Friday, December 4, 2009
All presentations take place in the Neilson Room at the Heldrich Hotel, 10 Livingston Avenue, New Brunswick, New Jersey 08901.
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Tomasz Bielecki
Hedging of a Credit Default Swaption in the CIR Default Intensity Model
An important issue arising in the context of credit default swap (CDS) rates is the construction of an appropriate model in which a family of options written on credit default swaps, referred to hereafter as credit default swaptions, can be valued and hedged. The goal of this work is to exemplify the usefulness of some abstract hedging results, obtained previously by the authors, for the valuation and hedging of the credit default swaption in a particular hazard process setup, namely, the CIR default intensity model. (Joint work with Monique Jeanblanc and Marek Rutkowski.) [Slides (pdf)]
Rene Carmona
Pricing Options on CO2 Emissions
Based on equilibrium analysis results, futures on EU Emission Allowances are risk neutral martingales with binary terminal values.This is in contrast with the standard Black Scholes models where underlying prices are risk neutral martingales converging to zero. We propose simple models of binary martingales, and we show how to price call options in these models which we fit to historical data by maximum likelihood. [Website]
John Chadam
The Inverse Boundary Crossing Problem for Diffusions
A summary of our work on the inverse boundary crossing problem for diffusions will be presented. To begin, the direct and inverse problems will be described in their probabilistic, PDE and integral equation settings. Our existence and uniqueness of the solution to the inverse problem in the PDE setting will be outlined. More recently a verification theorem was established showing that this solution solves the probabilistic version of the problem. Results on the initial behavior and continuity of the boundary will be described. Finally, a numerical scheme based on the equivalent integral equation formulation of the problem will be analyzed. (Joint work with Xinfu Chen, Lan Cheng and David Saunders.) [Slides (pdf)]
Rama Cont
Forward Equations and Mimicking Theorems for Semi-Martingale Models
In option pricing theory, partial (integro-)differential equations are usually associated with Markovian models, but since the seminal work of Dupire (1994) we know that Markovian projection methods can be used to derive PDEs for call option prices in many non-Markovian models where the price follows a Brownian martingale under the risk-neutral measure. We give conditions under which the flow of marginal distributions of a discontinuous semi-martingale X can be matched by a Markov process whose infinitesimal generator is expressed in terms of the local characteristics of X. Using this Markovian projection result, we derive a forward partial integro-differential equation for option prices in a large class of (non-Markovian) semi-martingale models with jumps. Our work extends the results of Gyongy (1986) and Dupire (1994) to the case of discontinuous semi-martingales. (Joint work with Amel Ben-Tata, Universite de Paris VI.) [Website]
Bruno Dupire
Functional Ito Calculus and PDE’s for Path-Dependent Options
We present an extension of Ito calculus to functionals and establish that under certain conditions, path-dependent option prices satisfy a PDE. We apply this result to compute the sensitivity of path-dependent option prices to perturbations of the local volatility surface. We show that the coefficients of the portfolio of European options with the same volatility risk profile can be obtained as the source term of a PDE. [Slides (ppt)]
Tom Hurd
Credit Risk via First Passage for Time Changed Brownian Motions
The first passage structural approach to credit risk, while very natural, is beset by technical difficulties that make it inflexible in practice. Time changed Brownian motions (TCBMs) offer a simple but mathematically interesting way to circumvent these technicalities and open the door to a number of innovations. After a quick sketch of the basic properties of TCBM models, I show that they can give an excellent fit to the dynamics of credit default swaps observed in the market. I then consider a more complex ``hybrid'' framework that can model the joint dynamics of equity and credit derivatives. [Slides (pdf)]
Elton Hsu
Differential Geometry, Heat Kernel, and Implied Volatility in Stochastic Volatility Models
In stochastic volatility models, the volatility is driven by a Brownian motion uniformly with the driving Brownian motion for the stock price and the price-volatility process is a multidimensional diffusion process. As such they determine a Riemannian geometry, which is in general hyperbolic. We will show that the near-expiry asymptotic behavior of the implied volatility can be characterized explicitly by the geometry near the unique shortest geodesic to the half plane determined by the strike price. In this respect, the second variation of the geodesic and the associated Jacobi field will play an important role, and the geometric analysis is relatively simple due to the hyperbolic nature of the price-volatility space. For some concrete models, this method yields some results obtained previously by other and often heuristic methods. [Website]
Andrey Itkin
Fractional PDE Approach for Numerical Solution of Some Jump-Diffusion Models
In mathematical finance a popular approach for pricing options under some Levy model is to consider underlying that follows a Poisson jump diffusion process. As it is well known this results in a partial integro-differential equation (PIDE) that usually does not allow an analytical solution while numerical solution brings some problems. In this paper we elaborate a new approach on how to transform the PIDE to some class of so-called pseudo-parabolic equations which are known in mathematics but are relatively new for mathematical finance. As an example we discuss several jump-diffusion models which Levy measure allows such a transformation.
Alternatively for some class of Levy processes, known as GTSP/KoBoL/SSM models, with the real dumping exponWebsiteent we show how to transform the corresponding PIDE to a fractional PDE. Fractional PDEs for the Levy processes with finite variation were derived by Boyarchenko and Levendorsky (2002) and later by Cartea (2007) using a characteristic function technique. In this paper we derive them in all cases, including processes with infinite variation, using a different technique - shift operators. Then to solve them we apply a matrix exponent formalism, and finally construct finite difference schemes proven to be a) unconditionally stable and b) at least of the second order of approximation in space and time. Higher order approximations in time are also considered. [Website]
Ioannis Karatzas
Probabilistic Aspects of Arbitrage
Consider the logarithm log(1/U(T,z)) of the highest return on investment that can be achieved relative to a market with Markovian weights, over a given time-horizon [0,T] and with given initial market weight configuration Z(0) = z. We characterize this quantity (i) as the smallest amount of relative entropy with respect to the Foellmer exit measure, under which the market weight process Z(.) is a diffusion with values in the unit simplex and the same covariance structure but zero drift; and (ii) as the smallest "total energy" expended by the respective drift, over a class of probability measures which are absolutely continuous with respect to the exit measure and under which Z(.) stays in the interior of the unit simplex at all times, a.s. The smallest relative entropy, or total energy, corresponds to the
conditioning of the exit measure on the event that Z(.) stays in the interior of the unit simplex throughout [0,T]; whereas, under this "minimal energy" measure, the portfolio generated by the function U(.,.) has the numeraire and relative log-optimality properties. This same portfolio also realizes the highest possible relative return on investment with respect to the market. (Joint work with D. Fernholz.) [Website]
Peter Laurence
Asymptotics for Time Inhomogeneous Local Volatility Models
We review recent work on highly accurate expansions for the implied volatility in time homogeneous local volatility models and discuss their application to SABR and Heston SV models. (Joint work with Gatheral, Hsu, Ouyang and Wang.) [Slides (pdf), website]
Chris Rogers
Optimal Stopping and Convex Regression
This talk presents a way to price and hedge numerically a range of Bermudan options which have a special structural property, namely that the value function is a convex function of the underlying assets. There are plenty of interesting examples, such as the min-put, or the max-call. The idea is to make a numerical approximation to the value function, expressing it as the maximum of a family of linear functionals. This allows a compact characterization of the approximate value function, which is well suited to calculating expectations, and which goes well together with the operation of taking the max of the stopping value with the expected future value. There are a number of challenging issues in devising an efficient computational scheme. [Website]
Hao Xing
Strict Local Martingale Deflators and Pricing American Call-Type Options
When the discounted stock price is a martingale under the risk neutral measure, it is well know that exercising the American call option at the terminal time is optimal. However, it may not be the case when the discounted stock price is a strict local martingale. In this talk, I will present how to price and optimally exercise American call-type options in markets which do not necessarily admit an equivalent local martingale measure. This resolves an open question proposed by Fernholz and Karatzas [Stochastic Portfolio Theory: A Survey, Handbook of Numerical Analysis, 15:89-168, 2009]. The relationship between the martingale property of diffusion processes and the uniqueness of classical solutions for Cauchy problems will be also discussed at the second half of the talk. (Joint work with Erhan Bayraktar and Kostas Kardaras.) [Slides (pdf]
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