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, Hongzhong Zhang
For an archive of past seminars, please click here.
“Forward-backward stochastic differential equations of McKean-Vlasov type and mean field games”
We review the mean eld game paradigm and demonstrate how the solutions of these models can be identied with solutions of forward – backward stochastic dierential equations (FBSDEs) of McKean-Vlasov type. We give existence and uniqueness results for a large class of these FBSDEs and discuss the similarities and the dierences with the solutions of the optimal control of McKean-Vlasov stochastic dierential equations.
“Diffusion scaling of a limit-order book model”
With the movement of trading away from the trading oor onto electronic ex-changes – and the accompanying rise in the volume of order submission – has come an increase in the need for tractable mathematical models of the whole limit order book. The problem is inherently high-dimensional and the most natural description of the dynamics of the order ows has them depend on the state of the book in a discontinuous way. We examine a popular discrete model from the literature and describe its limit under a diusion scaling inspired by queueing theory. Interesting features include a process that is either “frozen” or diusing according to whether another diusion is positive or negative. This is joint work with Christopher Almost and John Lehoczky
(moved to Feb 19th)
“Rough Volatility” (4:10pm-5:00pm, 903 SSW)
Starting from the observation that increments of the log-realized-volatility possess a remarkably simple scaling property, we show that log-volatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale. The resulting Rough Fractional Stochastic Volatility (RFSV) model is remarkably consistent with financial time series data. We then show how the RFSV model can be used to price claims on both the underlying and integrated volatility. . We analyze in detail a simple case of this model, the rBergomi model. In particular, we find that the rBergomi model fits the SPX volatility markedly better than conventional Markovian stochastic volatility models, and with fewer parameters. Finally, we show that actual SPX variance swap curves seem to be consistent with model forecasts, with particular dramatic examples from the weekend of the collapse of Lehman Brothers and the Flash Crash.
This is joint work with Christian Bayer, Peter Friz, Thibault Jaisson, and Mathieu Rosenbaum.
Johannes Ruf (5:05pm-5:55pm, 903 SSW)
“A Numeraire-Independent Version of the Fundamental Theorem of Asset Pricing”
The Fundamental Theorems of Asset Pricing are aptly-named results that show the relationship between absence of arbitrage and the martingale property. These theorems are fundamental to mathematical finance in that they provide the bridge between the mathematics and the finance: on the one side, the mathematical objects of stochastic processes and martingale measures; on the other the financial ideas of trading strategies and arbitrage. We aim to widen the bridge to cover cleanly the case when there are multiple financial assets, any of which may potentially lose all value relative to the others. To do this we shift away from having a pre-determined numeraire to a more symmetrical point of view where all assets have equal priority.
Joint work with Travis Fisher and Sergio Pulido.
So far, path-dependent volatility models have drawn little attention from both practitioners and academics compared with local volatility and stochastic volatility models. This is unfair: in this article we show that they combine benefits from both. Like the local volatility model, they are complete and can fit exactly the market smile of the underlying asset; smile calibration is achieved using the particle method. Like stochastic volatility models, they can produce rich implied volatility dynamics; for instance, they can generate large negative forward skews, even when they are calibrated to a flat smile. But path-dependent volatility models can even do better than that: thanks to their huge flexibility, they can actually produce spot-vol dynamics that are not attainable using stochastic volatility models, thus possibly preventing large mispricings; and they can also capture prominent historical patterns of volatility, such as volatility depending on the recent trend of the underlying asset, for instance. We give many examples and show many graphs to demonstrate their great capabilities.
The talk is based on my article that appeared in Risk magazine in Oct 2014. The preprint of the article is available at: ssrn.com/abstract=2425048.
“No-arbitrage conditions and hedging dualities under ambiguity”
In this talk an approach to robust versions of the basic theorems of mathematical finance is presented that is based on general representation results for increasing convex functionals.
“Optimality of doubly reflected Levy processes in singular control”
We consider a class of two-sided singular control problems. A controller either increases or decreases a given spectrally negative Levy process so as to minimize the total costs comprising of the running and control costs where the latter is proportional to the size of control. We provide a sufficient condition for the optimality of a double barrier strategy, and in particular show that it holds when the running cost function is convex. Using the fluctuation theory of doubly reflected Levy processes, we express concisely the optimal strategy as well as the value function using the scale function. Numerical examples are provided to confirm the analytical results.
“Systemic Risk: The dynamics under center clearing”
We develop a tractable model to resemble asset value processes of financial institutions, trading with the central clearinghouse for risk mitigating purposes. Each institution allocates assets between his loan book and trade account. The volatility of the traded portfolio depends both on his and the aggregate amount of trading capital. We show that there exists a unique equilibrium allocation profile when institutions adjust positions with the clearinghouse to perfectly hedge risk stemming from their loan books. We then analyze the dynamic equilibrium path. This shows a buildup of systemic risk, manifested through the increase of market concentration. The associated size externalities can be internalized via a self-funding systemic risk charge mechanism. We provide new testable predictions, including that (i) hedging becomes increasingly costly for an institution as his asset value increases, (ii) market shocks have smaller impact on allocation decisions than operational shocks, (iii) capital raising and centralized trading have opposite effects on market concentration.
A preprint of the paper is available here:
“Moral hazard in dynamic risk management”
We consider a contracting problem in which a principal hires an agent to manage a risky project. When the agent chooses volatility components of the output process and the principal observes the output continuously, the principal can compute the quadratic variation of the output, but not the individual components. This leads to moral hazard with respect to the risk choices of the agent. We identify a family of admissible contracts for which the optimal agent’s action is explicitly characterized, and, using the recent theory of singular changes of measures for It\^o processes, we study how restrictive this family is. In particular, in the special case of the standard Homlstrom-Milgrom model with fixed volatility, the family includes all possible contracts. We solve the principal-agent problem in the case of CARA preferences, and show that the optimal contract is linear in these factors: the contractible sources of risk, including the output, the quadratic variation of the output and the cross-variations between the output and the contractible risk sources.Thus, like sample Sharpe ratios used in practice, path-dependent contracts naturally arise when there is moral hazard with respect to risk management. In a numerical example, we show that the loss of efficiency can be significant if the principal does not use the quadratic variation component of the optimal contract.
(Joint with N. Touzi and D. Possamai)
“A Pure-Jump Market-Making Model for High-Frequency Trading using constrained FBSDEs”
We propose a new market-making model which incorporates a number of realistic features relevant for high-frequency trading. In particular, we model the dependency structure of prices and order arrivals with novel self- and cross-exciting point processes. Furthermore, instead of assuming that bid and ask prices can be adjusted continuously by the market maker, we formulate the market maker’s decisions as an optimal switching problem. The model also allows for over-trading risk, and the use market orders, which are modeled as impulse control, to get rid of excessive inventory. Because of the stochastic intensities of the cross-exciting point processes, the optimality condition appears to fall outside of the scope of classical Hamilton-Jacobi-Bellman quasi-variational inequalities, so we leverage a newly developed constrained forward backward stochastic differential equation (FBSDE) to solve the optimal control problem. The method’s implementability, which includes a Monte-Carlo requirement, is illustrated thanks to full-scale simulations. This is joint work in progress with Mr. Baron Law.
“Nonlinear Lévy processes with applications to superreplication and utility maximization under Knightian uncertainty”
We present a tractable framework for Knightian uncertainty, the so-called nonlinear Lévy processes, and use it to formulate and solve problems of robust utility maximization. Moreover, we show its connection to superreplication under uncertainty.
“A test for the rank of the volatility process: the random perturbation approach”
In this talk we present a test for the maximal rank of the matrix-valued volatility process in the continuous Ito semimartingale framework. Our idea is based upon a random perturbation of the original high frequency observations of an Ito semimartingale, which opens the way for rank testing. We develop the complete limit theory for the test statistic and apply it to various null and alternative hypotheses. The talk is based on joint work with Jean Jacod.