Mathematical Challenges of the Emissions Markets
We review recent attempts to provide rigorous mathematical analyses of the cap-and-trade schemes used as market mechanisms to control greenhouse gas emissions. We include equilibrium models as well as reduced form models proposed to price options. We conclude with a set of new mathematical results showing that some of the singular BSDEs occurring in the pricing of emission allowance can have pathological behaviors.

Functional Itô Calculus and Applications
We present an extension of Itô calculus to functionals of price paths. It leads to a Black-Scholes like PDE for path dependent options, even if the path dependency cannot be summarized by a finite number of state variables, with the classical Gamma/Theta (properly defined) trade-off. It also gives an alternative expression of the Clark-Ocone formula for the Martingale Representation Theorem and a non Markov extension of the Feynman-Kac formula.
We apply the functional Itô Formula to obtain the difference of price of an exotic option in two different models and deduce the sensitivity of the price to local deformations of the implied volatility surface. It leads to a decomposition of the volatility risk across strikes and maturities and the associated hedge in terms of a portfolio of European options.
We also apply it in the context of super-replication and show it leads to a refinement of the Kramkov optional decomposition by splitting the increasing process into a convex component and a time component, with trading interpretations.

Quadratic Variance Swap Models: Theory and Evidence
We introduce a quadratic term structure model for the variance swap rates. The latent multivariate state variable is shown to follow a quadratic process characterized by linear drift and quadratic diffusion functions. The univariate case turns out to be a parsimonious and flexible class of models. We provide a complete classification and canonical representation, and discuss model identification. We fit the model to the cross section of variance swap rates and returns of the S&P 500 Index, and perform a specification analysis. This is joint work with Elise Gourier and Loriano Mancini.

Multiscale Stochastic Volatility Models
Stochastic volatility modeling plays a central role in academic research as well as in practice for various markets: equity, fixed income, foreign exchange, credit, energy,... These models present the advantage of being very flexible and being able to fit the data (returns and options) quite well with two or three factors. Choice of one particular model, its calibration, and computation of exotic option prices are very challenging problems. Asymptotic methods have been proposed in various regimes (short or long maturities, small vol-vol, large strikes, ...).
In this talk, I will present a combination of regular and singular asymptotics for multiscale models where stochastic volatility factors are identified by their time scales. I will explain how we address the challenging problems mentioned above as well as the mathematical difficulties inherent to singular payoffs of financial contracts.

Joint work with George Papanicolaou, Ronnie Sircar, and Knut Solna.

Credit Ratings and Securitization
The rating agencies have come under criticism for assigning AAA ratings to the senior tranches of the ABSs and ABS CDOs that were created from subprime mortgages. This presentation will consider whether the criteria used by rating agencies involved appropriate measures of credit quality and whether the criteria were applied appropriately. It will propose a no-arbitrage condition that a credit quality measure should satisfy and examine whether the measures that were used satisfy this condition. It will also use one-and two-factor copula models to examine whether the AAA ratings assigned by rating agencies were ex ante reasonable. It will reach conclusions on the future of structured finance.

Electronic Market Making: Potential Profits and Research Opportunities
With the emergence of electronic markets for stocks and other securities, market makers migrated from the trading floor to computer terminals, each manually posting quotes in one or a handful of securities. But as it became apparent that speed was of the essence, market makers developed computer algorithms for automatically posting their quotes. Not only did speeds increase, but market makers became capable of managing more securities, trading costs decreased, and bid-ask spreads narrowed. This talk will describe the speaker's experience with a Chicago trading firm that does electronic market making. The issue is how to devise optimal algorithms for posting bid and ask quotes. In one sense it is a familiar issue: a trade-off between expected return and risk. But the real-time dynamics of bid and ask prices are very complex, due to the very high, but irregular, frequency of the updated quotes. So to address expected profits it is necessary to employ new statistical techniques that fall outside of the familiar frameworks of continuous-time and classical discrete-time models. And to deal with risk one has to confront numerous issues, some of which can be very subtle such as the presence of traders doing volatility or pairs trading. As far as optimization is concerned, there is little hope for "simple" approaches such as Markov decision theory, because an unmanageable number of state variables would be required in order for the underlying processes to be Markovian. More appealing are approaches like adaptive control, reinforcement learning, and genetic algorithms that deal with incomplete information about the underlying statistical dynamics. We conclude with a step in this direction.

Conic Finance and Accounting: The Static Case
We take up the modeling of two price markets developing closed forms for bid and ask prices. Applications include capital policy, debt valuation adjustments, new perspectives on stocks and bonds, and the formulation of balance sheets that explicitly recognize the value of the taxpayer put option as an asset owned and reported by the .rm. We further illustrate with the construction of conic hedges for multidimensional risks. The presentation closes with a decomposition of pro.ts and capital reserves as an actvity weighted integral of exposure charges for gains and losses over all quantile levels.