### Contributing Speakers

Maxim Bichuch
Worcester Polytechnic Institute
Portfolio optimization under convex incentive schemes
view abstract
Lijun Bo
Xidian University
Bilateral Credit Valuation Adjustment for Large Credit Derivatives Portfolios
view abstract
Agostino Capponi
Johns Hopkins University
Pricing Vulnerable Claims in a Lévy Driven Model
view abstract
Hector Chang
Columbia University
Hölder estimates for fully nonlinear parabolic integro-differential equations
view abstract
Jungmin Choi
East Carolina University
Approximation and application of the Musiela stochastic PDE in forward rate models
view abstract
Giovanni Di Crescenzo
Applied Communication Sciences, New Jersey
On strategy comparisons and sensitivity for a generalized El Farol problem
view abstract
Giancarlo Facchi
Pennsylvania State University
Optimal bidding strategies in a continuum limit order book
view abstract
Arash Fahim
Florida State University
Balancing small fixed and proportional transaction cost in trading strategies
view abstract
Ruoting Gong
Rutgers University, New Brunswick
High-order short-time expansions for ATM option prices under a tempered stable Lévy model
view abstract
Jun Hu
Tampere University of Technology
Series solutions to stochastic volatility models
view abstract
Ahmed Derar Islim
Florida State University, Tallahassee
Pricing exotics with sharp profiles using high resolution finite difference schemes
view abstract
Inna Khagleeva
University of Illinois at Chicago
Understanding jumps in the high-frequency VIX
view abstract
Damir Kinzebulatov
The Fields Institute for Mathematical Sciences, Toronto
Algorithmic Trading with Learning: Informed versus Uniformed
view abstract
Tim Leung
Columbia University
Implied Volatility of Leveraged ETF Options
view abstract
Xin Li
Columbia University
Optimal Timing for Mean Reversion Trading
view abstract
Peter Lin
Johns Hopkins University
A new trinomial recombination tree algorithm and its applications
view abstract
Matt Lorig
Princeton University
Pricing variance swaps on time-changed Markov processes.
view abstract
Oleksii Mostovyi
University of Texas, Austin
An approximation of utility maximization in incomplete markets
view abstract
Michael Oancea
University of Connecticut
Stochastic dominance option pricing under a multivariate diffusion of the underlying returns
view abstract
Michael O. Okelola
University of KwaZulu-Natal, South Africa
Solving a PDE associated with the pricing of power options with time dependent parameters
view abstract
Triet Pham
Rutgers University
Two person zero-sum game under feedback controls and path dependent Bellman-Isaa cs equations
view abstract
Dan Pirjol
JP Morgan, New York
Explosive behavior in log-normal short rate interest rate models
view abstract
Gordon Ritter
Multi-period portfolio choice and Bayesian dynamic models
view abstract
Alexandre Roch
ESG UQAM, Monteal, Canada
Term structure of interest rates with liquidity risk
view abstract
Alexander Shklyarevsky
Bank of America, New York
Mutual benefit of ODE, PDE, PIDE and related analytical approaches developed in and applied to physics and quantitative finance
view abstract
Michael Spector
Numerix
SABR spreads its wings
view abstract
Stephan Sturm
Worcester Polytechnic Institute, Massachusetts
Optimal incentives for delegated portfolio optimization
view abstract
Gu Wang
University of Michigan
Consumption in Incomplete Markets
view abstract
Mingxin Xu
University of North Carolina, Charlotte
Forward Stopping Rule Within HJM Framework
view abstract

#### 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.)

#### Abstract:

Nonlinear expecations are closely related to risk measures. In this talk, we will present the martingale problem under the framework of nonlinear expectations, analogous to that in a probability space in the seminal paper of Stroock and Varadhan (1969). We first establish an appropriate comparison theorem and the existence result for the associated state-dependent fully non-linear parabolic PDEs. We then construct the conditional expectation from the viscosity solution of the PDEs, and solve the martingale problem. As an application, we introduce the notion of weak solution of SDE under the non-linear expectation. This is a joint work with C. Pan and S. G. Peng.

#### 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.

#### 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.

#### Abstract:

The fractional Laplacian operator is the infinitesimal generator of the symmetric 2s-stable Markov process. Variations of such processes, such as normal and generalized tempered stable processes, are widely used in mathematical finance in models of asset prices with discontinuous paths. A problem of great interest, and which is still not well understood, is to establish that prices of American-style options on assets with discontinuous paths are solutions to evolutionary obstacle problems defined by suitable nonlocal operators, and to prove regularity of such solutions and of the free boundary, that is, the set of points where the value function of the American option coincides with its intrinsic value. We develop tools necessary to obtain such results when the infinitesimal generator of the underlying asset price process is the fractional Laplacian with drift with parameter 1/2 < s < 1. We develop a new monotonicity formula, which is used together with perturbation methods in Holder spaces, to obtain the optimal regularity of solutions. We apply this result to study the regularity of the free boundary in a neighborhood of regular points, and to obtain the stochastic representation of the solutions. The latter result allows us to interpret solutions to stationary obstacle problems defined by the fractional Laplacian with drift as prices of perpetual American options when the underlying asset price process is driven by a symmetric stable process with drift. This is joint work with Charles Epstein and Arshak Petrosyan.

#### 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.

#### Abstract:

Martingales play a fundamental role in the understanding of Mathematical Finance, while Local Martingales play a pathological role. In this talk we will illustrate how they arise naturally from stochastic integration, and from conditions involving the absence of arbitrage opportunities, changes of measure and relative arbitrage, pathologies in the Heston model and its extensions, financial bubbles, to the existence of illusory arbitrage opportunities. We will also present a method to construct examples of strict local martingales with jumps.

#### 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.

#### 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.

#### 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

#### 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

#### 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.

#### 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.

#### 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.

#### 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.

#### 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.

#### 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)

#### 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.

#### 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.

#### Abstract:

We prove stochastic representation formulae for solutions to the elliptic boundary value problem associated with a degenerate Markov diffusion process. The degeneracy in the diffusion coefficient is proportional to the a-power of the distance to the boundary of the half-space, where 0<a<1. This generalizes the well-known Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance and a paradigm for a degenerate diffusion process. The generator of this degenerate diffusion process with killing, is a second-order, degenerate-elliptic partial differential operator where the degeneracy in the operator symbol is proportional to the 2a-power of the distance to the boundary of the half-plane. Our stochastic representation formula provide the unique solution to the elliptic boundary value problem, when we seek solutions which are suitably smooth up to the boundary portion contained in the boundary of the half-plane. In the case when the full Dirichlet condition is given, our stochastic representation formula provide the unique solution which are not guaranteed to be any more than continuous up to the boundary portion in the boundary of the half-plane. This is the joint work with Paul Feehan and Jian Song.

#### 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.

#### 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.

#### Abstract:

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

#### 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.

#### Abstract:

We discuss the problem of understanding implied volatilities from options written on leveraged exchanged-traded funds (LETFs), with an emphasis on the relations between options on LETFs with different leverage ratios. First, we examine from empirical data the implied volatility surfaces for LETFs based on the S&P 500 index, and introduce the concept of "moneyness scaling" to enhance their comparison with non-leveraged ETF implied volatilities. Under a multiscale stochastic volatility framework, we apply asymptotic techniques to derive an approximation for both the LETF option price and implied volatility. The approximation formula reflects the role of the leverage ratio, and thus allows us to link implied volatilities of options on an ETF and its leveraged counterparts. Our result is applied to quantify matches and mismatches in the level and slope of the implied volatility skews for various LETF options using data from the underlying ETF option prices. This reveals some apparent biases in the leverage reflected in the different products, long and short with leverage ratios two times and three times.

#### Abstract:

This paper studies the problem of trading under mean-reverting price dynamics subject to transaction cost. We formulate an optimal double stopping problem to analyze the optimal timing to enter and subsequently exit the market, when prices are driven by an Ornstein-Uhlenbeck (OU), exponential OU, or Cox-Ingersoll-Ross (CIR) process. In the OU and CIR cases, the investor's optimal strategy is characterized by a lower level for entry and an upper level for exit. However, in the exponential OU case, we find that it is optimal to delay entry not only when the current price is high, but also when it is sufficiently close to zero. Both analytical and numerical results are provided to illustrate the dependence of timing strategies on model parameters such as mean-reversion level and transaction cost. As extensions, we further impose a stop-loss constraint or a minimum holding period, and examine their effects on the timing of trades.

#### 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.

#### 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.

#### Abstract:

We prove that the variance swap rate is just the price of a co-terminal European-style contract when the underlying is modeled as an exponential Markov process, time-changed by an arbitrary continuous stochastic clock, which has arbitrary correlation with the driving Markov process. The payoff function of the European contract that prices the variance swap satisfies an ordinary integro-differential equation, which depends only on the dynamics of the Markov process, not on the clock. We present examples of Markov processes whose payoff function can be computed explicitly. When the Markov process is a Lévy process, the European contract has a log-style payoff, which recovers the results Carr, Lee, and Wu (2011). This is joint work with Peter Carr and Roger Lee.

#### Abstract:

The implementation of utility-maximization methods for the optimal portfolio choice rely on proper calibration of the model, and, in particular, on the correct estimation of the parameters of the stock-price dynamics. We analyze the effect of a misspecification in the parametric description of the stock-price evolution on the value function of utility-maximization problem for rational economic agent, whose preferences are described by a utility function of the "power" type, with p < 0. In the framework of an incomplete financial market where the stock price is modeled by a continuous semimartingale, we perform an asymptotic analysis of the value function with respect to a small perturbation of the finite-variation part of the price process. We establish a first-order expansion formula and bound the error of our approximation. The implications of our result, such as an approximation of the less tractable models by the more tractable ones, are illustrated by specific examples. The talk is based on the joint work with Gordan Zitkovic.

#### 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.

#### 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.

#### 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.

#### 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.

#### 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.

#### Abstract:

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

#### 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 (Worcester Polytechnic Institute).

#### Abstract:

An agent maximizes isoelastic utility from consumption with infinite horizon in an incomplete market, in which state variables are driven by diffusions. We first provide a general verification theorem, which links the solution of the Hamilton-Jacobi-Bellman equation to the optimal consumption and investment policies. To tackle the intractability of such problems, we propose approximate policies, which admit an upper bound, in closed-form for their utility loss. The approximate policies have closed form solutions in common models, and become optimal if the market is complete, or utility is logarithmic.

#### 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.

#### 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.

#### 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.

#### Abstract:

In recent times, the Lie group approach has been employed in the solution of time dependent PDEs. This method proves successful in providing exact solutions to these PDEs - even in cases where solutions did not previously exist. In this presentation, we look at the particular case of the PDE which models the power option. Using Lie symmetry analysis, we obtain the Lie point symmetries of the power option PDE and demonstrate an algorithmic method for finding solutions to the equation. We not only present results obtained via this approach for the constant parameter scenario but we also employ the approach for the solution of the time dependent parameter case. (Joint with K. S. Govinder and J. G. O'Hara.)

#### Abstract:

We describe a novel approach to the study of multi-period portfolio selection problems with time varying alphas, trading costs, and constraints. We show that, to each multi-period portfolio optimization problem, one may associate a dual'' Bayesian dynamic model. The dual model is constructed so that the most likely sequence of hidden states is the trading path which optimizes expected utility of the portfolio. The existence of such a model has numerous implications, both theoretical and computational. Sophisticated computational tools developed for Bayesian state estimation can be brought to bear on the problem, and the intuitive theoretical structure attained by recasting the problem as a hidden state estimation problem allows for easy generalization to other problems in finance. We discuss optimal hedging for derivative contracts as a special case. (Joint with Petter Kolm.)

#### Abstract:

We introduce the feedback control setting to study two person zero-sum stochastic differential games. In standard literature, the open-loop setting is typically used, which requires the game to be set up under the strategy versus control framework. The main drawback of this approach is the asymmetry of information between the two players. The feedback control allows us to consider the game under the control versus control setting, which preserves the symmetry. Under natural conditions, we show the game value exists. We also allow for non-Markovian structure, and thus the game value is a random process. We characterize the value process as the unique viscosity solution of the corresponding path dependent Bellman-Isaacs equation, a notion recently introduced by Ekren-Keller-Touzi-Zhang and Ekren-Touzi-Zhang. This is joint work with Jianfeng Zhang.

#### Abstract:

We obtain explicit expressions for prices of vulnerable claims written on a stock whose predefault dynamics follows a Lévy-driven SDE. The stock defaults to zero with a hazard rate intensity being a negative power of the stock price. We recover the characteristic function of the terminal log price as the solution of a complex valued infinite dimensional system of first order ordinary differential equations. We provide an explicit eigenfunction expansion representation of the characteristic function in a suitably chosen Banach space, and use it to price defaultable bonds and stock options. We present numerical results to demonstrate the accuracy and efficiency of the method. (Joint with Stefano Pagliarani and Tiziano Vargiolu.)

#### Abstract:

We try to develop an approximate analytic formula for a general stochastic volatility model. Starting with option pricing PDE, we apply parameter expansion to the option price and transform the PDE to a set of inhomogeneous heat equations, whose solution is known exactly. Then option price can be written as an converging sum of infinite analytic functions. Comparison with popular numerical approaches verifies the validity of the analytic solution.

#### Abstract:

We revisit the optimal stopping problem using Heath-Jarrow-Morton (HJM) approach. The HJM method was originally introduced to model the fixed-income market by Heath et al. (1992). More recently, it was implemented in equity market models by Schweizer and Wissel (2008), and Carmona and Nadtochiy (2008,2009). Prior work has mainly focused on European derivative pricing, while in this paper we apply the HJM philosophy to American derivative pricing with a focus toward solving optimal stopping problems in general. As a counterpart to forward rate for the fixed-income market and forward volatility for the equity market, we introduce forward drift for the optimal stopping problem. The standard results for HJM-type models are confirmed for the forward drift dynamics: the drift condition and the spot consistency condition. More interestingly, we discover a forward stopping rule that is fundamentally different from the classical stopping rule based on backward induction. We illustrate this difference in two benchmark models: a binomial example for American option pricing and a Black-Sholes example for the optimal time to sell a stock. In addition to the minimal optimal stopping time, we characterize the maximal optimal stopping time in the forward approach. (Joint with Wenhua Zou.)

#### Abstract:

We will revisit Hölder estimates for some non local problems we worked recently with G. Dvila. They arise in stochastic optimal control driven by purely jump processes. Each one of them can be considered as an extension of the Krylov-Safanov regularity theory for fully non linear second order equations. We will start by motivating these equations, then we will see how the proofs work for simpler models and finally discuss how those ideas can be adapted to the nonlocal setting.

#### Abstract:

A dynamic spending and investment model allows for spending shortfall aversion through a utility function that entails scaling of spending by a fractional power of past peak spending. The past peak spending is the current reference or target spending. Under the closed form solution the wealth to target ratio follows a diffusion process. At the lowest levels of the ratio, up to a point, the spending rate and weight of the risky asset are fixed fractions of wealth, as prescribed by Merton. Beyond that point, at the higher levels of wealth to target ratio, spending is constant and the weight of the risky asset increases with wealth. Wealth to spending ratio has an upper bound at which increases in the spending rate (and the target) offset wealth increases. (Joint work with Gur Huberman and Dan Ren)

#### Abstract:

We obtain an explicit formula for the bilateral counterparty valuation adjustment of a credit default swaps portfolio referencing an asymptotically large number of entities. We perform the analysis under a doubly stochastic intensity framework, allowing for default correlation through a common jump process. The key insight behind our approach is an explicit characterization of the portfolio exposure as the weak limit of measure-valued processes associated to survival indicators of portfolio names. We validate our theoretical predictions by means of a numerical analysis, showing that counterparty adjustments are highly sensitive to portfolio credit risk volatility as well as to default correlation. To appear in Finance and Stochastics. (Joint with Agostino Capponi.)

#### Abstract:

The stochastic alpha beta rho (SABR) model introduced in Hagan, Lesniewski & Woodward (2001) and Hagan et al (2002) is widely used by practitioners to capture the volatility skew and smile effects of interest rate options. Traditional methods for the stochastic alpha beta rho model tend to focus on expansion approximations that are inaccurate in the long maturity ‘wings’. However, if the Brownian motions driving the forward and its volatility are uncorrelated, option prices are analytically tractable. In the correlated case, model parameters can be mapped to a mimicking uncorrelated model for accurate option pricing. (Joint with Alexander Antonov and Michael Konikov.)