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Erik Ekström
The Black-Scholes equation in stochastic volatility models
We study existence and uniqueness of solutions to the Black-Scholes equation for stochastic volatility models and for local volatility models.
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Holger Kraft
Asset Allocation and Liquidity Breakdowns: What if Your Broker Does not Answer the Phone?
This paper analyzes the portfolio decision of an investor facing
the threat of illiquidity. In a continuous-time setting, the efficiency loss
due to illiquidity is addressed and quantified. We show that the efficiency
loss for a logarithmic investor with 30 years until the investment horizon
is a significant 22.7% of current wealth if the illiquidity part of the
model is calibrated to the Japanese data of the aftermath of WW II.
Furthermore, it is demonstrated that the threat of illiquidity can change
the demand for risky securities tremendously and that a logarithmic
investor will not behave myopically anymore.
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Rama Cont (joint with Andreea Minca)
Recovering credit portfolio loss rates from CDO tranches: solution of an inverse problem via intensity control
The calibration of pricing models for portfolio credit derivatives such as CDOs involves the construction of a risk-neutral jump intensity for the loss process which is compatible with a set of observations of market spreads for CDO tranches. We propose an efficient and stable algorithm to solve this inverse problem using an approach based on intensity control. Given a set of observations of market spreads for CDO tranches, we construct a risk-neutral default intensity process for the portfolio underlying the CDO which matches these observations, by looking for the risk neutral loss process 'closest' --in the sense of relative entropy-- to a prior loss process, verifying the calibration constraints. We formalize the problem in terms of minimization of relative entropy with respect to the prior under calibration constraints and use convex duality techniques to solve the problem. The dual problem is shown to be an intensity control problem, characterized in terms of a nonlinear Hamilton Jacobi system of differential equations which we represent in terms of a nonlinear transform of a linear system and thus easily solved. Our method allows to construct an implied intensity process which leads to CDO tranche spreads consistent with the observations. We illustrate our method ITRAXX index data: our results reveal strong evidence for the dependence of loss transitions rates on the past number of defaults, thus offering quantitative evidence for ``contagion effects" in the risk--neutral loss process.
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Mike Tehranchi (joint with Chris Rogers)
No-arbitrage implied volatility dynamics
We study the implications of a no-arbitrage assumption on the possible shapes and dynamics of the volatility surface implied by the Black-Scholes formula. In particular, we prove that the implied volatility surface flattens at long maturities in a rather precise manner. Furthermore, the long implied volatility cannot fall, in complete analogy with the Dybvig-Ingersoll-Ross theorem on long zero coupon rates. Finally, we show that a conjecture of Steve Ross on the impossibility of parallel movements of the implied volatility surface is true for a large class of models.
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Josef Teichmann
Convexity Theorems in Interest Rate Theory
We show several multi-dimensional extensions of results by E. Ekström and J. Tysk on convexity in interest rate theory.
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Srdjan Stojanovic
Foreign exchange rates and foreign exchange derivatives
We present some theoretical, modeling, and empirical results in regard to foreign exchange rates and foreign exchange derivatives.
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Benjamin Bruder (joint with H. Pham)
Optimal investment and hedging with execution delay
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Ljudmila Bordag
Nonlinear option pricing models for illiquid markets: invariant properties and solutions
Several models for the pricing of derivative securities in illiquid markets are discussed. A typical form of nonlinear partial differential equations arising from these investigation is studied. The scaling properties of these equations are discussed. Explicit solutions for some of the models are obtained and studied.
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Marco Avellaneda
Stock pinning on option expiration date with power price-demand elasticity
Stock pinning near option expiration dates has been documented empirically by many authors (Nelken 2000, Ni, Pearson & Poteshman 2003) and is well-known to market practitioners. In 2002 Avellaneda and Lipkin (AL) proposed a mechanism to explain the pinning of stock prices around strikes on option expiration dates, based on modelling the demand for deltas by market-markers, or delta-hedgers, who are long premium. The AL model is a Langevin-type equation based on a linear price-impact function that models the change in price caused by an order of a given size. In this paper, we generalize the AL model allowing for non-linear price-impact functions. The latter have been proposed by several researchers, eg. Farmer, Lillo & Mantegna, Bouchaud and Potters, Gabaix, etc., in the context of market microstructure and order-book dynamics. Following these authors, we study stock pinning in a market with price-demand function d ln P~Q^p, where p is a positive exponent. Our model indicates that pinning takes place only if p>0.5, and that the pinning probability satisfies ln (PinningProb )~ - 1/(2p-1) as p approaches 0.5 from above. This result may be interesting because it implies that impact models with low values for p, which have been reported by some authors, are incompatible with pinning, whereas higher values are not. This is joint work with Michael D. Lipkin (Columbia & American Stock Exchange) and Gennady Kasyan (NYU).
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Francesca Biagini (joint with A. Cretarola)
Local Risk-Minimization for Defaultable Markets
We study the local risk-minimization approach for defaultable markets in a general framework where the asset's behavior and the default time may influence each other. We find the Foellmer-Schweizer decomposition in this general setting, when a recovery payment is due either at time of maturity or at time of default.
We compute it explicitly in two particular cases, when default time depends on the risky asset's behavior and when only a dependence of discounted asset price on default time is occurring.
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Tomas Björk (joint with M.H.A. Davis and C. Landen)
Optimal investment under partial information
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Boualem Djehiche
Optimal strategies to sustain profitability of producing a commodity
We address the issue of finding a strategy to sustain structural
profitability of an investment project whose production activity
depends on the market price of a number of underlying commodities.
Depending on the fluctuating prices of these commodities, the
activity will either continue until the project's profitability
reaches a critical low level at which it is stopped and starts again
when it becomes profitable. But, if the structural non-profitability
remains for a while, the investment project will face the risk to
default and be definitely closed. We suggest a general probabilistic
set up to model profitability as a function of the market price of a
set of commodities, and find the related optimal strategy to sustain
it, under the constraint that the project faces the risk of
defaulting when being non-profitable under a fixed finite time
interval. Extension to the multiple regimes case is also discussed.
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John Chadam
Free Boundary Problems in Mathematical Finance
We provide a unified approach to studying a wide variety of free boundary problems that arise in mathematical finance. For the most part, the main ideas will be presented in the simplest case of the early exercise boundary for the American put option on a geometric Brownian motion. In addition to discussing the existence and uniqueness of the solution to the problem, and the convexity of the free boundary, we will describe several fast and accurate numerical and analytical approximations for the location of these early exercise boundaries. The same approach can be used to treat similar problems with more general underliers such as jump diffusion processes. We will also mention how the techniques can be carried over to treat other classes of free boundary problems such as the inverse first crossing problem of the default barrier of a credit process as well as the pricing of mortgage prepayment options. Various parts of this work are joint efforts with Xinfu Chen (Pittsburgh) and David Saunders (Pittsburgh and Waterloo) as well as a long list of students, practitioners and foreign collaborators.
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Denis Talay (joint with C. Blanchet, R. Gibson, B. de Saporta, E. Tanre)
Stochastic Control and Technical Analysis in Finance
We will justify that statistical procedures cannot provide calibration of financial models with good accuracies, so that model risk cannot be avoided in finance. We will then propose a mathematical framework to study technical analysis algorithms used by practitioners, and compare their performances to strategies issued from stochastic control. We will also present a recent result on a particular HJB equation which arises in our study.
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Bruno Dupire
Forward PDE and discrete local volatility
From the option prices of all strikes and maturities it is possible to obtain the local volatilities consistent with them. The reality gives us only a finite set of option prices and it is common practice to interpolate them to recreate the desired continuum. However, it may generate some arbitrage and the local volatilities obtained this way are highly dependent on the precise interpolation scheme, as differentiating twice is required. We present the concept of discrete local volatility and make use of the stripping PDE in alternative variables. The PDE is nonlinear and we relate a finite difference version to a certain average of the continuous local volatilities and use them as the central pillar of an interpolation scheme.
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Special session: From theory to practice
Presented by Sungard FRONT ARENA
Sungard FRONT ARENA delivers an integrated cross-asset front to back trading solution to the finance industry.
In our special session we will talk about the requirements on todays financial software and the Finite Difference PDE solver implemented in FRONT ARENA.
Different usage of the software, such as e.g. quoting, risk-managment and hedging, have different demands on accuracy and speed of the computations and when numerical methods are used such trade-offs must be considered. We will also give some insight into our PDE solver for equities and the considerations to make it fast and accurate. Considerable effort has been put into producing smooth Greeks, also for more demanding cases such as barrier and American options.
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Imran Habib Biswas
Error estimates for finite difference-quadrature approximations for integro-PDEs associated with controlled jump-diffusion
We study the problem of error estimates for finite difference-quadrature schemes approximating viscosity solutions of nonlinear degenerate integro-PDEs with variable diffusion coefficients. The relevant equations can be viewed as Bellman equations associated to a class of controlled jump-diffusion (Levy) processes. Our results cover both finite and infinite activity cases.
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Maciej Klimek
Empirical data and modelling of financial and economic processes
It has been known for years that commonly used stochastic models of financial and economic processes do not reflect precisely the statistical properties of empirical data. Nevertheless - most of the time - the combination of convenience and reasonable accuracy easily outweighs potential disadvantages. In mature financial markets, it is the proper accounting for rare events that may still present some significant modelling difficulties. However, recent research indicates that in developing markets the discrepancies between models and reality pose much more fundamental problems.
The purpose of this talk is to outline a work-in-progress on a new approach to adaptive time series modelling that hopefully will bring us a step closer to realistic representation of empirical data. The research is conducted jointly with Yasunori Okabe and Masaya Matsuura. It has evolved from earlier work of Okabe and his Japanese colleagues. At the current stage, it relies on frame theory for natural geometric language and on polynomial approximation for more realistic modelling of conditional expectation. Early results are encouraging.
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Johan Tysk (joint with Erik Ekström)
Convexity theory for the term structure equation
We study convexity and model parameter monotonicity properties for prices of bonds and bond options when the short rate is modeled by a diffusion process. We provide sharp conditions on the model parameters under which convexity of the price in the short rate is guaranteed. Under these conditions the price is decreasing in the drift and increasing in the volatility of the short rate. We also study convexity properties of the logarithm of the price, and find simple conditions on the coefficients that guarantee that the price is log-convex or log-concave, respectively.
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Nizar Touzi
Second order BSDE and fully nonlinear PDEs