Double Mean Reversion in FX
In this paper, we propose a new model for Foreign Exchange rate that accounts for multiple time scale. We review dynamic and calibration to Vanilla and Exotic short-dated market. The effect of the ATM decorrelation on first and second generation exotics is explored.

Market indifference prices

We develop an equilibrium based model for the trades of a large investor with a group of market makers at their indifference prices. Using the theory of saddle functions form convex duality, we show how to compute expansions of derivative prices around their Black-Scholes valuation, illustrating how these depend on the market makers' risk tolerances. This is joint work with Dmitry Kramkov.

On Correlation and Default Clustering in Credit Markets
We establish Markovian models in the Heath-Jarrow-Morton paradigm that permit an exponential affine representation of riskless and risky bond prices while offering significant flexibility in the choice of volatility structures. Estimating models in our family is typically no more difficult than in the workhorse affine family. Besides diffusive and jump-induced default correlations, defaults can impact the credit spreads of surviving firms, allowing for a greater clustering of defaults. Numerical implementations highlight the importance of incorporating interest rate-credit spread correlations, credit spread impact factors, and the full credit spread curve when building a unified framework for pricing credit derivatives.

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Effiicient Risk Estimation via Nested Simulation
We analyze the computational problem of estimating financial risk in a nested simulation. In this approach, an outer simulation is used to generate financial scenarios and an inner simulation is used to estimate future portfolio values in each scenario. We propose a new algorithm to estimate risk which sequentially allocates computational effort in the inner simulation based on marginal changes in the risk estimator in each scenario. Theoretical and numerical results are given to show that the risk estimator has a faster convergence order compared to the conventional uniform inner sampling approach.

Pricing Variance Swaps on Time Changed Lévy Processes
We generalize to time-changed Lévy processes (including a wide range of jump dynamics) the "model-free'' valuation of variance swaps previously investigated only in the case of continuous underlying dynamics.

Processes of class Sigma, last passage times and drawdowns
We propose a general framework to study last passage times, suprema and drawdowns of a large class of stochastic processes. Our approach is based on processes of class Sigma. We provide three general representation results that allow to deduce the laws of a variety of interesting random variables such as running maxima, drawdowns and maximum drawdowns of suitably stopped processes. As an application we discuss the pricing and hedging of options that depend on the running maximum of an underlying price process.

Exponential Hedging under Variable Horizons
This paper addresses four main issues intimately related to the exponential hedging when the horizon may vary in the set of bounded stopping times. The first contribution deals with explicit and complete characterization/parametrization of the class of exponential forward performances. The second main contribution focuses on the horizon-unbiased exponential hedging problem. Precisely, for a given dynamic payoff B = (B_t, t>0), this problem consists of finding an admissible optimal strategy that minimizes the risk in the exponential utility framework up to any stopping time bounded by a constant T. Here, our goal is to explicitly describe the optimal strategy when it exists, and fully describe the payoff B for which this problem admits solution. The effect of the change of numeraire on this problem as well as its relationship to the exponential forward performances are illustrated. Both contributions (the first and the second) rely heavily on the concept of minimal entropy-Hellinger martingale density (MEHM density hereafter) and its variations. The third contribution discusses the mathematical structures of the horizon-dependence in the optimal portfolio and strategy for an exponential hedging problem. This extends the work of Choulli and Schweizer (2009) to the case of exponential utilities. The last contribution analyzes the optimal sale problem for the exponential utility and the semimartingale framework. Again, here the MEHM density concept plays a crucial role in our methodology and this enhances -once more- our belief that this Hellinger tool is tailor-made to deal with the horizon's effect in exponential hedging.

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On the Relative Pricing of long Maturity S&P 500 Index Options and CDX Tranches

Analysis of Fourier transform valuation in Lévy models
A systematic analysis of the conditions such that Fourier transform valuation formulas are valid is provided. The interplay between the conditions on the payoff function and the driving process is investigated. Plain vanilla as well as exotic path-dependent options such as one-touch or lookback options and equity default swaps are considered. A number of results are extended to the multidimensional setting to include basket products and calculation of Greeks by Fourier transform methods is discussed.

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Optimal Securitization of Credit Portfolios via Impulse Control
We study the optimal loan securitization policy of a commercial bank which is mainly engaged in lending activities. For this we propose a stylized dynamic model which contains the main features affecting the securitization decision. In line with reality we assume that there are non-negligible fixed and variable transaction costs associated with each securitization. The fixed transaction costs lead to a formulation of the optimization problem in an impulse control framework. We prove viscosity solution existence and uniqueness for the quasi-variational inequality associated with this impulse control problem. Iterated optimal stopping is used to find a numerical solution of this PDE, and numerical examples are discussed.

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Exact Simulation of Jump-Diffusion Processes
This paper develops a method for the exact simulation of a one-dimensional jump-diffusion process. The drift, diffusion coefficient and intensity are arbitrary functions of the state. We illustrate the effectiveness of the method with examples from credit/equity, including the analysis of options on defaultable stocks.

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Pricing Contingent Capital with Continuous Conversion
Among the solutions proposed to the problem of banks "too big to fail" is contingent capital in the form of debt that converts to equity when the firm's capital ratio falls below a threshold. We analyze the dynamics of such a security with continuous conversion and derive closed form expressions for its value when the firm's assets are modeled as geometric Brownian motion and the conversion trigger is an asset-based capital ratio. A key step in the analysis is an explicit formula for the fraction of equity held by the original holders of the contingent capital debt.

 

We develop a model where dividend payout, investment and financing decisions are made by managers who attempt to maximize the rents they take from the firm. But the threat of intervention by outside shareholders constrains rents and forces rents and dividends to move in lockstep. Managers are risk-averse, and their utility function allows for habit formation. We show that dividends follow Lintner's (1956) target-adjustment model. We provide closed-form, structural expressions for the payout target and the partial adjustment coefficient. Risk aversion causes managers to underinvest, but habit formation mitigates the degree of underinvestment. Changes in corporate borrowing absorb fluctuations in earnings and investment.

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Modeling Dependent Jumps: A Time Change Approach
We show how to construct multi-dimensional Markov processes with dependent jumps via time changes, how to solve the resulting models via the spectral method, and discuss a range of applications in mathematical finance.

Joint Risk Neutral Laws and Hedging
Complex positions on multiple underliers are hedged using the options surface of all underliers. Hedging objectives follow Cherny and Madan (2010) by minimizing ask prices for which post hedge residual risks are acceptable at prespeci.ed levels. It is shown that such hedges require one to use a risk neutral law on the set of underlying risks. We propose a joint risk neutral law for multiple underliers and estimate it from multiple option surfaces. Under our joint law asset returns are a linear mixture of independent Lévy components.
Data on the independent components are estimated by an application of independent components analysis on time series data for the underlying returns. A comparison of the the risk neutral law with the statistical law shows that risk neutral correlations dominate their statistical counterparts. Hedges signicantly reduce ask prices.

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