric one, the cost associated to traffic bans, and the net purchase cost. After determining the optimal switching time and the minimal cost function for a general diffusive opportunity cost, we specialize to the case of a mean-reverting process. In such a setting, we provide a model calibration on real data from Italy, and we study the dependency of the optimal switching time with respect to the model's parameters. Moreover, we study the effect of traffic bans and incentive on the expected optimal switching time. We observe that incentive and traffic bans on fossil-fueled transport can be used as effective tools in the hand of the policymaker to encourage the adoption of electric vehicles, and hence to reduce air pollution.

Simulation of conditional expectations under fast mean-reverting stochastic volatility models
Andrei Cozma,Christoph Reisinger

In this short paper, we study the simulation of a large system of stochastic processes subject to a common driving noise and fast mean-reverting stochastic volatilities. This model may be used to describe the firm values of a large pool of financial entities. We then seek an efficient estimator for the probability of a default, indicated by a firm value below a certain threshold, conditional on common factors. We first analyse the convergence of the Euler--Maruyama scheme applied to the fast Ornstein--Uhlenbeck SDE for the volatility, and show that the first order strong error is robust with respect to the mean reversion speed (only) if the step size is scaled appropriately. Next, we consider approximations where coefficients containing the fast volatility are replaced by certain ergodic averages (a type of law of large numbers), and study a correction term (of central limit theorem-type). The accuracy of these approximations is assessed by numerical simulation of pathwise losses and the estimation of payoff functions as they appear in basket credit derivatives.

The Causal Learning of Retail Delinquency
Yiyan Huang,Cheuk Hang Leung,Xing Yan,Qi Wu,Nanbo Peng,Dongdong Wang,Zhixiang Huang

This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.

The Thermodynamic Approach to Whole-Life Insurance: A Method for Evaluation of Surrender Risk
Jirô Akahori,Yuuki Ida,Maho Nishida,Shuji Tamada

We introduce a collective model for life insurance where the heterogeneity of each insured, including the health state, is modeled by a diffusion process. This model is influenced by concepts in statistical mechanics. Using the proposed framework, one can describe the total pay-off as a functional of the diffusion process, which can be used to derive a level premium that evaluates the risk of lapses due tothe so-called adverse selection. Two numerically tractable models are presented to exemplify the flexibility of the proposed framework.