Mathematical Finance Seminar – Fall 2015

probfinmathSchedule for Fall 2015

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, Philip Protter, Marcel Nutz, Yuchong Zhang




Johannes Muhle-Karbe (ETH and U Michigan)

NOTE: Friday, 10:30 am, Room 507 Math

TITLE: Hedging with small uncertainty aversion

ABSTRACT: We study the pricing and hedging of derivative securities with uncertainty about the volatility of the underlying asset. Rather than taking all models from a prespecified class equally seriously, we penalise less plausible ones based on their “distance” to a reference local volatility model. In the limit for small uncertainty aversion, this leads to explicit formulas for prices and hedging strategies in terms of the security’s cash gamma.


Xuedong He (Columbia)

Title: Processing Consistency in Non-Bayesian Inference

Abstract: We propose a coherent inference model which is obtained by distorting the prior density in Bayes’ rule and replacing the likelihood with a so-called pseudo-likelihood. This model includes the existing non-Bayesian inference models as special cases and implies new models of base-rate neglect and conservatism. We prove a sufficient and necessary condition under which the coherent inference model is processing consistent, i.e., implies the same posterior density however the samples are grouped and processed retrospectively. We further show that processing consistency does not imply the Bayes’ rule by proving a sufficient and necessary condition under which the coherent inference model can be obtained by applying the Bayes’ rule to a false stochastic model. This is a joint work with Di Xiao.


Daniel Schwarz (Carnegie Mellon)

TITLE: Existence of Radner equilibria with price constraints


We prove the existence of complete Radner equilibria in an economy in which the prices of a subset of traded securities are constrained to follow exogenously given dynamics. Applications of the framework include the pricing of derivative securities when the underlying is given by an incomplete financial model, such as, for example, a stochastic volatility model.


Kostas Kardaras (LSE)

TITLE: Equilibrium in risk-sharing games

ABSTRACT: The large majority of risk-sharing transactions involve few agents, each of whom can heavily influence the structure and the prices of securities. This paper proposes a game where agents’ strategic sets consist of all possible sharing securities and pricing kernels that are consistent with Arrow-Debreu sharing rules. First, it is shown that agents’ best response problems have unique solutions, even when the underlying probability space is infinite. The risk-sharing Nash equilibrium admits a finite-dimensional characterisation and it is proved to exist for general number of agents and be unique in the two-agent game. In equilibrium, agents choose to declare beliefs on future random outcomes different from their actual probability assessments, and the risk-sharing securities are endogenously bounded, implying (amongst other things) loss of efficiency. In addition, an analysis regarding extremely risk tolerant agents indicates that they profit more from the Nash risk-sharing equilibrium as compared to the Arrow-Debreu one.

(Joint work with Michail Anthropelos)


No seminar (Minerva Lectures)


Andreea Minca (Cornell)

“Dynamics and Stability of Debt Capacity”

We propose a dynamic model that explains the build-up of short term debt when the creditors are strategic and have different beliefs about the prospects of the borrowers’ fundamentals. We define a dynamic game among creditors, whose outcome is the short term debt. As common in the literature, this game features multiple Nash equilibria. We give a refinement of the Nash equilibrium concept that leads to a unique equilibrium. For the resulting debt-to-asset process of the borrower we define a notion of stability and find the debt ceiling which marks the point when the borrower becomes illiquid. We show existence of early warning signals of bank runs: a bank run begins when the debt-to-asset process leaves the stability region and becomes a mean-fleeing sub-martingale with tendency to reach the debt ceiling. Our results are robust across a wide variety of specifications for the distribution of the capital across creditors’ beliefs. (joint with J. Wissel)


Josef Teichmann (ETH)

“On surprising relations between Americans and Europeans”

Abstract: Following work of Jourdain—Martini we shed some light on a surprising relationship between American and European options motivated by questions from Finance, Analysis and Numerics.

*11/09/2015 Jean Jacod (Paris 6) Joint with Statistics Seminar

Jan Vecer (Charles University in Prague; Frankfurt School of Finance and Management)

Title: New results for the distribution of the average of the geometric Brownian motion


Finding the distribution of the arithmetic average of the stock price that follows a geometric Brownian motion is a widely studied problem in quantitative finance. The average of the stock price appears in evaluations of the cash flows, the pricing of Asian options or in perpetuities, but there is no simple analytical formula that would give a density of the stock price average. In this talk, we extend the work of Marc Yor (1993) who found a representation of the density in terms of a Laplace transform and noticed that the corresponding pricing formula for the Asian option (option on the average of the price) resembles the Black-Scholes formula for the plain vanilla option. We are able to prove that the pricing formula indeed admits the Black-Scholes representation, but under a special and previously unknown martingale measure that is associated with the “average asset” taken as a numeraire. The Black-Scholes representation of the pricing formula gives immediately the hedging portfolio as the probability that the contract is in the money under this new martingale measure. We give both the Laplace transform representation of the average price distribution under this new measure and the corresponding partial differential equation characterization. Both approaches give us the chance to obtain the distribution numerically. Interestingly, the distribution (under the all relevant martingale measures) of the perpetual average admits a closed form solution (which is related to the inverse gamma distribution) and thus we get interesting new solutions of the Black-Scholes partial differential equations for the price of the average.

11/26/2015 No seminar (Thanksgiving)

Christa Cuchiero (University of Vienna)

“A new perspective on the fundamental theorem of asset pricing for large financial markets.”

In the context of large financial markets we formulate the notion of no asymptotic free lunch with vanishing risk (NAFLVR), under which we can prove a version of the fundamental theorem of asset pricing (FTAP) in markets with an (even uncountably) infinite number of assets, as it is for instance the case in bond markets. We work in the general setting of admissible portfolio wealth processes as laid down by Y. Kabanov [3] under a relaxed concatenation property and adapt the FTAP proof variant obtained in [1] for the classical small market situation to large financial markets. In the case of countably many assets, our setting includes the large financial market model considered by M.De Donno et al. [2] and its abstract integration theory. The notion of (NAFLVR) turns out to be an economically meaningful “no arbitrage” condition (in particular not involving weak-*-closures), and, (NAFLVR) is equivalent to the existence of a separating measure. Finally, we briefly discuss a version of the fundamental theorem of asset pricing under restricted and delayed information for large financial markets in discrete time.

The talk is based on joint work with Irene Klein and Josef Teichmann.


[1] C. Cuchiero and J. Teichmann. A convergence result in the Emery topology and a variant of the proof of the fundamental theorem of asset pricing. Finance and Stochastics, 19(4):743-761, 2015.

[2] M. De Donno, P. Guasoni, and M. Pratelli. Super-replication and utility maximization in large


Kavita Ramanan (Brown)

“Greeks for reflected Brownian motions”

Abstract: The need to compute Greeks, or sensitivities of prices of derivatives to a change in underlying parameters, has led to a vast body of literature on sensitivity analysis of diffusions. On the other hand, these methods do not directly carry over to reflected Brownian motions (SRBMs) which arise in many areas, including math finance (e.g., currency exchange rate targe-tzone models, asset pricing models with caps or supports), economics (Leontief systems) and queuing theory.

We establish pathwise differentiability of a large class of SRBMs in convex polyhedra. In particular, we prove existence of directional derivatives of the extended Skorohod map and show that they can be characterized as solutions to a certain time-inhomogeneous extended Skorokhod reflection problem.   We also obtain results on sensitivity of steady-state functionals of the RBM. This is joint work with David Lipshutz.