Mathematical Finance Seminar – Spring 2017

probfinmathSchedule for Spring 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, Philip Protter, Marcel Nutz, Yuchong Zhang


Pete Kyle (Maryland)

“Dimensional Analysis and Market Microstructure Invariance”


This paper combines dimensional analysis, leverage neutrality, and a principle of market microstructure invariance to derive scaling laws expressing transaction costs functions, bid-ask spreads, bet sizes, number of bets, and other financial variables in terms of dollar trading volume and volatility. The scaling laws are illustrated using data on bid-ask spreads and number of trades for Russian stocks. These scaling laws provide useful metrics for risk managers and traders; scientific benchmarks for evaluating controversial issues related to high frequency trading, market crashes, and liquidity measurement; and guidelines for designing policies in the aftermath of financial crisis.


Thaleia Zariphopoulou (Austin)

“Optimal asset allocation under forward performance criteria”

Abstract: Optimal asset allocation models require model commitment, exogenous performance criteria and single horizon specification. As a result, there is limited flexibility in incorporating learning, revision of risk preferences and rolling horizons, all ubiquitous elements for practical relevance. I will discuss these shortcomings and introduce a new approach for optimality and performance measurement. This approach is built “forward in time”, and alleviates the above limitations. It gives, however, rise to ill-posed problems and, in particular, to an ill-posed stochastic pde. I will describe a family of solutions to this forward spde, and discuss various applications.


Wei Zhou (JP Morgan)

Title: Optimal Liquidation of Child Limit Orders


The present paper studies the optimal placement problem of a child order. In practice, short term momentum indicators inferred from order book data play important roles in order placement decisions. In the present work, we first propose to explicitly model the short term momentum indicator, and then formulate the order placement problem as an optimal multiple stopping problem. In contrast to the common formulation in the existing literature, we allow zero refracting period between consecutive stopping times to take into account the common practice of submitting multiple orders at the same time. This optimal multiple stopping problem will be explored over both infinite and finite horizon. It is shown that the optimal placement of a child order and the optimal replacement of outstanding limit orders by the market ones are determined by first passage times of the short term momentum indicator across a sequence of time-dependent boundaries. The aggressiveness of the optimal order replacement strategy is also examined via several numerical examples.

In particular, our work illustrates that the optimal order replacement strategy is more aggressive when the bid-ask spread is smaller, when the impact from the momentum indicator is larger, or when the remaining time becomes shorter. All these decision-making behaviour predicted by our model are natural and agree with empirical studies in the existing literature.


Mykhaylo Shkolnikov (Princeton)

Title: A random surface description of the capital distribution in large markets

Abstract: We study the capital distribution in the context of the first-order models of Fernholz and Karatzas. We find that when the number of companies becomes large the capital distribution fluctuates around the solution of a porous medium PDE according to a linear parabolic SPDE with additive noise. Such a description opens the path to modeling the capital distribution surfaces in large markets by systems of a PDE and an SPDE and to understanding a variety of market characteristics and portfolio performances therein. (Joint work with Praveen Kolli.)


Sebastian Herrmann (U Michigan)

“Model Uncertainty, Recalibration, and the Emergence of Delta-Vega Hedging”

We study option pricing and hedging with uncertainty about a Black-Scholes reference model which is dynamically recalibrated to the market price of a liquidly traded vanilla option. For dynamic trading in the underlying asset and this vanilla option, delta-vega hedging is asymptotically optimal in the limit for small uncertainty aversion. The corresponding indifference price corrections are determined by the disparity between the vegas, gammas, cannas, and volgas of the non-traded and the liquidly traded options. This is joint work with Johannes Muhle-Karbe (University of Michigan).


Yacine Ait-Sahalia (Princeton) “High Frequency Market Making”


Viktor Todorov (Northwestern)

“Nonparametric Option-based Volatility Estimation”

Abstract: In this talk we first review the different methods for recovering volatility non-parametrically from high-frequency return data. We then derive analogues of some of these methods for recovering volatility from options written on the underlying asset. The option data is observed with error and we prove the consistency of the option-based volatility estimators. We further derive a Central Limit Theorems for the estimators. The limiting distribution is mixed-Gaussian and depends on the quality of the option data on the given date as well as on the overall state of the economy. We compare the option and return based volatility estimators and present numerical experiments documenting the superior performance of the former.


No seminar (Spring Recess)


Julien Guyon (Bloomberg)

“Bounds for VIX Futures Given S&P 500 Smiles”


We derive sharp bounds for the prices of VIX futures using the full information of S&P 500 smiles. To that end, we formulate the model-free sub/superreplication of the VIX by trading in the S&P 500 and its vanilla options as well as the forward-starting log-contracts. A dual problem of minimizing/maximizing certain risk-neutral expectations is introduced and shown to yield the same value. The classical bounds for VIX futures given the smiles only use a calendar spread of log-contracts on the S&P 500. We analyze for which smiles the classical bounds are sharp and how they can be improved when they are not. In particular, we introduce a tractable family of functionally generated portfolios which often improves the classical spread while still being tractable, more precisely, determined by a single concave/convex function on the line. Numerical experiments on market data and SABR smiles show that the classical lower bound can be improved dramatically, whereas the upper bound is often close to optimal.

3/30/17 No seminar

Christoph Czichowsky (LSE)

“Portfolio Optimisation, Transaction Costs, Shadow Prices and Fractional Brownian Motion”
Abstract: While absence of arbitrage in frictionless financial markets (i.e. without transaction costs) requires price processes to be semimartingales, non-semimartingales can be used to model prices in an arbitrage-free way, if proportional transaction costs are taken into account. In this talk, I will present an overview over several results that provide a way how to use non-semimartingale price processes such as the fractional Black-Scholes model in portfolio optimisation under proportional transaction costs by establishing the existence of a so-called shadow price. This is a semimartingale price process, taking values in the bid ask spread, such that frictionless trading for that price process leads to the same optimal strategy and utility as the original problem under transaction costs. The talk is based on joint work with Walter Schachermayer.


Christoph Frei (Alberta)

“Managing Counterparty Risk in OTC Markets”

Abstract: We study how counterparty risk affects trading decisions in over-the-counter (OTC) markets. Banks first manage their default risk, and then decide on credit default swap (CDS) trading volumes to hedge against an aggregate risk factor. Because counterparty risk introduces an asymmetry between protection buyers and sellers, our model predicts that perfect risk sharing is only done between safe banks, while riskier banks still maintain diverse post-trade exposures. We show that the costly actions exerted by banks to reduce their default risk are not socially optimal. Surprisingly, we find that banks may choose to reduce their default probabilities below the socially optimal level, depending on the imposed trade size limits and costs of risk management. The model produces new empirical predictions including (i) intermediation is done by relatively safe banks with medium initial exposure, (ii) banks with high initial exposures are net buyers of CDSs, and banks with low initial  exposures are the main net sellers but only if they have sufficiently low default risk, (iii) heterogeneity in post-trade exposures is higher for riskier and smaller for safer banks. These predictions are borne out by bilateral exposures data from the CDS market. The talk is based on joint work with Agostino Capponi and Celso Brunetti.


Special Start Time 5:15 – 6:15

Umut Cetin (LSE)

“Diffusion transformations, Black-Scholes equation and optimal stopping”

We develop a new class of path transformations for one-dimensional diffusions that are tailored to alter their long-run behaviour from transient to recurrent or vice versa. This immediately leads to a formula for the  distribution of the first exit times of diffusions, which is recently characterised by Karatzas and Ruf as the minimal solution of an  appropriate Cauchy problem under more stringent conditions. These transformations also turn out to be instrumental in characterising the stochastic solutions of Cauchy problems defined by the generators of strict local martingales, which are well-known for not having unique solutions even though one restricts   the solutions  to have  linear growth. Using an appropriate diffusion transformation we show that the aforementioned stochastic solution is the unique classical  solution of an alternative Cauchy problem with suitable boundary conditions. This in particular  resolves the long-standing issue of non-uniqueness with the Black-Scholes equations in derivative pricing in the presence of  bubbles.  Finally, we use these path transformations to propose a unified framework for solving explicitly the optimal stopping problem for one-dimensional diffusions with discounting, which in particular is relevant for the pricing and the optimal exercise boundaries of perpetual American options.


Persi Diaconis (Stanford)

Columbia Joint Probability Colloquium

“The Mathematics of Spatial Mixing”

Abstract: Questions of spatial mixing arise when cards (or dominoes or Mahjong tiles) are ‘smooshed’ around on the table with two hands. It also occurs in coloring molten glass. In joint work with Soumik Pal, we have (a) a reasonable model (b) some practical examples and data (c) a new technique which give a first quantitative analysis of ‘how long to smoosh to mix things (d) some sophisticated math which shows that everything makes sense. This talk is aimed at a no-specialist audience.