### Conference Program

All presentations take place in the Neilson Room at the

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

8:50

**Paul Feehan** and

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

Welcome remarks

9:00

**Robert Almgren** (Quantitative Brokers and New York University)

Option Hedging with Market Impact

view abstract
9:40

**Xin Guo ** (University of California, Berkeley)

Optimal execution: with and without Limit Order Books

view abstract
*Waksman Room*: Interest rate derivative modeling.

*Dickson Room*: Option pricing I

*Ellis Room*: Financial engineering

*Kilmer Room*: Transaction costs and optimal investment

*Neilson Room*: Asymptotic expansions and implied volatility surfaces

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

11:20

**David Hobson** (Warwick University)

Optimal timing for the sale of an indivisible asset in an incomplete market

view abstract
12:00

**Martin Keller-Ressel** (Technische Universität Berlin)

Convex order properties of discrete and continuous realized variance & applications to options on variance

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

*Chair: Daniel Ocone*

1:50

**Sergei Levendorskiĭ** (University of Leicester)

Efficient Laplace and Fourier inversions and Wiener-Hopf factorization in financial applications

view abstract
2:30

**Victor Nistor** (Pennsylvania State University)

Short time asymptotic of option prices in time-dependent
stochastic volatility models

view abstract
*Waksman Room*: Computational finance

*Dickson Room*: Option pricing II

*Ellis Room*: PDEs and SPDEs in finance

*Kilmer Room*: Game theory and optimal trading strategies

*Neilson Room*: Portfolio optimization

4:10-4:30 Coffee Break (Kelly Kiosk)

*Chair: Jian Song*

4:30

**Camelia Pop** (University of Pennsylvania)

Degenerate-parabolic partial differential equations with unbounded coefficients, martingale problems, and a mimicking theorem for Itô processes

view abstract
5:10

** Philip Protter** (Columbia University)

Expansion of filtrations via stochastic processes, and insider trading

view abstract
5:50

**Gordan Žitković** (University of Texas at Austin)

Facelifting in mathematical finance

view abstract
6:30-7:30pm Cocktail Reception (Neilson Room)

**Speaker**: Robert Almgren

**Title**: Option Hedging with Market Impact

#### Abstract:

We present a delta-hedging result for a large investor whose trades generate market impact proportional to the rate of trading. This impact model is more realistic than a fixed bid-offer spread. In a regime of small impact, the solution is to trade towards the classic Black-Scholes hedge portolio, at a rate that is proportional to the degree of mishedge and to liquidity. We discuss a number of implications of the model including intraday volume patterns and stock pinning. We also consider the extension to the "broker problem" where trading is allowed only in one direction. (Joint work with Tianhui Li of Princeton University.)

**Speaker**: Xin Guo

**Title**: Optimal execution: with and without Limit Order Books

#### Abstract:

There is a growing body of research works
on trading strategies for big orders over a period of time with various
assumptions of price impact. These works mostly focus on a macroscopic
timescale. On the millisecond timescale the price is no longer well
defined and the state of the order book contains important information.
More importantly, one of the key issues at this timescale is the order
placement problem, which is different from the optimal execution one. We will discuss associated optimization problems and the mathematical
difficulties in analyzing these problems.

**Speaker**: David Hobson

**Title**: Optimal timing for the sale of an indivisible asset in an incomplete market

#### Abstract:

Consider an agent with a single unit of an indivisible asset to sell, the price of which fluctuates over time. The aim of the agent is to maximise utility of consumption over time. In addition to the indivisible asset the agent has outside wealth and she is free to invest this wealth on a financial market. When should the agent sell the indivisible asset? What should her investment and consumption strategies be, both before and after she sells the asset? We set up the problem as a stochastic control problem. The solution has some natural and expected features, but there are also some suprising consequences. Joint work with Vicky Henderson.

**Speaker**: Sergei Levendorskii

**Title**:Efficient Laplace and Fourier inversions and Wiener-Hopf factorization in financial applications

#### Abstract:

In the talk, it will be explained how a family of conformal changes of variables in integrals in formulas for prices, sensitivities and probability distribution functions can greatly increase accuracy and speed of calculations of oscillatory integrals in formulas for prices and sensitivities of options, calculation of probability distributions, and Monte Carlo simulations (gain in speed 10-100 times). The techniques is applicable to Lévy models, Heston model with jumps, affine term structure models, models with subordination, regime-switching models, European, American, barrier options with discrete and continuous monitoring, lookbacks, CDS, CDS and exotic options with CVA, Asians.

**Speaker**: Camelia Pop

**Title**: Degenerate-parabolic partial differential equations with unbounded coefficients, martingale problems, and a mimicking theorem for Itō processes

#### Abstract:

We solve four intertwined problems, motivated by mathematical finance, concerning degenerate-parabolic partial differential operators and degenerate diffusion processes. First, we consider a parabolic partial differential equation on a half-space whose coefficients are suitably Holder continuous and allowed to grow linearly in the spatial variable and which becomes degenerate along the boundary of the half-space. We establish existence and uniqueness of solutions in weighted Holder spaces which incorporate both the degeneracy at the boundary and the unboundedness of the coefficients. Second, we show that the martingale problem associated with a degenerate elliptic differential operator with unbounded, locally Holder continuous coefficients on a half-space is well-posed in the sense of Stroock and Varadhan. Third, we prove existence, uniqueness, and the strong Markov property for weak solutions to a stochastic differential equation with degenerate diffusion and unbounded coefficients with suitable "older continuity properties. Fourth, for an Itō process with degenerate diffusion and unbounded but appropriately regular coefficients, we prove existence of a strong Markov process, unique in the sense of probability law, whose one-dimensional marginal probability distributions match those of the given Itō process. Joint work with Paul Feehan, Rutgers University.

**Speaker**: Martin Keller-Ressel

**Title**: Convex order properties of discrete and continuous realized variance & applications to options on variance

#### Abstract:

We consider a square-integrable semimartingale with conditionally independent increments and symmetric jump measure, and show that its discrete realized variance dominates its quadratic variation in increasing convex order. The result has immediate implications to the pricing of options on realized variance. For a class of models including time-changed Lévy models and Sato processes with symmetric jumps our results show that options on variance are typically underpriced, if quadratic variation is substituted for the discretely sampled realized variance. This talk is based on joint work with Claus Griessler.

**Speaker**: Philip Protter

**Title**: Expansion of filtrations via stochastic processes, and insider trading

#### Abstract:

Recently there has been a sequence of scandals involving insider trading, and insider knowledge, the most spectacular of which are the Galleon Group in New York, and the LIBOR scandal in London. There is a long tradition of modeling insider trading in finance using the expansion of filtrations, either initially, or progressively by making a random time into a stopping time. We alternatively propose expanding a filtration dynamically with a stochastic process, and develop a (rather complicated) technique for doing so. However we see this as a beginning, and this new technique requires refinement and improvement. Nevertheless we can apply it to insider trading examples, and it is appropriate for continuing activities such as the Galleon Group and LIBOR (as opposed to one time activities such as the Martha Stewart example). We show how it can change the risk neutral measure of an insider, and how in some cases it can introduce arbitrage opportunities. The talk is based on ongoing joint work with Younes Kchia of the ANZ Bank.

**Speaker**: Victor Nistor

**Title**: Short time asymptotic of option prices in time-dependent
stochastic volatility models

#### Abstract:

We study the short time asymptotic of option prices by
considering the short time solutions of degenerate parabolic equations
with time-dependent coefficients. The method relies on a general
computation using perturbative (Dyson series) expansions similar to
those used in Theoretical Physics. We are able to approximate
efficiently the perturbative integrals by using commutator tricks
relying on Campbell-Hausdorff-Backer formula. We apply our result to
time-dependent stochastic volatility models. This is joint work with
Wen Cheng, Nicolas Costanzino, John Liechty, Anna Mazzucato, and Xiao
Han.

**Speaker**: Gordan Zitkovic

**Title**: Facelifting in mathematical finance

#### Abstract:

Superreplication of contingent claims in incomplete markets often involves a 'facelift': the payoff function is replaced by an envelope in an appropriate class and the new, facelifted, payoff is priced using complete-market methods. More generally, in optimal stochastic control theory, the facelift appears in the form of a discontinuity of the value function at the terminal time and typically arises when the control set is unbounded. In that case, the Hamiltonian may take infinite values and the facelift typically consist of replacing the terminal payoff by a smallest finite-Hamiltonian majorant.
When utility-based pricing is used instead of superreplication, no facelift is expected, thanks to the smoothing effect of the utility function. Indeed, the Hamiltonian applied to the terminal payoff is always finite. Yet, there is a facelift as soon as the claim being priced is non-replicable. We show this unexpected fact using control-theoretic methods and relate it to the appearance of finitely-additive dual minimizers in the problem of utility maximization with a random endowment. We also analyze the corresponding Hamilton-Jacobi-Bellman equation and provide necessary analytic conditions on its terminal condition for the absence of a facelift. This is joint work with Kasper Larsen.

**Speaker**: Rafe Mazzeo

**Title**:
Wright-Fisher diffusions and related degenerate operators

#### Abstract:

I will discuss recent and ongoing work with C. L. Epstein regarding the fine regularity properties of a class of degenerate elliptic and parabolic operators on domains or manifolds with boundaries and corners. Although these problems came to our attention from population genetics, they are similar to certain equations of interest in finance, including the Heston equation. I will describe our setting and the methods we use as well as some other potential methods of interest

**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**: Amel Bentata

**Title**: Short time asymptotics for semimartingales and an application for short maturity index options in a multivariate jump-diffusion model

#### Abstract:

We extend and unify the short-time asymptotics of the marginal laws of a stochastic process to the more general case when ξ is a d-dimensional discontinuous semimartingale with jumps. We compute the leading term in the asymptotics in terms of the local characteristics of the semimartingale. In contrast to previous derivations, our approach is purely based on Ito calculus, and makes no use of the Markov property or independence of increments. We derive in particular the asymptotic behavior of call options with short maturity in a semimartingale model: whereas the behavior of out-of-the-money options is found to be linear in time, the short time asymptotics of at-the-money options is shown to depend on the fine structure of the semimartingale. Our multidimensional setting allows to treat examples which are not accessible using previous results (e.g the index process). We propose an analytical approximation for short maturity index options, generalizing the approach by Avellaneda & al 03 to the multivariate jump-diffusion case.

**Speaker**: Maxim Bichuch

**Title**: Portfolio optimization under convex incentive schemes

#### Abstract:

We consider a utility maximization problem of terminal wealth from the point of view of a portfolio manager paid by convex incentives. Even though the manager's utility function is concave, the result is a non-concave optimization problem that does not fit into the classical portfolio optimization theory. Using duality theory, we prove existence and uniqueness of the optimal wealth in general (incomplete) markets. This is a joint work with Stephan Sturm.

**Speaker**: Ovidiu Calin

**Title**: Transience of Brownian motion with constraints

#### Abstract:

The transience and recurrence properties of Brownian motion have been extensively studied on Riemanian manifolds. However, these type of problems are still open in the case of sub-Riemannian manifolds. In this case the diffusion is degenerate and moves along a non-integrable distribution, which is defined by some non-holonomic constraints. We shall discuss the transience of Brownian motion that is constraint to move along the Heisenberg, Grushin, and Martinet distributions.

**Speaker**: Jungmin Choi

**Title**: Approximation and application of the Musiela stochastic PDE in forward rate models

#### Abstract:

We consider Musiela equation of the forward rates, which is a hyper-bolic stochastic differential equation. A weak formulation of the problem using the SUPG (Streamline Upwind Petrov Galerkin) method is analyzed. Error analysis of the method yields estimates for the convergence rates. Computational examples are provided that illustrate not only the discretization methods used, but the type of results relevant to bond pricing that can be obtained from the equation.

**Speaker**: Giovanni Di Crescenzo

**Title**: On strategy comparisons and sensitivity for a generalized El Farol problem

#### Abstract:

The El Farol bar problem is a game theory problem proposed in 1994 by B. Arthur as an example of a framework to investigate bounded rationality modeling in economics. In its most basic formulation, this problem has no optimal pure strategy (in that the existence of such a strategy would bring to a contradiction), has a unique and symmetric Nash equilibrium mixed strategy, but other theoretical advances are considered very hard in the literature. We study a variant of this problem, with a larger range for the capacity parameter, and where, most importantly, players are allowed to study the previous history of all game executions before making their next choice. We report results from extensive simulation experiments that compare the success of different classes of player strategies. In some cases, our simulation experiments provide concrete estimates that confirm our calculus and probability-based modeling and analysis. Our simulation and analysis results build on strategies presented in a graduate class project, where our strategy resulted the top winning one and was awarded a project of the year award.

**Speaker**: Arash Fahim

**Title**: Balancing small fixed and proportional transaction cost in trading strategies

#### Abstract:

In financial markets, transaction cost appears in two forms, i.e. proportional to the amount of transaction and a fixed flat rate cost. In previous studies, there are several results about small proportional transaction cost. In the present work, we heuristically study the affect of both types of transaction cost by focusing on a portfolio optimization problem, however, a similar framework can be applied to other problems, such as option pricing and investment-consumption problem. Here we assume the presence of fixed transaction cost and that there is a balance between fixed and proportional transaction cost, such that none of them dominates the other, asymptotically. We find out that the deviation of value function, when the fixed transaction cost is $\epsilon$, from the Merton value function, without transaction cost, is of order $\epsilon^\frac{1}{2}$ which is different from the pure proportional cost of $\epsilon^\frac{2}{3}$. Based on this, we propose an expansion for the value function in terms of powers of $\epsilon^\frac{1}{2}$. Joint work with Jose V. Alcala (University of Michigan)

**Speaker**: Pierre Garreau

**Title**: A consistent spectral element framework for option pricing under general Lévy processes

#### Abstract:

We derive a consistent spectral element framework to compute the price of vanilla derivatives when the dynamic of the underlying follows a general Lévy process. The representation of the solution with Legendre polynomials allows to naturally approximate the convolution integral with high order quadratures. We use a third order implicit/explicit approximation to integrate in time. The method is spectrally accurate in space for the solution and the greeks, and third order accurate in time. The spectral element framework does not require the approximation of the Lévy measure nor the lower truncation of the convolution integral as commonly seen in Finite Difference schemes.

**Speaker**: Giancarlo Facchi

**Title**: Optimal bidding strategies in a continuum limit order book

#### Abstract:

Aim of this paper is to study a continuum model of the limit order book, viewed as a noncooperative game for n players. An external buyer asks for a random amount X>0 of a certain asset. This external agent will buy the amount X at the lowest available price, as long as this price does not exceed a given upper bound P. One or more sellers offer various quantities of the same asset at different prices, competing to fulfill the incoming order. Having observed the prices asked by his competitors, each seller must determine an optimal strategy, maximizing his expected payoff. Of course, when other sellers are present, asking a higher price for the asset reduces the probability of selling it. In our model we assume that the i-th seller owns an amount $\kappa_i$ of stocks. He can put all of it on sale at a given price, or offer different portions at different prices. In general, his strategy will thus be described by a measure $\mu_i$ on $[0, P]$. Here $\mu_i([0, p])$ denotes the total amount of stock put on sale by the $i$-th player at a price $\leq p$. We analyze in detail two different scenarios. If $\left(\ln \mathbb P[X>s]\right)^{\prime\prime} \geq 0$, then the Nash equilibrium exists and can be explicitly determined. We show that the all the optimal strategies (except at most one) consist of measures which are absolutely continuous with respect to the Lebesgue measure. If $\left(\ln \mathbb P[X>s]\right)^{\prime\prime} < 0$, a Nash equilibrium does not exist, and the competition between sellers does not settle near any equilibrium state.

**Speaker**: Ruoting Gong

**Title**: High-order short-time expansions for ATM option prices under
a tempered stable Lévy model

#### Abstract:

The short-time asymptotic behavior of option prices for a variety of models with
jumps has received much attention in recent years. In the present talk, a novel
second-order approximation for ATM option prices under an exponential tempered
stable model, a rich class of Lévy processes with desirable features for financial modeling, is derived. This result is then extended to a model with an additional independent Brownian component. Our method of proof is based on an integral representation of the option price involving the tail probability of the log-return process under the share measure and a suitable change of probability measure under which the process becomes stable. Out approach is sufficiently general to cover a wide class of Lévy processes which satisfy the latter property and whose Lévy densities can be “closely” approximated by a stable density near the origin. The asymptotic behavior of the corresponding Black-Scholes implied volatilities is also addressed. Our numerical results show that first-order term typically exhibits rather poor performance and that the second-order term significantly improves the approximation’s accuracy.

**Speaker**: Yu Gu

**Title**: Local versus non-local forward equations for option prices

#### Abstract:

When the underlying asset is a continuous martingale, call option prices solve the Dupire equation, a forward parabolic PDE in the maturity and strike variables. By contrast, when the underlying asset is described by a discontinuous semimartingale, call price solve a partial integro-diﬀerential equation (PIDE), containing a nonlocal integral term. We show that the two classes of equations share no common solution: a given set of option prices is either generated from a continuous martingale ("diﬀusion") model or from a model with jumps, but not both. In particular, our result shows that Dupire's inversion formula for reconstructing local volatility from option prices does not apply to option prices generated from models with jumps.

**Speaker**: Yu-Jui Huang

**Title**: Robust maximization of asymptotic growth under covariance uncertainty

#### Abstract:

This paper resolves a question proposed in Kardaras and Robertson (2012): how to invest in a robust growth-optimal way in a market where precise knowledge of the covariance structure of the underlying assets is unavailable. Among an appropriate class of admissible covariance structures, we characterize the optimal trading strategy in terms of a generalized version of the principal eigenvalue of a fully nonlinear elliptic operator and its associated eigenfunction, by slightly restricting the collection of non-dominated probability measures.

**Speaker**: Ahmed Derar Islim

**Title**: Pricing exotics with sharp profiles using high resolution finite difference schemes

#### Abstract:

We price exotic options with sharp profiles and calculate the Greeks using high resolution finite difference schemes. These approximations detect the discontinuous profiles automatically using non-linear limiter functions and then add just enough volatility locally to smooth discontinuities and produce non-oscillatory prices and Greeks.

**Speaker**: Andrea Karlova

**Title**: Volatility surfaces generated by stable distributions

#### Abstract:

In the talk we present approximate formulas for volatility smiles generated by stable laws. We discuss its properties and quality of numerical implementation.

**Speaker**: Inna Khagleeva

**Title**: Understanding jumps in the high-frequency VIX

#### Abstract:

In this article, a comprehensive nonparametric study of jumps in the VIX is conducted by examining high-frequency data on the VIX and the S&P 500 futures from 1992 to 2010. It is found that jumps in the VIX occur 18.5 times more often than jumps in the S&P 500. Further, it is shown that the behavior of jumps in the VIX, not simultaneous to jumps in the S&P 500, is so unusual that most of them are likely to be pseudo-jumps. Specifically, they occur too frequently, rarely correspond to any economic event, and do not contribute very much to the leverage effect. Importantly, the frequency of these pseudo-jumps has a monotonically decreasing time trend that is independent of market conditions but consistent with the overall improvement of the quality of the option data. The results of this study have important implications for other studies based on the VIX data, because the pseudo-jumps might considerably distort the inference about volatility dynamics. For example, the jumpiness of volatility might be overstated and the leverage effect might be understated.

**Speaker**: Tim Leung

**Title**: Pricing American options with event risk under Lévy models

#### Abstract:

We study the valuation of American options with event risk. These contracts can potentially be terminated exogenously before the holder's voluntary exercise. Practical examples of these contracts include (i) defautable American options, and (ii) early-exercisable employee stock options with job termination risk. Working with exponential Lévy underlying price dynamics, we analyze the underlying free-boundary problems, and present a number of numerical approaches, such as finite-differnece and Fourier transform methods, to compute the option value, optimal exercise boundary, as well as the associated Greeks. Analytically and numerically, we find that under higher event risk it is optimal for the holder to voluntarily accelerate exercise.

**Speaker**: Xin Li

**Title**: An Optimal Timing Approach to Mean-Reversion Trading

#### Abstract:

This paper studies the optimal entry and exit timing for
trading with mean reversion in price dynamics. This leads to the
analysis of a optimal double stopping problem under time-homogeneous
diffusions, including Ornstein-Uhlenbeck (OU), exponential OU, and
Cox-Ingersoll-Ross price processes. We rigorously derive the optimal
price levels for entry and exit respectively, and analyze their
dependence on various model parameters such as mean-reversion level. As extensions, we further impose a stop-loss constraint or a minimal
holding period, and examine their impact on the optimal trading
strategies. Numerical results are provided to illustrate the optimal
strategies.

**Speaker**: Peter Lin

**Title**: A new trinomial recombination tree algorithm and its applications

#### Abstract:

When pricing a derivative with no closed-form formula, simulation is a major tool in valuation. Nonetheless, Monte Carlo simulation is inefficient for those derivatives with early exercising times or path-dependent payoffs. In this work, we ask for a trinomial recombination tree to reduce simulation dimensionality for general Ornstein-Uhlenbeck process, which is very widely used in interest-rate modeling and derivatives pricing. This algorithm consequently overcomes the challenges in two major interest-rate tree methodologies: Hull-White algorithm cannot guarantee a recombination-tree structure when volatility is decreasing over time; Black-Derman-Toy algorithm only works on a specific Ornstein-Uhlenbeck dynamics. An application in pricing AA rated callable corporate bonds is given.

**Speaker**: Ruihua Liu

**Title**: Optimal investment and consumption with proportional transaction costs in regime-switching model

#### Abstract:

This presentation is concerned with an infinite-horizon problem of optimal investment and consumption with proportional transaction costs in continuous-time regime-switching models. An investor distributes his wealth between a risky asset (a stock) and a risk-less asset (a bond) and consumes at a non-negative rate from the bond account. The market parameters (the interest rate, the appreciation rate and the volatility rate of the stock) are assumed to depend on a continuous-time Markov chain with finite number of states (also known as regimes). The objective of the optimization problem is to maximize the expected discounted total utility of consumption. For this optimal control problem, the Hamilton-Jacobi-Bellman (HJB) equation is given by a system of m coupled variational equalities where m is the total number of regimes. For a class of HARA (hyperbolic absolute risk aversion) type utility functions, we establish some fundamental properties of the value function and show that the value function is a viscosity solution of the HJB equation. We then treat a power utility function and derive qualitative properties of the optimal trading strategy and the value function.

**Speaker**: Matt Lorig

**Title**: Exact implied volatility expansions

#### Abstract:

We derive an exact implied volatility expansion for any model whose European call price can be expanded analytically around a Black-Scholes call price. Two examples of our framework are provided (i) exponential Lévy models and (ii) CEV-like models with local stochastic volatility and local stochastic jump-intensity.

**Speaker**: Oleksii Mostovyi

**Title**: Optimal investment with intermediate consumption and random endowment

#### Abstract:

"We consider a problem of optimal investment with intermediate consumption and random endowment in an incomplete semimartingale model of a financial market. We establish the key assertions of the utility maximization theory assuming that both primal and dual value functions are finite in the interiors of their domains as well as that random endowment at maturity can be dominated by the terminal value of a self-financing wealth process. In order to facilitate verification of these conditions, we present
alternative, but equivalent conditions, under which the conclusions of the theory hold."

**Speaker**: Michael Oancea

**Title**: Stochastic dominance option pricing under a multivariate diffusion of the underlying returns

#### Abstract:

We present a new method of pricing plain vanilla call and put options when the underlying asset returns follow a stochastic volatility process. The method is based on stochastic dominance insofar as it does not need any assumption on the utility function of a representative investor apart from risk aversion. This approachdevelops discrete time multiperiod reservation write and reservation purchase bounds on option prices. The bounds are evaluated recursively and the limiting forms of the bounds are found as time becomes continuous. We discuss the implications of this result on the pricing of volatility risk. Joint work with S. Perrakis.

**Speaker**: Dan Pirjol

**Title**: Hogan-Weintraub singularity and phase transition in the Black, Derman, Toy model

#### Abstract:

It is well-known that short-rate interest rate models with log-normally distributed rates in continuous time are afflicted with divergences which result from infinite accumulation factors in a finite time (the Hogan-Weintraub singularity). Examples of such models are the Dothan model and the Black-Karasinski model. We show explicitly the appearance of this singularity in the Black, Derman, Toy model, which is the discrete time version of the Dothan model, in the limit of a very small time step. A novel singular behavior is shown to appear in the BDT model at large volatility, which is similar to a phase transition in condensed matter physics.

**Speaker**: Alexandre Roch

**Title**: Term structure of interest rates with liquidity risk

#### Abstract:

We develop an arbitrage pricing theory for liquidity risk and price impacts on fixed income markets. We define a liquidity term-structure of interest rates by hypothesizing that liquidity costs arise from the quantity impact of trading of bonds with different maturities on the interest rates and the associated risk-return premia. We derive no arbitrage conditions which gives a number of theoretical relation satisfied between the impact on risk premia and the volatility structure of the term structure and prices. We calculate the quantity impact of trading a zero-coupon on prices of zero-coupons of other maturities and represent this quantity as a supermartingale. We give conditions under which the market is complete, and show that the replication cost of an interest rate derivative is the solution of a quadratic backward stochastic differential equation. Joint work with Robert Jarrow.

**Speaker**: Hasanjan Sayit

**Title**: Absence of arbitrage in a general framework

#### Abstract:

Cheridito (Finance Stoch. 7: 533-553, 2003) studies a financial market that consists of a money market account and a risky asset driven by a fractional Brownian motion (fBm). It is shown that arbitrage pos-sibilities in such markets can be excluded by suitably restricting the class of allowable trading strategies. In this note, we show an analogous result in a multi-asset market where the discounted risky asset prices follow more general non-semimartingale models. In our framework, investors are allowed to trade between a risk-free asset and multiple risky assets by following simple trading strategies that require a minimal deterministic waiting time between any two trading dates. We present a condition on the discounted risky asset prices that guarantee absence of arbitrage in this setting. We give examples that satisfy our condition and study its invariance under certain transformations.

**Speaker**: Alexander Shklyarevsky

**Title**: Mutual benefit of ODE, PDE, PIDE and related analytical approaches developed in and applied to physics and quantitative finance

#### Abstract:

We present a comprehensive methodology and approach to tackle ordinary differential equations (ODE), partial differential equations (PDE), partial integro-differential equations (PIDE) and related topics analytically. These approaches are used in both physics and quantitative finance with mutual benefit, both theoretically and practically. In our presentation, we will show that these analytical methodologies are making both research in physics and research and its implementation in quantitative finance much more efficient and are critical to substantial advances in physics and quantitative finance, as well as assure a trading and risk optimization success across asset classes.

**Speaker**: Jian Song

**Title**: Feynman-Kac Formula for SPDE driven by fractional Brownian motion.

#### Abstract:

In the talk, I will introduce the Feynman-Kac Formula for SPDE driven by fractional Brownian motion, and show its long term behavior.

**Speaker**: Stephan Sturm

**Title**: Optimal incentives for delegated portfolio optimization

#### Abstract:

We study the problem of an investor who hires a fund manager to manage his wealth. The latter is paid by an incentive scheme based on the performance of the fund. Manager and investor have different risk aversions; the manager may invest in a financial market to form a portfolio optimal for his expected utility whereas the investor is free to choose the incentives -- taking only into account that the manager is paid enough to accept the managing contract. We discuss the problem of existence of optimal incentives in general semimartingale models and give an assertive answer for some classes of incentive schemes. This is joint work with Maxim Bichuch (Princeton University).

**Speaker**: Gu Wang

**Title**: High-water marks and private investments

#### Abstract:

A hedge fund manager, who receives performance fees proportional to the fund’s profits, invests optimally for both the fund and his own account, as to maximize the expected power utility of personal wealth. If separate and constant investment opportunities are available for each account, it is optimal for the manager to hold a constant fraction of the fund in risky assets, which corresponds to an effective risk aversion between one and the manager’s own risk aversion. For the personal account, the optimal policy is to accumulate performance fees in safe assets, and invest remaining wealth in a constant portfolio corresponding to the manager’s risk aversion. Under the optimal policy, the manager’s welfare is the maximum between the welfare he would obtain from either keeping fees in safe assets only, or investing his personal wealth alone. This result is robust to correlation between investment opportunities in both accounts, suggesting that the manager does not tend to hedge exposure to the funds’ performance with personal investments.

**Speaker**: Tao Wu

**Title**: Pricing and hedging the smile with SABR: Evidence from the interest rate caps market

#### Abstract:

This is the first comprehensive study of the SABR (Stochastic Alpha-Beta-Rho) model (Hagan et. al (2002)) on the pricing and hedging of interest rate caps. We implement several versions of the SABR interest rate model and analyze their respective pricing and hedging performance using two years of daily data with seven different strikes and ten different tenors on each trading day. In-sample and out-of-sample tests show that the fully stochastic version of the SABR model exhibits excellent pricing accuracy and more importantly, captures the dynamics of the volatility smile over time very well. This is further demonstrated through examining delta hedging performance based on the SABR model. Our hedging result indicates that the SABR model produces accurate hedge ratios that outperform those implied by the Black model.

**Speaker**: Tianyao Yue

**Title**: Pricing binary options and their sensitivities under CEV using spectral element method

#### Abstract:

Because binary option has its unique property of discontinuous payoff at maturity, classical finite difference method (FDM) produces oscillation in the numerical solutions especially for the Greeks. Spectral element method (SEM) is introduced to solve the partial differential equation (PDE) of the option to achieve high convergence rate and avoid such oscillation phenomenon around discontinuous points. A European binary option under constant elasticity of variance (CEV) is studied and computed with this approach. The numerical results of the price and Greeks show the spectral element method is an efficient alternative method for exotic options with discontinuous payoffs.

**Speaker**: Wenhua Zou

**Title**: A unified treatment of derivative pricing and forward
decision problems within HJM framework

#### Abstract:

We study the HJM approach which was originally introduced in the fixed income market by David Heath, Robert Jarrow and Andrew Morton and later was implemented in the case of European option market by Martin Schweizer, Johannes Wissel, Rene Carmona and Sergey Nadtochiy. The main contribution of this thesis is to apply HJM philosophy to the American option market. We derive the absence of arbitrage by a drift condition and compatibility between long and short rate by a spot consistency condition. In addition, we introduce a forward stopping rule which is significantly different from the classical stopping rule which requires backward induction. When It\^{o} stochastic differential equation are used to model the dynamics of underlying asset, we discover that the drift part instead of the volatility part will determine the value function and stopping rule. As counterpart to the forward rate for the fixed income market and implied forward volatility and local volatility for the European option market, we introduce the forward drift for the American option market.