Mathematical Finance Seminar Fall 2017

Schedule for Fall 2017

Seminars are on Thursdays
Time: 4:10pm – 5:25pm
Location: Columbia University, 903 SSW (1255 Amsterdam Ave, between 121st and 122nd Street)

Organizers: Ioannis Karatzas, Marcel Nutz, Philip Protter, Yuchong Zhang

MAFN Seminar Archive

9/14/17

 Antoine Jacquier (Baruch Collge, CUNY – Imperial)

Title: Perturbation analysis of sub/super hedging problems

Abstract: We investigate the links between various no-arbitrage conditions and the existence of pricing functionals in general markets, and prove Fundamental Theorem of Asset Pricings therein. When the market consists only of European Call options, we refine the analysis by weakening the no-arbitrage conditions, and in particular, in the case of finitely many traded options, by showing how to complete the market according to precise implied volatility extrapolation. We finally perform a rigorous perturbation analysis of the infinite-dimensional (primal and dual) optimisation problems, and highlight, with numerical evidence, the influence of smile extrapolation on the bounds of exotic options. Joint work with Sergey Badikov and Mark H.A. Davis.

9/21/17

 Ibrahim Ekren (Michigan)

Title: Portfolio choice with transient and temporary transaction costs

In this talk we study the problem of optimal portfolio choice with transient and temporary transaction costs. In a general Markovian model the objective of the agent is to maximise the discounted value of the future excess return penalised for risk. We establish an expansion of the value function of this problem when the transaction costs go to zero. We obtain a characterisation of the asymptotically optimal portfolio. The talk is based on a joint work with Johannes Muhle-Karbe.

9/28/17  No Seminar
10/5/17

 Xiaolu Tan (Paris Dauphine)

Title: Mean Field Games with dynamic population

Abstract: We study mean field games in an open economy with immigration and branching behavior in the population. Technically, we use branching processes to model the dynamic of the population. Joint work with Julien Claisse and Zhenjie Ren.

10/12/17

 Jin Ma (USC)

Title: Optimal Dividend and Investment Problems under Sparre Andersen Model

Abstract: This talk concerns an open problem in Actuarial Science: the optimal dividend and investment problems Sparre Andersen model, that is, the claim frequency is a renewal process. The main feature of the problem is that the underlying reserve dynamics, even in its simplest form, is no longer Markovian. By using the backward Markovization technique we recast the problem in a Markovian framework with an added random “clock”, from which we validate the dynamic programming principle (DPP). We will then show that the corresponding (dynamic) value function is the unique constrained viscosity solution, and discuss the possible optimal strategy or epsilon-optimal strategy. This talk is based on the joint works with Lihua Bai and Xiaojing Xing

10/19/17

 Blanka Horvath (Imperial)

Title: Functional central limit theorems for rough volatily models

Abstract

We extend Donsker’s approximation of Brownian motion to fractional Brownian motion with any Hurst exponent (including the ’rough’ case H < 1/2), and Volterra-like processes. Some of the most relevant consequences of our ‘rough Donsker (rDonsker) Theorem’ are convergence results (with rates) for discrete approximations of a large class of rough models. This justifies the validity of simple and easy-to-implement Monte-Carlo methods, for which we provide detailed numerical recipes. We test these against the current benchmark hybrid scheme of Bennedsen, Lunde, and Pakkanen and find remarkable agreement (for a large range of values of H). This rDonsker Theorem further provides a weak convergence proof for the hybrid scheme itself, and allows to construct binomial trees for rough volatility models, the first available scheme (in the rough volatility context) for early exercise options such as American or Bermudan.

The talk is based on joint work with A. Jacquier and A. Muguruza.

10/26/17

 Michael Ludkovski (UC Santa Barbara)

Title: Capacity Expansion Games with Application to Competition in Power Generation Investments

Abstract: We consider competitive capacity investment for a duopoly of two distinct producers. The producers are exposed to stochastically fluctuating costs and interact through aggregate supply. Capacity
expansion is irreversible and modeled in terms of timing strategies characterized through threshold rules. Because the impact of changing costs on the producers is asymmetric, we are led to a non-zero-sum timing
game describing the transitions among the discrete investment stages. Working in a continuous-time diffusion framework, we characterize and analyze the resulting Nash equilibrium and game values. Our analysis
quantifies the dynamic competition effects and yields insight into dynamic preemption and over-investment in a general asymmetric setting. A case-study considering the impact of fluctuating emission costs on power
producers investing in green and fossil-fuel plants is also presented.

Joint work with Rene Aid (Paris Dauphine) and Liangchen Li (UCSB).

11/2/17

 Kay Giesecke (Stanford)

Title: Deep Learning for Mortgage Risk

Abstract

We develop a deep learning model of multi-period mortgage risk and use it to analyze an unprecedented dataset of 3.5 billion origination and monthly performance records for over 120 million mortgages originated across the US between 1995 and 2014. A parallel GPU cloud computing approach yields nonparametric estimators of term structures of conditional probabilities of prepayment, foreclosure and various states of delinquency. The estimators incorporate the dynamics of a large number of loan-specific as well as economic and demographic covariates at national, state, county and zip-code levels. They highlight the importance for mortgage risk of local economic conditions, in particular state unemployment. The behavior of mortgage risk can vary strongly depending upon the geographic region. The correlation between loans increases with geographic proximity. Moreover, the relationship between factors and mortgage risk is often highly nonlinear. Higher-order interactions between multiple factors are prevalent. By incorporating these nonlinear effects, our deep learning estimators offer superior out-of-sample predictions of multi-period mortgage risk at loan- and pool-levels. The estimators enable the analysis of mortgage securities such as MBS and CRT as well as the selection of performing mortgage investment portfolios.

11/9/17

 Yu-Jui Huang (Colorado)

Title: 
Optimal Equilibrium for Time-inconsistent Stopping Problems
 
Abstract:
For time-inconsistent control/stopping problems, it is known that one should employ an equilibrium strategy, formulated in an intertemporal game between current and future selves. Such strategies, however, are not unique. This gives rise to two unsettled problems: (i) How do we find all equilibria? (ii) If we found all equilibria, how do we select the appropriate one to use? For stopping problems under non-exponential discounting, we develop a new method, called the iterative approach, to resolve both (i) and (ii). First, we formulate equilibria as fixed points of an operator, which represents strategic reasoning that takes into account future selves’ behaviors. Under appropriate regularity conditions, every equilibrium can be found through a fixed-point iteration. When the state process is one-dimensional, we further establish the existence of an optimal equilibrium, which generates larger value than any other equilibrium does at all times. To the best of our knowledge, this is the first time a dominating subgame perfect Nash equilibrium is shown to exist in the literature of time-inconsistency. Our theory is illustrated explicitly in several real options models. 
11/16/17  No Seminar
11/23/17  No Seminar

11/29/17

*Room 703 Hamilton

 Damiano Brigo (Imperial)

“Optimizing S-shaped utility and risk management:  ineffectiveness of VaR and ES constraints”

(joint work with John Armstrong, King’s College London)

Abstract

We consider market players with tail-risk-seeking behaviour as exemplified by the S-shaped utility introduced by Kahneman and Tversky. We argue that risk measures such as value at risk (VaR) and expected shortfall (ES) are ineffective in constraining such players. We show that, in many standard market models, product design aimed at utility maximization is not constrained at all by VaR or ES bounds: the maximized utility corresponding to the optimal payoff is the same with or without ES constraints. By contrast we show that, in reasonable markets, risk management constraints based on a second more conventional concave utility function can reduce the maximum S-shaped utility that can be achieved by the investor, even if the constraining utility function is only rather modestly concave. It follows that product designs leading to unbounded S-shaped utilities will lead to unbounded negative expected constraining utilities when measured with such conventional utility functions. To prove these latter results we solve a general problem of optimizing an investor expected utility under risk management constraints where both investor and risk manager have conventional concave utility functions, but the investor has limited liability. We illustrate our results throughout with the example of the Black-Scholes option market. These results are particularly important given the historical role of VaR and that ES was endorsed by the Basel committee in 2012-2013.  Joint work with John Armstrong (King’s College London)

 12/7/17  

Gaoyue Guo (Oxford)

Title: Some numerical aspects of (martingale) optimal transportation

Abstract: Martingale optimal transport (MOT), a version of the optimal transport (OT) with an additional martingale constraint on transport’s dynamics, is an optimisation problem motivated by, and contributing to model-independent pricing problems in quantitative finance. Compared to the OT, numerical solution techniques for MOT problems are close to non-existent, relative to the theory and applications. In fact, the martingale constraint destroys the continuity of the value function, and thus renders any of the usual OT approximation techniques unusable. With Obloj, we proved that the MOT value could be approximated by a sequence of linear programming (LP) problems to which we apply the entropic regularisation. Further, we obtain in dimension one the convergence rate, which, to the best of our knowledge, is the first estimation of convergence rate in the literature. In the second part, we consider a semi-discrete Wasserstein distance of order 2, which could be solved by means of Voronoi diagram — which is a static object in computational geometry. Inspired by a criterion in statistic physics, we may construct a sequence of probability distributions and we aim to show its convergence to some limit related to the minimal energy.