Research articles for the 2019-05-19

A Comment on "Estimating Dynamic Discrete Choice Models with Hyperbolic Discounting" by Hanming Fang and Yang Wang
Jaap H. Abbring,Øystein Daljord

The recent literature often cites Fang and Wang (2015) for analyzing the identification of time preferences in dynamic discrete choice under exclusion restrictions (e.g. Yao et al., 2012; Lee, 2013; Ching et al., 2013; Norets and Tang, 2014; Dub\'e et al., 2014; Gordon and Sun, 2015; Bajari et al., 2016; Chan, 2017; Gayle et al., 2018). Indeed, Fang and Wang's Proposition 2 claims generic identification of a dynamic discrete choice model with hyperbolic discounting. However, this claim uses a definition of "generic" that does not preclude the possibility that a generically identified model is nowhere identified. To illustrate this point, we provide two simple examples of models that are generically identified in Fang and Wang's sense, but that are, respectively, everywhere and nowhere identified. We conclude that Proposition 2 is void: It has no implications for identification of the dynamic discrete choice model. We show how its proof is incorrect and incomplete and suggest alternative approaches to identification.

A Dynkin game on assets with incomplete information on the return
Tiziano De Angelis,Fabien Gensbittel,Stéphane Villeneuve

This paper studies a 2-players zero-sum Dynkin game arising from pricing an option on an asset whose rate of return is unknown to both players. Using filtering techniques we first reduce the problem to a zero-sum Dynkin game on a bi-dimensional diffusion $(X,Y)$. Then we characterize the existence of a Nash equilibrium in pure strategies in which each player stops at the hitting time of $(X,Y)$ to a set with moving boundary. A detailed description of the stopping sets for the two players is provided along with global $C^1$ regularity of the value function.

A Nonlocal Approach to The Quantum Kolmogorov Backward Equation and Links to Noncommutative Geometry
Will Hicks

The Accardi-Boukas quantum Black-Scholes equation can be used as an alternative to the classical approach to finance, and has been found to have a number of useful benefits. The quantum Kolmogorov backward equations, and associated quantum Fokker-Planck equations, that arise from this general framework, are derived using the Hudson-Parthasarathy quantum stochastic calculus. In this paper we show how these equations can be derived using a nonlocal approach to quantum mechanics. We show how nonlocal diffusions, and quantum stochastic processes can be linked, and discuss how moment matching can be used for deriving solutions.

Capital Regulation under Price Impacts and Dynamic Financial Contagion
Zachary Feinstein

We construct a continuous time model for price-mediated contagion precipitated by a common exogenous stress to the banking book of all firms in the financial system. In this setting, firms are constrained so as to satisfy a risk-weight based capital ratio requirement. We use this model to find analytical bounds on the risk-weights for assets as a function of the market liquidity. Under these appropriate risk-weights, we find existence and uniqueness for the joint system of firm behavior and the asset prices. We further consider an analytical bound on the firm liquidations, which allows us to construct exact formulas for stress testing the financial system with deterministic or random stresses. Numerical case studies are provided to demonstrate various implications of this model and analytical bounds.

Cointegration in high frequency data
Simon Clinet,Yoann Potiron

In this paper, we consider a framework adapting the notion of cointegration when two asset prices are generated by a driftless It\^{o}-semimartingale featuring jumps with infinite activity, observed synchronously and regularly at high frequency. We develop a regression based estimation of the cointegrated relations method and show the related consistency and central limit theory when there is cointegration within that framework. We also provide a Dickey-Fuller type residual based test for the null of no cointegration against the alternative of cointegration, along with its limit theory. Under no cointegration, the asymptotic limit is the same as that of the original Dickey-Fuller residual based test, so that critical values can be easily tabulated in the same way. Finite sample indicates adequate size and good power properties in a variety of realistic configurations, outperforming original Dickey-Fuller and Phillips-Perron type residual based tests, whose sizes are distorted by non ergodic time-varying variance and power is altered by price jumps. Two empirical examples consolidate the Monte-Carlo evidence that the adapted tests can be rejected while the original tests are not, and vice versa.

Continuous-time Duality for Super-replication with Transient Price Impact
Peter Bank,Yan Dolinsky

We establish a super-replication duality in a continuous-time financial model where an investor's trades adversely affect bid- and ask-prices for a risky asset and where market resilience drives the resulting spread back towards zero at an exponential rate. Similar to the literature on models with a constant spread, our dual description of super-replication prices involves the construction of suitable absolutely continuous measures with martingales close to the unaffected reference price. A novel feature in our duality is a liquidity weighted $L^2$-norm that enters as a measurement of this closeness and that accounts for strategy dependent spreads. As applications, we establish optimality of buy-and-hold strategies for the super-replication of call options and we prove a verification theorem for utility maximizing investment strategies.

Equivalence Between Time Consistency and Nested Formula
Henri Gérard,Michel de Lara,Jean-Philippe Chancelier

You are a financial analyst. At the beginning of every week, you are able to rank every pair of stochastic processes starting from that week up to the horizon. Suppose that two processes are equal at the beginning of the week. Your ranking procedure is time consistent if the ranking does not change between this week and the next one. In this paper, we propose a minimalist definition of Time Consistency (TC) between two (assessment) mappings. With very few assumptions, we are able to prove an equivalence between Time Consistency and a Nested Formula (NF) between the two mappings. Thus, in a sense, two assessments are consistent if and only if one is factored into the other. We review the literature and observe that the various definitions of TC (or of NF) are special cases of ours, as they always include additional assumptions. By stripping off these additional assumptions, we present an overview of the literature where the contribution of each author is enlightened.

Is Volatility Rough ?
Masaaki Fukasawa,Tetsuya Takabatake,Rebecca Westphal

Rough volatility models are continuous time stochastic volatility models where the volatility process is driven by a fractional Brownian motion with the Hurst parameter smaller than half, and have attracted much attention since a seminal paper titled "Volatility is rough" was posted on SSRN in 2014 showing that the log realized volatility time series of major stock indices have the same scaling property as such a rough fractional Brownian motion has. We however find by simulations that the impressive approach tends to suggest the same roughness irrespectively whether the volatility is actually rough or not; an overlooked estimation error of latent volatility often results in an illusive scaling property. Motivated by this preliminary finding, here we develop a statistical theory for a continuous time fractional stochastic volatility model to examine whether the Hurst parameter is indeed estimated smaller than half, that is, whether the volatility is really rough. We construct a quasi-likelihood estimator and apply it to realized volatility time series. Our quasi-likelihood is based on the error distribution of the realized volatility and a Whittle-type approximation to the auto-covariance of the log-volatility process. We prove the consistency of our estimator under high frequency asymptotics, and examine by simulations its finite sample performance. Our empirical study suggests that the volatility is indeed rough; actually it is even rougher than considered in the literature.

On the consistency of jump-diffusion dynamics for FX rates under inversion
Federico Graceffa,Damiano Brigo,Andrea Pallavicini

In this note we investigate the consistency under inversion of jump diffusion processes in the Foreign Exchange (FX) market. In other terms, if the EUR/USD FX rate follows a given type of dynamics, under which conditions will USD/EUR follow the same type of dynamics? In order to give a numerical description of this property, we first calibrate a Heston model and a SABR model to market data, plotting their smiles together with the smiles of the reciprocal processes. Secondly, we determine a suitable local volatility structure ensuring consistency. We subsequently introduce jumps and analyze both constant jump size (Poisson process) and random jump size (compound Poisson process). In the first scenario, we find that consistency is automatically satisfied, for the jump size of the inverted process is a constant as well. The second case is more delicate, since we need to make sure that the distribution of jumps in the domestic measure is the same as the distribution of jumps in the foreign measure. We determine a fairly general class of admissible densities for the jump size in the domestic measure satisfying the condition.

What Do Mutual Fund Managers' Private Portfolios Tell Us About Their Skills?
Ibert, Markus
I collect a registry-based dataset on the personal portfolios of Swedish mutual fund managers. The managers who invest personal money in the very same funds they professionally manage outperform the managers who do not. The main results are consistent with a Berk and Green (2004) equilibrium in which fund managers, in contrast to regular investors, are certain about their ability to generate abnormal returns, or lack thereof, and invest their personal wealth accordingly.