Speaker: Peter Bank

Title: Convex duality and intertemporal consumption choice

Abstract:

We show how to develop a theory of convex duality for optimal investment in incomplete markets when utility is obtained from a consumption stream, rather than from terminal wealth. As it turns out, different notions of convexity for functions and for stochastic processes are needed to address this problem. Joint work with Helena Kauppila.

Speaker: Erhan Bayraktar

Title: Valuation equations for stochastic volatility models

Abstract:

We study the valuation partial differential equation for European contingent claims in a general framework of stochastic volatility models. The standard Feynman-Kac theorem cannot be directly applied because the diffusion coefficients may degenerate on the boundaries of the state space and grow faster than linearly. We allow for various types of model behavior; for example, the volatility process in our model can potentially reach zero and either stay there or instantaneously reflect, and asset-price processes may be strict local martingales under a given risk-neutral measure. Our main result is an extension of the standard Feynman-Kac theorem in the context of stochastic volatility models. Sharp results on the existence and uniqueness of classical solutions to the valuation equation are obtained using a combination of probabilistic and analytical techniques. The role of boundary conditions is also discussed. Joint work with with Kostas Kardaras and Hao Xing. Available at http://arxiv.org/abs/1004.3299.

Speaker: Douglas Borden

Title: Stochastic control and automated market-making

Abstract:

Trading decisions in Automated Market-Making involve a subtle interplay between expected price movement, transaction costs, market impact and risk. And these decisions need to be made in milliseconds, millions of times a day. Standard practice is to derive a set of trading rules, with the rule parameters optimized through simulation backtesting. In this talk I present a different approach to the decision process in Automated Market-Making, making use of techniques from Stochastic Control Theory.

Speaker: Panagiota Daskalopoulos

Title: Free-boundary problems and degenerate diffusion

Abstract:

We will discuss existence and optimal regularity of solutions to free-boundary problems related to degenerate diffusion. Free-boundary problems presenting similar analytical challenges often appear in mathematical finance in the context of option pricing problems for stochastic volatility models.

Speaker: Emmanuel Gobet

Title: Expansion formulas for pricing

Abstract:

In this talk, we present a recent methodology based on stochastic analysis and Malliavin calculus to derive tractable approximations of option prices in various models. Regarding the models, our results cover the case of local volatility models, including or not Gaussian jumps and Gaussian interest-rate models, Heston models. We also handle the case of stocks paying affine dividends at discrete times, Asian or basket options, etc. Error estimates are provided. Numerical results illustrate the relevancy of the expansions. Applications to real-time pricing/calibration are discussed as well.

Speaker: Igor Halperin

Title: Option pricing and hedging by risk minimization

Abstract:

We discuss the risk minimization approach to the pricing and hedging of options under incomplete market scenarios in the presence of transaction costs. We present a practical framework which assumes a multifactor stochastic model for the underlying assets.

Speaker: Kumar Muthuraman

Title: Moving boundary methods for solving free-boundary problems

Abstract:

Several classes of stochastic control problems reduce free-boundary PDE problems with the use of dynamic programing arguments. These include optimal stopping, singular control and impulse control problems. Several classical examples including American option pricing and portfolio optimization with transaction costs belong to these classes. This talk describes a computational method that solves free-boundary problems by converting them into a sequence of fixed-boundary problems, that are much easier to solve. We will illustrate application on a set of classical problems, of increasing difficulty and will also see how the method can be adapted to efficiently handle problems in large dimensions.

Speaker: Andrea Pascucci

Title: Kolmogorov equations and asian options

Abstract:

The talk will present a survey of the theory of partial differential equations of Kolmogorov type arising in physics and in mathematical finance. These evolutionary equations, which are generally non-uniformly parabolic, are naturally associated to stochastic models with memory. Financial derivatives with dependence on the past provide some typical examples: in particular, Asian options of European and American style will be discussed.

Speaker: Thaleia Zariphopoulou

Title: An Approximation Scheme for Investment Performance Processes in Incomplete Markets

Abstract:

An approximation scheme for the maximal expected utility in an incomplete market will be presented. The market incompleteness comes from a stochastic factor affecting the dynamics of the stock price. The scheme yields an intuitively pleasing decomposition of the value function process at each splitting step. Specifically, in the first sub-step, the current utility is being adjusted while the component of the investment opportunity set that is perfectly correlated with the stock remains unchanged. In the second sub-step the reverse happens. This "orthogonal" decomposition highlights how dynamic preferences and investment decisions behave in terms on the imperfect correlation and, moreover, explains some effects of the stochastic factors on the risk preferences and the investment decisions.

Speaker: Jianfeng Zhang

Title: Second order backward stochastic differential equations

Abstract:

In this talk we introduce the recent development on Second Order BSDEs which, in the Markovian case, provide a probabilistic representation of viscosity solution to fully nonlinear parabolic PDEs, e.g. the HJB equations. The theory is closely related to and has applications in various areas, including: numerical methods for fully nonlinear PDEs, super-hedging contingent claims under volatility uncertainty or under liquidity risk, dynamic risk measures under volatility uncertainty, stochastic optimization with volatility control, stochastic target problems with Gamma constraints, forward utilities, and two person repeated games. The theory is also closely related to the so called G-expectation, a nonlinear expectation proposed by Peng in recent years. In particular, a martingale under G-expectation can be viewed as the solution to a "linear" second order BSDE. Our main tool will be the quasi-sure analysis, which involves a family of mutually singular probability measures. The talk is based on joint work with Mete Soner and Nizar Touzi.

Speaker: Johan Tysk

Title: Boundary behavior of densities for non-negative diffusions

Abstract:

It is well-known that the transition density of a di ffusion process solves the corresponding Kolmogorov forward equation. If the state space has finite boundary points, then naturally one also needs to specify appropriate boundary conditions when solving this equation. However, many processes in finance have degenerating diffusion coefficients, and for these processes the density may explode at the boundary. We describe a simple symmetry relation for the density that transforms the forward equation into a backward equation, the boundary conditions of which being much more straightforward to handle. This relation allows us to derive new results on the precise asymptotic behavior of the density at boundary points where the diffusion degenerates. This is joint work with Erik Ekström

Speaker: Wen Cheng

Title: Closed-form asymptotics and numerical approximations of 1D parabolic equations with applications to option pricing

Abstract:

We construct closed-form asymptotic formulas for the Green's function of parabolic equations (e.g. Fokker-Planck equations) with variable coefficients in one space dimension. The approximate kernels are derived by applying the Dyson-Taylor commutator method that we have recently developed for short-time expansions of n-dimensional heat kernels. We then utilize these kernels to obtain closed-form pricing formulas for European call options. The validity of such approximations to large time is extended using a bootstrap scheme. We prove explicit error estimates in weighted Sobolev spaces, which we test numerically and compare to other methods. This is a joint work with N. Costanzino, J. Liechty, A. Mazzucato and V. Nistor.

Speaker: Huibin Cheng

Title: Regularity of the free boundary for the American put option

Abstract:

We show the free boundary of the American put option with dividend payment is C^\infty.

Speaker: Youngna Choi

Title: Financial crisis dynamics: Attempt to define a market instability indicator

Abstract:

Historically a financial crisis has resulted from heavily leveraged overinvestment, which makes the economic agents susceptible to decrease in revenue. In the 2007-2009+ financial crisis securitization interconnected the market agents, therefore a small default in one segment spread to the entire system, causing a systemic risk. We use classical perturbation theory of dynamical systems to analyze the bifurcation mechanism in the 2007-2009+ financial crisis and extend the result to general financial crises. Market instability can be monitored by measuring the spectral radius of the Jacobian matrix of the flow of funds dynamical system. Our contribution is to provide an actual way of measuring how close to chaos the market is.

Speaker: Ionut Florescu

Title: Numerical solutions to partial integro-differential equations appearing in financial mathematics

Abstract:

We study the numerical solutions for integro-differential parabolic problems modeling a process with jumps and stochastic volatility. This work continues the previous existence study (Florescu and Mariani, 2010) by implementing a numerical scheme appearing in the proof of the existence result. This new algorithm is compared with another (also new but more traditional) discretization in time and both are applied to calculate numerical solutions. The algorithms are implemented in PDE2D, a general purpose, partial differential equation solver.

Speaker: Yu-Jui Huang

Title: On the multi-dimensional controller-and-stopper games

Abstract:

We consider a zero-sum stochastic differential controller-and-stopper game in which the state process is a controlled jump-diffusion evolving in a multi-dimensional Euclidean space. In this game, the controller affects both the drift and the volatility terms of the state process. Under appropriate conditions, we show that the lower value function of this game is a viscosity solution to an obstacle problem for a Hamilton-Jacobi-Bellman equation, by generalizing the weak dynamic programming principles introduced in Bouchard and Touzi (2010). This is joint work with Erhan Bayraktar, available at http://arxiv.org/abs/1009.0932.

Speaker: Kasper Larsen

Title: Horizon dependence of utility optimizers in incomplete models

Abstract:

This paper studies the utility maximization problem with changing time horizons in the incomplete Brownian setting. We first show that the primal value function and the optimal terminal wealth are continuous with respect to the time horizon $T$. Secondly, we exemplify that the expected utility stemming from applying the T-horizon optimizer on a shorter time horizon S, S < T, may not converge as S ↑ T to the T-horizon value. Finally, we provide necessary and sufficient conditions preventing the existence of this phenomenon.

Speaker: Min Dai

Title: Optimal Trend Following Trading Rules

Abstract:

We are concerned with optimal trend following trading rules in a bull-bear switching market, where the market switching is unobservable. We formulated it as an optimal stopping problem which is described by a system of variational inequalities. The optimal long and liquidating times are given in terms of a sequence of stopping times determined by two threshold curves. Numerical experiments are conducted to validate the theoretical results and demonstrate how they perform in a marketplace.

Speaker: Tim Siu-Tang Leung

Title: Optimal timing to buy options in incomplete markets

Abstract:

We study the timing of derivative purchases in incomplete markets. In our model, an investor attempts to maximize the spread between her model price and the offered market price through optimally timing her purchase. Both the investor and the market value the options by risk-neutral expectations but under different equivalent martingale measures representing different market views. We show that the structure of the resulting optimal stopping problem depends on the interaction between the respective market price of risk and the option payoff. In particular, a crucial role is played by the delayed purchase premium that is related to the stochastic bracket between the market price and the buyer's risk premia. Explicit characterization of the purchase timing is given for two representative classes of Markovian models: (i) defaultable equity models with local intensity; (ii) diffusion stochastic volatility models. Several numerical examples are presented to illustrate the results. Our model is also applicable in the related contexts of hedging long-dated options and quasi-static hedging.

Speaker: Johannes Ruf

Title: Hedging under arbitrage

Abstract:

Explicit formulas for optimal trading strategies in terms of minimal required initial capital are derived to replicate a given terminal wealth in a continuous-time Markovian context. To achieve this goal this talk does not assume the existence of an equivalent local martingale measure. Instead a new measure is constructed under which the dynamics of the stock price processes simplify. It is shown that delta hedging does not depend on the "no free lunch with vanishing risk assumption. However, in the case of arbitrage the problem of finding an optimal strategy is directly linked to the non-uniqueness of the partial differential equation corresponding to the Black-Scholes equation. The recently often discussed phenomenon of "bubbles" is a special case of the setting in this talk.

Speaker: Camelia Pop

Title: Stochastic representations of solutions to degenerate variational equalities and inequalities

Abstract:

We consider the generator of the Heston stochastic volatility process and establish existence and uniqueness results of classical solutions for the elliptic and parabolic PDEs and obstacle problems with Dirichlet conditions. We provide the stochastic representation of the solutions and discuss the relationship between the behavior of the process and the Dirichlet boundary condition needed to obtain a well-posed problem. In addition, using the existence and uniqueness result of weak solutions for variational equalities and inequalities for the Heston generator, recently obtained by P. Feehan and P. Daskalopoulos, we prove stochastic representation results for the elliptic and parabolic variational equality and inequality in this setting. This is joint work with P. Feehan.

Speaker: Johannes Ruf

Title: Hedging under arbitrage

Abstract:

Explicit formulas for optimal trading strategies in terms of minimal required initial capital are derived to replicate a given terminal wealth in a continuous-time Markovian context. To achieve this goal this talk does not assume the existence of an equivalent local martingale measure. Instead a new measure is constructed under which the dynamics of the stock price processes simplify. It is shown that delta hedging does not depend on the ``no free lunch with vanishing risk'' assumption. However, in the case of arbitrage the problem of finding an optimal strategy is directly linked to the non-uniqueness of the partial differential equation corresponding to the Black-Scholes equation. The recently often discussed phenomenon of ``bubbles'' is a special case of the setting in this talk.

Speaker: Hasanjan Sayit

Title: Arbitrage-free models in markets with transaction costs

Abstract:

We study no-arbitrage conditions in a market with multiple risky assets and proportional transaction costs. We present a condition which is sufficient for the market to be arbitrage-free and investigate its properties. In particular, we provide examples of price processes that are not semimartingales but are consistent with absense of arbitrage.

Speaker: Alexander Shklyarevsky

Title: Analytical approaches to the solution of PDEs and PIDEs and their application to pricing and risk-managing derivative securities and their portfolios

Abstract:

We would like to present an overview of analytical approaches to the solution of Partial Differential Equations (PDEs) and Partial Integro-Differential Equations (PIDEs) in light of their application to pricing and risking derivative securities and their portfolios and giving examples of structured and non-structured products. These analytical approaches include using certain Integral Transforms, Non-Integral Transforms, Operator Theory and other Functional Analysis methodologies and are actively being applied by the author in both the Financial Industry and the cutting edge Physics, as well as presented at leading Financial Industry conferences

Speaker: Stephan Sturm

Title: On the implied volatility surface of stochastic volatility models under indifference pricing

Abstract:

The study of the implied volatility surface of stochastic volatility models and in particular it's asymptotics for small time and extreme strikes is a major topic in current research. This is not a purely academic exercise, but has practical relevance as e.g. the implied volatility of far out of the money put options contains information on trader's fear of huge crashes in the stock market. Since stochastic volatility models are in general incomplete, one has also to fix the pricing mechanism employed. In the present talk we will focus on the implied volatility surface under indifference pricing via dynamic convex risk measures given as solutions of quadratic BSDEs. We derive a characterization of the implied volatility in terms of the solution of Cauchy problem of a nonlinear PDE and provide a small time to maturity expansion. This procedure allows to choose convex risk measures in a parametrized class such that the asymptotic volatility smile under indifference pricing can be matched with the market smile.

Speaker: Tao Wu

Title: An equilibrium model with buy and hold investors

Abstract:

This the first study to analyze the effects of buy and hold investors on equilibrium security price dynamics. The empirical literature suggests that many investors follow buy and hold strategies by rarely changing asset allocations due to information costs or other frictions. Similar strategies are documented for institutional investors. A buy and hold investor effectively faces an incomplete market and di?ers in her pricing of risk from a dynamic asset allocator. The equilibrium is solved through the construction of a representative agent with state-dependent utility. The fraction of the stock held by the buy and hold investor emerges as an additional state variable. The equilibrium quantities are obtained by solving a coupled system of PDEs. In contrast to most previous literature, stock return volatility is solved endogenously in this paper. A simple calibration of our model shows that the economy with buy and hold investors can simultaneously produce a low interest rate and a high Sharpe ratio for the stock. In addition, the buy and hold economy can deliver stock return volatility more than twice that in the limited participation economy, because the stock price is more sensitive to dividend shocks in the buy and hold economy. Moreover, the buy and hold economy achieves this while keeping interest rate volatility at reasonably low levels at the same time.

Speaker: Maxim Bichuch

Title: Asymptotic Analysis for Optimal Investment with Transaction Costs in Finite Time

Abstract:

We consider an agent who invests in a stock and a money market account with the goal of maximizing the utility of his investment at the final time T in the presence of a proportional transaction cost. We will consider the case when the utility function is a power function. We provide a heuristic and a rigorous derivation of the asymptotic expansion of the value function in powers of the transaction cost. We also obtain a "nearly optimal" strategy, whose utility asymptotically matches the leading terms in the value function.