Mathematical Finance Seminar Series

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Schedule for Fall 2021

Seminars are on Thursdays
Time: 4:10pm - 5:25pm
Attention: All talks are available online, via Zoom. Select talks take place in hybrid mode. In-person participation is only available to Columbia affiliates with building access.

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Meeting ID: 933 7454 7641

Passcode: bachelier

Organizers: Ioannis Karatzas, Marcel Nutz, Philip Protter, Xiaofei Shi, Johannes Wiesel


*This talk will take place in hybrid mode.

Paolo Guasoni (Dublin)

Title: Rogue Traders

Abstract: Investing on behalf of a firm, a trader can feign personal skill by committing fraud that with high probability remains undetected and generates small gains, but that with low probability bankrupts the firm, offsetting ostensible gains. Honesty requires enough skin in the game: if two traders with isoelastic preferences operate in continuous-time and one of them is honest, the other is honest as long as the respective fraction of capital is above an endogenous fraud threshold that depends on the trader’s preferences and skill. If both traders can cheat, they reach a Nash equilibrium in which the fraud threshold of each of them is lower than if the other one were honest. More skill, higher risk aversion, longer horizons, and greater volatility all lead to honesty on a wider range of capital allocations between the traders.


Ciamac Moallemi (Columbia GSB)

“Risk-Sensitive Optimal Execution via a Conditional Value-at-Risk Objective”

We consider a liquidation problem in which a risk-averse trader tries to liquidate a fixed quantity of an asset in the presence of market impact and random price fluctuations. When deciding the liquidation strategy, the trader encounters a trade-off between the transaction costs incurred due to market impact and the volatility risk of holding the position. Our formulation begins with a continuous-time and infinite horizon variation of the seminal model of Almgren and Chriss (2000), but we define as the objective the conditional value-at-risk (CVaR) of the implementation shortfall, and allow for dynamic (adaptive) trading strategies. In this setting, remarkably, we are able to derive closed-form expressions for the optimal liquidation strategy and its value function.

Our results yield a number of important practical insights. We are able to quantify the benefit of adaptive policies over optimized static (pre-committed) policies. The relevant improvement depends only on the level of risk aversion, and grows without bound as the trader becomes more risk neutral. For moderate levels of risk aversion, the optimal dynamic policy outperforms the optimal static policy by 5–15%, and outperforms the optimal volume weighted average price (VWAP) policy by 15–25%. This improvement is achieved through dynamic policies that exhibit “aggressiveness-in-the-money”: trading is accelerated when price movements are favorable (to minimize risk), and is slowed when price movements are unfavorable (to minimize transaction costs). Overall, the optimal dynamic policies exhibit much better performance in the right tail of worst outcomes, relative to optimal static policies.

From a mathematical perspective, our analysis exploits the dual representation of CVaR to convert the problem to a continuous-time, zero sum dynamic game. In this setting, we leverage the idea of the state-space augmentation, recently applied to certain discrete-time Markov decision processes with a CVaR objective. We obtain a partial differential equation describing the optimal value function, which is separable and a special instance of the Emden–Fowler equation. This leads to a closed-form solution. As our problem is a special case of a continuous-time linear-quadratic-Gaussian control problem with a CVaR objective, these results may be interesting in broader settings.

This is joint work with Seungki Min (KAIST) and Costis Maglaras (Columbia).


Kostas Spiliopoulos (Boston University)

Title: The pricing of contingent claims and optimal positions in asymptotically complete markets


We study utility indifference prices and optimal purchasing quantities for a contingent claim, in an incomplete semi-martingale market, in the presence of vanishing hedging errors and/or risk aversion. Assuming that the average indifference price converges to a well defined limit, we prove that optimally taken positions become large in absolute value at a specific rate. We draw motivation from and make connections to Large Deviations theory, and in particular, the celebrated Gärtner-Ellis theorem. We analyze a series of well studied examples where this limiting behavior occurs, such as fixed markets with vanishing risk aversion, the basis risk model with high correlation, models of large markets with vanishing trading restrictions and the Black-Scholes-Merton model with either vanishing default probabilities or vanishing transaction costs. Lastly, we show that the large claim regime could naturally arise in partial equilibrium models.



Erhan Bayraktar (University of Michigan)

Title: Countercyclical Unemployment Benefits: General Equilibrium Analysis of Transition Dynamics

Abstract: We analyze the general equilibrium effects of countercyclical unemployment benefit policies. Our heterogenous-agent model features costly job search with imperfect insurance of unemployment risk and individual savings. Our model predicts: (1) the additional unemployment under a countercyclical policy relative to that under an acyclical policy to be a superlinear function of the aggregate shock’s size, (2) a higher unemployment rate sensitivity to UI policy changes when individual savings are relatively low. Our estimates of the effects of UI policy changes are based on transition dynamics; we show these estimates to be substantially different from estimates based on steady-state analyses. Joint work with Indrajit Mitra of the Federal Reserve Board of Atlanta and Jingjie Zhang of University of International Business and Economics (Beijing).


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Haoyang Cao (Alan Turing Institute)

Title: Identifiability in inverse reinforcement learning

Abstract: Inverse reinforcement learning attempts to reconstruct the reward function in a Markov decision problem, using observations of agent actions. As already observed in Russell [1998] the problem is ill-posed, and the reward function is not identifiable, even under the presence of perfect information about optimal behavior. We provide a resolution to this non-identifiability for problems with entropy regularization. For a given environment, we fully characterize the reward functions leading to a given policy and demonstrate that, given demonstrations of actions for the same reward under two distinct discount factors, or under sufficiently different environments, the unobserved reward can be recovered up to a constant. We also give general necessary and sufficient conditions for reconstruction of time-homogeneous rewards on finite horizons, and for action-independent rewards, generalizing recent results of Kim et al. [2021] and Fu et al. [2017].


*This talk will take place in hybrid mode.

Martin Larsson (CMU)

High-dimensional open markets in stochastic portfolio theory

Stochastic portfolio theory studies investments in large equity markets. Such investments are frequently confined to an “open market”: a high capitalization investment-grade subset of a much broader equity universe. We develop models for open markets which (i) are consistent with a given invariant distribution of relative market capitalizations, (ii) lead to explicit growth-optimal portfolios, (iii) are robust to the dimensionality and specific characteristics of lower-capitalization stocks outside the investment-grade subset, and (iv) serve as a worst-case model for a robust asymptotic growth maximization problem that incorporates model ambiguity. (Joint work with David Itkin.)


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University Holiday - No Seminar

Damir Filipović (École polytechnique fédérale de Lausanne and Swiss Finance Institute)

Title: Stripping the Discount Curve – a Robust Machine Learning Approach

Abstract: We propose a non-parametric method, kernel ridge regression, for estimating the discount curve from treasury securities, with regularization penalties in terms of the smoothness of the approximating discount curve. We provide analytical solutions, which are straight-forward to implement. We then apply our method on a large data set of U.S. Treasury securities to extract term structure estimates at daily frequency. The resulting term structure estimates closely matches benchmarks from the literature but have smaller pricing errors.

This is joint work with Kay Giesecke, Markus Pelger, and Ye Ye.


*This talk will take place in hybrid mode.

Viktor Todorov (Northwestern)

Title: Systematic Jump Risk

Abstract: In this paper we develop tests for detecting systematic jump risk in asset prices of general form and we further propose nonparametric estimates for it. The inference is based on a panel of high-frequency asset returns, with both the sampling frequency and the size of the cross-section increasing asymptotically. The feasible limit theory developed in the paper utilizes the different asymptotic role played by diffusive versus jump risk and systematic versus idiosyncratic risk in statistics that involve cross-sectional averages of suitably chosen transforms of the high-frequency price increments. The rate of convergence of the statistics is determined by the two asymptotically increasing dimensions of the panel, without imposing restrictions on their relative size. In an empirical application, using the developed tools, we document the existence of systematic jump risk, that is not spanned by traditional systematic risk factors, and we further show that this risk commands a nontrivial risk premium.