Abstracts and Slides

Daniel Bauer

The Marginal Cost of Risk and Capital Allocation in a Property and Casualty Insurance Company

We develop a multi-period profit maximization model for a property and casualty (P&C) insurance company and use it for determining the marginal cost of risk for different lines of business and resulting economic capital allocations. In contrast to previous literature, our model features a loss structure that matches the characteristics of a P&C company, comprising so-called “short-tailed” and “long-tailed” business lines with different expected settlement terms. As one key contribution, our theoretical and numerical results show that lines with different terms are assessed differently, depending on the company’s financial situation. This is joint work with Qiheng Guo and George Zanjani.

Shreya Bose

Kyle-Back models with risk aversion and non-Gaussian beliefs

We show that the existence of equilibrium in the Kyle-Back models can be characterized by considering a system of forward Fokker-Planck equation and a system of backward quasilinear parabolic partial differential equations coupled via an optimal transport type constraint at maturity. We also study the properties of the equilibrium obtained for small enough risk aversion parameter. In our model, the insider has exponential type utility and the belief of the market maker on the distribution of the price at final time can be non-Gaussian. This is based on a joint work with Dr. Ibrahim Ekren.

Sébastien Bossu

Static Replication of Dispersion & Multi-Asset Options

The Carr-Madan spanning formula, which underlies the computation of the VIX, decomposes any single-asset European option into a portfolio of vanilla calls and puts. The formula may be extended to replicate multi-asset European options, such as worst-of/best-of and dispersion options, with a portfolio of vanilla basket calls that are comparatively more liquid and easier to price.

René Carmona

Self-financing condition, trader inventories and optimal execution in the high frequency markets

Using NASDAQ ITCH data, we argue that the standard self financing condition of the Black-Scholes theory needs to be modified to account for trading behavior in the high frequency markets. A consequence of this empirical study was the realization that the time evolutions of traders’ inventories should not be modeled by differentiable functions of time — as in most theoretical models -- but with unbounded variations functions instead. We use econometric statistical tests to confirm the presence of a Brownian motion term in the inventories of traders on the Toronto Stock Exchange. Finally, we extend the theoretical analysis of a known optimal execution model to include this Brownian component, and we compare the actual behaviors to some day-traders to the “optimal” behavior suggested by the model.

The results presented in this talk were obtained in collaboration with Kevin Webster and Laura Leal.

Jonathan Chavez-Casillas

Modeling a limit order book model with time dependent rates under two scenarios

In this talk, we will formulate a model for a level-1 Limit Order Book, in the same spirit as the model discussed by Cont and deLarrard (2014). However, as empirical evidence shows, the intensity function of the price process driving the arrivals of orders is non-constant. To account for that fact, the assumption of homogeneous Poisson arrivals is relaxed and more general non-homogeneous Poisson arrivals are considered. Besides, two similar cases will be considered: when the clock driving the arrivals of orders is resettled after each price change (mimicking the model of Cont and deLarrard) and when it isn't. The latter case is justified by considering a periodic arrival rate each day or week. These models are part of an effort to understand the macroscopic long-run price dynamics and the different limiting price processes to which we may arrive when the orderbook is modeled from the market micro-structure.

Asaf Cohen

Markovian Equilibria In Ergodic Many-Player Games and Mean-Field Games.

We consider a symmetric stochastic game with weak interactions between many players. Time is continuous, the number of states is finite, and costs are ergodic. We prove the existence of a unique Nash equilibrium in the game and show that its limiting behavior (as the number of players goes to infinity) is governed by the unique mean-field equilibrium of the corresponding mean-field game. This is joint work with Ethan Zell.

Zhenyu Cui

A general approach for solving behavior optimal investment problems with non-concave utility and probability weighting

We consider the optimal investment problem with both probability distortion/weighting and general non-concave utility functions with possibly finite number of inflection points. Our model contains the model under cumulative prospect theory (CPT) as a special case, which has an inverse S-shaped probability weighting and a S-shaped utility function (i.e. one inflection point). We propose a step-wise relaxation method and have applied it to solve in closed-form several representative examples in mathematical behavioral finance including the CPT model, Value-at-Risk based risk management (VAR-RM) model with probability distortions, Yaari’s dual model and the Browne's goal reaching model. We obtain a closed-form optimal trading strategy for a special example of the CPT model, where a “distorted” Merton line has been shown exactly. The slope of the “distorted” Merton line is given by an inflation factor multiplied by the standard Merton ratio, and an interesting finding is that the inflation factor is solely dependent on the probability distortion rather than the non-concavity of the utility function.

Matthew Dixon

Kriging the Local Volatility Surface

We present a finite dimensional Gaussian process (GP) regression for no-arbitrage interpolation and modeling the local volatility surface. In this setup, the MAP estimate serves the purpose of identifying the locations of the most likely arbitrages in the data and quantifying them. Furthermore, Hamiltonian Monte Carlo can be used to efficiently sample from the posterior price surface and provide UQ of the local volatility surface. We demonstrate the performance of this approach relative to SSVI and a NN approach on SPX options, and in particular the importance of embedding no-arbitrage constraints into the ML for 10x sharper uncertainty quantification where data coverage in the option surface is poor. This is ongoing joint work with Stéphane Crépey, Areski Cousin, and Mike Ludkovski.

Dena Firoozi

A Mean Field Game Approach to Equilibrium Pricing in Compliance Markets

Compliance markets are designed to incentivize clean production. In a renewable energy certificate (REC) market a regulatory body imposes a lower bound on the amount of energy each regulated firm must generate via renewable means, providing them with a certificate for each MWh generated. Regulated firms seek to navigate the market to minimize the cost imposed on them, by modulating their REC generation and trading activities. As such, the REC market can be viewed through the lens of a large stochastic game with heterogeneous agents, where agents interact through the market price of the certificates.

We study this stochastic game by solving the mean-field game (MFG) limit with sub-populations of heterogeneous agents. Moreover, we endogenize REC price through market clearing. Using techniques from variational analysis, we characterize firms' optimal controls as the solution of McKean-Vlasov (MV) FBSDEs and determine the equilibrium SREC price. We establish the existence and uniqueness of a solution to this MV-FBSDE, and further prove that the MFG strategies have the Epsilon-Nash property for the finite player game. Finally, we develop a numerical scheme for solving the MV-FBSDEs and conclude by demonstrating how firms behave in equilibrium using simulated examples.

Paul Glasserman

Linear Classifiers Under Infinite Imbalance, With Application to Credit Risk

We study the behavior of linear discriminant functions for binary classification in the infinite-imbalance limit, where the sample size of one class grows without bound while the sample size of the other remains fixed. The coefficients of the classifier minimize an expected loss specified through a weight function. We show that for a broad class of weight functions, the intercept diverges but the rest of the coefficient vector has a finite limit under infinite imbalance, extending prior work on logistic regression. The limit depends on the left tail of the weight function, for which we distinguish three cases: bounded, asymptotically polynomial, and asymptotically exponential. The limiting coefficient vectors reflect robustness or conservatism properties in the sense that they optimize against certain worst-case alternatives. In the bounded and polynomial cases, the limit is equivalent to an implicit choice of upsampling distribution for the minority class. We apply these ideas in a credit risk setting, with particular emphasis on performance in the high-sensitivity and high-specificity regions. This is joint work with Mike Li.

Aparna Gupta

Designing Risk-free Service for Renewable Wind and Solar Resources

Renewable generation is inherently stochastic. We develop a risk mitigation strategy by applying principles of securitization to the stochastic generation of wind and solar resources. We demonstrate the design of a risk-free tranche for the renewable assets as a risk-free service the renewable resource can offer in the day-ahead market. The risk-free tranche needs to be comparably risk-free relative to a benchmark for the power markets, against which we evaluate the tranche’s risk-reward performance. Analyzing the risk profiles of renewable power generation on different days of a year, we identify the critical determinants of the risk-free tranche and develop a valuation framework to determine the renewable resources’ bidding strategy in the day-ahead market. Analogous to a risk-free instrument in the financial markets, we create a risk-free benchmark for power markets in terms of a combined-cycle natural gas generator. The risk-return of the risk-free benchmark is used to evaluate the performance of the designed risk-free tranche. We find that our risk-free tranche for renewable wind and solar assets outperforms the risk-free benchmark in terms of their respective risk-return trade-offs. Hence renewable energy producers can leverage the designed risk-free tranche to place competitive bids and participate in the day-ahead market at par with the conventional generators. Adoption of such risk management strategies can help renewable producers to graduate from being mere price takers in the power markets.

David Itkin

Open Markets in Stochastic Portfolio Theory and Rank Polynomial Processes

Abstract: Stochastic portfolio theory is a framework to study large equity markets over long time horizons. In such settings investors are often confined to trading in an "open market" setup consisting of only the assets with high capitalizations. We relax the notion of an open market and develop a tractable framework to study them under mild structural conditions on the market. Within this framework we introduce a large parametric class of processes called rank polynomial models. Their properties include (i) the induced capital distribution curve is stable, (ii) there are explicit expressions for the growth-optimal portfolios, (iii) the results are robust with respect to the dimension of the market and (iv) these models serve as worst-case models for a robust asymptotic growth problem under model ambiguity. This is based on joint work with Martin Larsson.

Robert Jarrow

Interest Rate Swaps: A Comparison of Compounded Daily versus Discrete Reference Rates

This paper studies the hedging effectiveness of interest rate swaps using different reference rates for eliminating interest rate risk from floating rate loans. Two reference rates are studied. The first rate's maturity, Δ , matches the payment interval of floating rate loans. The second has an incompatible maturity Δ /N . The prime examples are LIBOR and SOFR, respectively. We show that the Δ -based swap provides a good static hedge, but the Δ /N -based swap does not. Although dynamic hedging with the Δ /N -based swap is possible under some conditions, it both introduces model risk and increases transaction costs, making it a less practical alternative.

Donghan Kim

Open markets

An open market is a subset of a larger equity market, composed of a certain fixed number of top‐capitalization stocks. Though the number of stocks in the open market is fixed, their composition changes over time, as each company's rank by market capitalization fluctuates. When one is allowed to invest also in a money market, an open market resembles the entire “closed” equity market in the sense that the market viability (lack of arbitrage) is equivalent to the existence of a numéraire portfolio (which cannot be outperformed). When access to the money market is prohibited, the class of portfolios shrinks significantly in open markets; in such a setting, we discuss how to construct functionally generated stock portfolios and the concept of the universal portfolio. This talk is based on joint work with Ioannis Karatzas.

Yerkin Kitapbayev

Optimal capital structure with stochastic variable costs.

We examine the optimal capital structure of a firm with stochastic revenues, stochastic variable costs, and fixed costs. In this two-state variable setting with stochastic operating leverage, we establish an Early Default Premium (EDP) formula for the value of equity and derive an integral equation for the endogenous default boundary, a function of variable costs. The value of debt, the endogenous coupon, the optimal leverage ratio and the credit spread are solved for. The impact of taxes, fixed costs and bankruptcy costs is assessed. This is joint work with Jerome Detemple and Kristoffer Glover.

Steven Kou

Bitcoin Mining and Electricity Consumption

Abstract: We develop a dynamic equilibrium Bitcoin mining model to characterize miners’ optimal entry and exit strategy with technology innovation. The model can jointly calibrate to the co-movements of computing power, electricity consumption of Bitcoin network, and mining revenue. Our model reveals how miners’ electricity consumption is endogenously affected by the Bitcoin price.

Kiseop Lee

Optimal execution with liquidity risk in a diffusive order book market

We present the optimal order placement strategy with the presence of a liquidity cost. In this problem, a stock trader wishes to clear her large inventory by a predetermined time horizon T. A trader uses both limit and market orders, and a large market order faces an adverse price movement caused by the liquidity risk. First, we study a single period model where the trader places a limit order and/or a market order at the beginning. We show the behavior of optimal amount of market order, m*, and optimal placement of limit order, y*, under different market conditions. Next, we extend it to a multi-period model, where the trader makes sequential decisions of limit and market orders at multiple time points.

Camelia Minoiu

Expansionary Yet Different: Credit Supply and Real Effects of Negative Interest Rate Policy

We show that negative interest rate policy (NIRP) has expansionary effects on bank credit supply through a portfolio rebalancing channel. By shifting down and flattening the yield curve, NIRP differs from rate cuts just above the zero lower bound and has effects similar to quantitative easing (QE). For identification, we exploit the European Central Banks' NIRP and the Italian credit register, and, for external validity, European and U.S. datasets. NIRP affects more those banks with higher ex-ante liquid assets, including net interbank positions. More exposed banks reduce liquid assets, expand credit supply, especially to financially-constrained firms, and cut loan rates, inducing firms to increase investment and the wage bill.

Sveinn Olafsson

Proof-of-Work Cryptocurrencies: Does Mining Technology Undermine Decentralization?

Does the proof-of-work protocol serve its intended purpose of supporting decentralized cryptocurrency mining? To address this question, we develop a game-theoretical model where miners first invest in hardware to improve the efficiency of their operations, and then compete for mining rewards in a rent-seeking game. We argue that because of capacity constraints faced by miners, centralization in mining is lower than indicated by both public discourse and recent academic work. We show that advancements in hardware efficiency do not necessarily lead to larger miners increasing their advantage, but rather allow smaller miners to expand and new miners to enter the competition. Our calibrated model illustrates that hardware efficiency has a small impact on the cost of attacking a network, while the mining reward has a significant impact. This highlights the vulnerability of smaller and emerging cryptocurrencies, as well as of established cryptocurrencies transitioning to a fee-based mining reward scheme. This is joint work with Agostino Capponi and Humoud Alsabah.

Max Reppen

Cryptocurrency mining: centralization in a continuum agent model

We propose a model to study the question of how centralization of reward and computational power occur in the Bitcoin-like cryptocurrencies. A continuum of miners compete against each other for mining rewards by increasing their computational power. This leads to a novel mean field game type model of jump intensity control, which we solve explicitly for miners maximizing exponential utility, and handle numerically in the case of miners with power utilities. We show that the heterogeneity of their initial wealth distribution leads to greater imbalance of the reward distribution, and increased wealth heterogeneity over time, or a "rich get richer" effect. This concentration phenomenon is aggravated by a higher bitcoin price, and reduced by competition. Additionally, an advanced miner with cost advantages such as access to cheaper electricity, contributes a significant amount of computational power in equilibrium. Hence, cost efficiency can also result in the type of centralization seen among miners of cryptocurrencies.

Ajay Subramanian

Credit Risk Sharing and Credit Market Regulation

We show how aggregate risk influences credit default swap (CDS) markets and CDS regulation in a tractable general equilibrium framework. For low aggregate risk, unregulated CDS trading leads to full bondholder insurance and efficient investment in productive rms. The efficient allocation can, therefore, be implemented via transfers alone. For intermediate aggregate risk, the unregulated CDS economy overinvests in risky rms. A margin requirement on CDS sellers is necessary for efficiency. When aggregate risk is high, the CDS market breaks down. A margin requirement restores equilibrium and efficiency, but it must be maximally stringent and accompanied by a capital requirement on CDS sellers.

Ruodu Wang

E-backtesting risk measures

Expected Shortfall (ES) is the most important risk measure in finance and insurance. One of the most challenging tasks in risk modeling practice is to backtest ES forecasts provided by financial institutions, based only on daily realized portfolio losses without imposing specific models. Recently, the notion of e-values has gained attention as potential alternatives to p-values as measures of uncertainty, significance and evidence. We use e-values to construct a model-free backtest of ES, which can be naturally generalized to many other risk measures and statistical quantities.

Bin Zou

Optimal Fee Structure of Variable Annuities

We study the design of fee structures of variable annuities as a stochastic control problem, in which an insurer is allowed to choose the fee structure in any form that satisfies the budget constraint, and seeks an optimal one to maximize its business objective. Under the no surrender assumption, we show that the optimal fee structure is of barrier type with a time-dependent free boundary. The insurer's optimal strategy is to charge fees if and only if the account value of variable annuities hits the free boundary from below.