### Conference Program

All presentations take place in the Neilson Room at the

Heldrich Hotel, 10 Livingston Avenue, New Brunswick, New Jersey 08901.

8:00 Breakfast and Registration (Neilson Foyer)

*Chair: Paul Feehan*

8:50

**Paul Feehan** and

**Dan Ocone** (Rutgers, The State University of New Jersey)

Welcome remarks

9:00

**Erhan Bayraktar** (University of Michigan)

Valuation equations for stochastic volatility models

view abstract
9:40

**Panagiota Daskalopoulos ** (Columbia University)

Free-boundary problems and degenerate diffusion

view abstract
10:20

**Andrea Pascucci** (Università di Bologna)

Kolmogorov equations and asian options

view abstract
11:00-11:20 Coffee Break (Kelly Kiosk)

11:20

**Douglas Borden** (Knight Capital Group)

Stochastic control in automated market-making

view abstract
12:00

**Igor Halperin** (JP Morgan)

Option pricing and hedging by risk minimization

view abstract
12:40-1:50 Buffet lunch at Hotel restaurant (Heldrich Hotel)

*Chair: Daniel Ocone*

1:50

**Emmanuel Gobet** (École Polytechnique, Paris)

Expansion formulas for pricing

view abstract
2:30

**Kumar Muthuraman** (McComb School of Business, University of Texas at Austin)

Moving boundary methods for solving free-boundary problems

view abstract
3:10

**Jianfeng Zhang** (University of Southern California)

Second order backward stochastic differential equations

view abstract
3:50-4:10 Coffee Break (Kelly Kiosk)

*Chair: Jian Song*

*Neilson Room*: American Options and Stochastic Control

*Janeway Room*: Implied Volatility and Hedging

*Meyer Room*: Numerical Solution of Partial Integro-Differential
Equations

*Bishop Room*: Incomplete Markets

*Waldron Room*: Transaction Costs and Optimal Investment

5:00

**Peter Bank** (Technische Universität Berlin)

Convex duality and intertemporal consumption choice

view abstract
5:40

**Thaleia Zariphopoulou** (University of Oxford and University of Texas at Austin)

An Approximation Scheme for Investment Performance Processes in Incomplete Markets

view abstract
6:20

**Johan Tysk** (Uppsala Universitet, Sweden)

Boundary behavior of densities for non-negative diffusions

view abstract
7:00-8:00pm Cocktail Reception (Kelly Room)

**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.