Geometric Analysis Conference

Speaker: Ben Weinkove

Title: The complex Monge-Ampere equation I

Abstract:

I will give an introduction to Yau's Theorem on the existence of solutions to the complex Monge-Ampere equation on compact Kahler manifolds. I will also describe some more recent developments on this equation. This will be an introductory and expository mini-course, aimed at graduate students and recent PhDs.

Speaker: Ben Weinkove

Title: The complex Monge-Ampere equation II

Abstract:

I will give an introduction to Yau's Theorem on the existence of solutions to the complex Monge-Ampere equation on compact Kahler manifolds. I will also describe some more recent developments on this equation. This will be an introductory and expository mini-course, aimed at graduate students and recent PhDs.

Speaker: William Wylie

Title: An introduction to Ricci solitons I

Abstract:

I will give a brief introduction to Ricci solitons, which are a natural generalization of Einstein metrics coming from the Ricci flow. We'll discuss classification results and examples in low dimensions and for spaces with large amounts of symmetry. This will be an expository mini-course aimed at graduate students and recent Ph.D. students.

Speaker: William Wylie

Title: An introduction to Ricci solitons II

Abstract:

I will give a brief introduction to Ricci solitons, which are a natural generalization of Einstein metrics coming from the Ricci flow. We'll discuss classification results and examples in low dimensions and for spaces with large amounts of symmetry. This will be an expository mini-course aimed at graduate students and recent Ph.D. students.

Speaker: Christine Breiner

Title: Minimal and Constant Mean Curvature Surfaces

Abstract:

In this talk, we will give a brief introduction to minimal and constant mean curvature (CMC) surfaces. We will discuss the variational definitions and important partial differential equations that arise in the study of these surfaces. We will also discuss some existence and uniqueness results.

Speaker: Christine Breiner

Title: Minimal and Constant Mean Curvature Surfaces

Abstract:

In this talk, we will give a brief introduction to minimal and constant mean curvature (CMC) surfaces. We will discuss the variational definitions and important partial differential equations that arise in the study of these surfaces. We will also discuss some existence and uniqueness results.

Speaker: Christine Breiner

Title: Gluing constructions for CMC surfaces

Abstract:

Until the late 1980's, the only examples of CMC surfaces in Euclidean space were the sphere, the Wente torus, and the surfaces of Delaunay. In 1990, Kapouleas developed a gluing construction for CMC surfaces that produced infinitely many new examples. We discuss a generalized gluing construction for surfaces and hypersurfaces that builds on and extends the ideas of Kapouleas. This work is joint with Kapouleas.

Speaker: Huyên Pham

Title: Robust feedback switching control problem

Abstract:

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Speaker: William Wylie

Title: Comparison Geometry for manifolds with density

Abstract:

In this talk we'll discuss the extension of the theory of weighted Ricci curvature to the range of "negative" synthetic dimension. Specifically, we prove a new generalization of the Cheeger-Gromoll splitting theorem where we obtain a warped product splitting under the existence of a line. We'll also discuss how this result is related to some other new natural geometric quantities on manifolds with density including a concept of weighted sectional curvature and a weighted torsion-free connection.

Speaker: Jeffrey Streets

Title: Generalized Kähler-Ricci flow

Abstract:

In joint work with G. Tian I introduced a geometric flow which preserves ``generalized Kähler geometry." I will discuss long time existence and convergence results for this flow which lead to topological rigidity/classification results for generalized Kähler manifolds.

Speaker: Mu-Tao Wang

Title: Stability of Lagrangian curvature flows

Abstract:

I shall present some new stability theorems of curvature flows of Lagrangian submanifolds in both the cotangent bundle case and the Calabi-Yau case. The talk will be based on joint work with Knut Smoczyk and Mao-Pei Tsui and joint work with Chung-Jun Tsai.

Speaker: Xiaodong Cao

Title: Bochner-Weitzenbock formula for gradient Ricci solitons

Abstract:

In this talk, we will briefly survey on some Bochner-Weitzenbock type formulae in Riemannian geometry, which leads to a formula for the self-dual Weyl tensor on 4-dimensional gradient Ricci solitons. We will also discuss some of their applications to geometry and topology.

Speaker: Yannick Sire

Title: Bounds on eigenvalues on riemannian manifolds

Abstract:

I will describe several recent results with N. Nadirashvili where we construct extremal metrics for eigenvalues of the Laplace-Beltrami operator on riemannian surfaces. This involves the study of a Schrodinger operator. As an application, one gets isoperimetric inequalities on the 2-sphere for the third eigenvalue of the Laplace Beltrami operator.

Speaker: Ben Weinkove

Title: Gauduchon metrics with prescribed volume form

Abstract:

We prove that on any compact complex manifold one can find Gauduchon metrics with prescribed volume form. This is equivalent to prescribing the Chern-Ricci curvature of the metrics, and thus solves a conjecture of Gauduchon from 1984. This is a joint work with Gabor Szekelyhidi and Valentino Tosatti.

Speaker: Ram Sharan Adhikari

Title: A weak simpson method for a class of stochastic differential equation and numerical stability results

Abstract:

TBD

Speaker: Maxim Bichuch

Title: Arbitrage-free pricing of XVA

Abstract:

TBD

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: Dan Pirjol

Title: Explosive behavior in discrete time log-normal interest rate models

Abstract:

Interest rates models with log-normally distributed rates simulated in discrete time display explosive behavior, which is different and more complex than that of the better studied continuous time case. This phenomenon is studied in detail for the expectation of the money market account, and the Eurodollar futures prices in the Black-Derman-Toy model. The conditions under which the explosions occur are presented.

Speaker: Birgit Rudloff

Title: Measures of systemic risk

Abstract:

Systemic risk refers to the risk that the financial system is susceptible to failures due to the characteristics of the system itself. The tremendous cost of this type of risk requires the design and implementation of tools for the efficient macroprudential regulation of financial institutions. We propose a novel approach to measuring systemic risk. Key to our construction is a rigorous derivation of systemic risk measures from the structure of the underlying system and the objectives of a financial regulator. Systemic risk is measured by the set of allocations of additional capital that lead to acceptable outcomes. We explain the conceptual framework and the definition of systemic risk measures, provide an algorithm for their computation, and illustrate their application in numerical case studies. We apply our methodology to systemic risk aggregation extending Chen, Iyengar & Moallemi (2013) and to network models as suggested in the seminal paper of Eisenberg & Noe (2001) and their generalizations as in Cifuentes, Shin & Ferrucci (2005). This is joint work with Zach Feinstein and Stefan Weber.

Speaker: Konstantinos Spiliopoulos

Title: Indifference pricing for contingent claims: large deviations effects

Abstract:

We study utility indifference prices and optimal purchasing quantities for a non-traded contingent claim in an incomplete semi-martingale market with vanishing hedging errors, making connections with the theory of large deviations. We concentrate on sequences of semi-complete markets where for each n the claim hn admits the decomposition hn = Dn +Yn where Dn is replicable and Yn is completely unhedgeable in that the indifference price of Yn for an exponential investor is its certainty equivalent. Under broad conditions, we may assume that Yn vanishes in accordance with a large deviations principle as n grows. In this setting, we identify limiting indifference prices as the position size becomes large, and show the prices typically are not the unique arbitrage free price in the limiting market. Furthermore, we show that optimal purchase quantities occur at the large deviations scaling, and hence large positions endogenously arise in this setting. This is joint work with Scott Robertson.

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: Alexander Shklyarevsky

Title: Certain developments in ODE, SDE, PDE, PIDE and related analytical approaches and their applications to physics and quantitative finance and insurance

Abstract: We discuss certain latest developments in methodology and approaches to solve ordinary differential equations (ODE), stochastic differential equation (SDE), partial differential equations (PDE), partial integrodifferential equations (PIDE) and related objects analytically. These approaches are used in both Physics and Quantitative Finance and Insurance both theoretically and in practical applications. An additional advantage is that the approaches developed in Physics could be often applied in Quantitative Finance and Insurance and vice versa. In our presentation, we will show that these analytical methodologies are making both research and its implementation in Physics and research and its implementation in Quantitative Finance and Insurance much more efficient and are critical to substantial advances in both Physics and Quantitative Finance and Insurance.

Speaker: Agnes Tourin

Title: A dynamic model for pairs trading strategies

Abstract:

The profitability of pairs trading strategies has been extensively studied in the empirical literature. However, there are still relatively few attempts to model dynamic pairs trading strategies. In this talk, I will present a family of models based on the theory of stochastic control that lead to analytical formulae for the optimal trading strategies. Some preliminary experiments in the stock and bitcoin markets suggest that the computed strategies perform well and are comparable to those used by practitioners.

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: Kim Weston

Title: Stability of utility maximization in nonequivalent markets

Abstract:

Stability of the utility maximization problem with random endowment and indifference prices is studied for a sequence of financial markets in an incomplete Brownian setting. Our novelty lies in the nonequivalence of markets, in which the volatility of asset prices (as well as the drift) varies. Degeneracies arise from the presence of nonequivalence. In the positive real line utility framework, a counterexample is presented showing that the expected utility maximization problem can be unstable. A positive stability result is proven for utility functions on the entire real line.

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: Mackenzie Wildman

Title: A gaussian markov alternative to fractional brownian motion for pricing financial derivatives

Abstract:

Replacing Black-Scholes’ driving process, Brownian motion, with fractional Brownian motion allows for incorporation of a past dependency of stock prices but faces a few major downfalls because its implementation allows for the occurrence of arbitrage in the financial market. I will discuss the development, testing, and implementation of a simplified alternative to using fractional Brownian motion for pricing derivatives. By relaxing the assumption of past independence of Brownian motion but retaining the Markovian property, we are developing a competing model that retains the mathematical simplicity of the standard Black-Scholes model but also has the improved accuracy of allowing for past dependence. This is achieved by replacing Black-Scholes’ underlying process, Brownian motion, with the Dobric-Ojeda process. This is joint work with Daniel Conus and Vladimir Dobric.

Speaker: Siyan Zhang

Title: Option pricing under SABR model with mean reversion

Abstract:

SABR model is a stochastic volatility model that is widely used in financial industry. However, it may be more reasonable to impose a mean reversion term to volatility. In this talk, I’ll discuss this revised SABR model and price European call option under this model. Though there’s no evidence showing that we can write down the exact solution, in this talk I’ll give an exact formula of an approximation solution by ideas of semigroup and Duhamel’s formula.

Speaker: Oleksii Mostovyi

Title: The University of Texas at Austin

Abstract:

In the framework of an incomplete financial market where the stock price dynamics are modeled by a continuous semimartingale, an explicit first-order expansion formula for the power investor’s value function - seen as a function of the underlying market price of risk process - is provided and its second-order error is quantified. The numerical examples illustrating the accuracy of the method are also given. This talk is based on the joint work with Kasper Larsen and Gordan Zitkovic.

Speaker: Mustapha Pemy

Title: Optimal algorithms for trading large positions

Abstract:

In this paper, we are concerned with the problem of efficiently trading a large position on the market place. If the execution of a large order is not dealt with appropriately this will certainly break the price equilibrium and result in large losses. Thus, we consider a trading strategy that breaks the order into small pieces and execute them over a predetermined period of time so as to minimize the overall execution shortfall while matching or exceeding major execution benchmarks such as the volume-weighted average price (VWAP). The underlying problem is formulated as a discrete-time stochastic optimal control problem with resource constraints. The value function and optimal trading strategies are derived in closed-form. Numerical simulations with market data are reported to illustrate the pertinence of these results.

Speaker: Andrey Sarantsev

Title: Infinite Atlas Model

Abstract:

Consider an infinite system of Brownian particles on the real line. The (currently) bottom particle moves as a Brownian motion with drift one. All other particles move as driftless Brownian motions. Pal and Pitman (2008) proved that the gaps between adjancet particles have stationary distribution which is product of exponentials with rates two. We prove that if we start with the gaps stochastically larger than this stationary distribution, then the gap process weakly converges to this stationary distribution.

Speaker: Asaf Cohen

Title: Risk Sensitive Control of the Lifetime Ruin Problem

Abstract:

We study a risk sensitive control version of the lifetime ruin probability problem. We consider a sequence of investments problems in Black-Scholes market that includes a risky asset and a riskless asset. We present a differential game that governs the limit behavior. We solve it explicitly and use it in order to find an asymptotically optimal policy.(joint work with Erhan Bayraktar)

Speaker: Oleksii Mostovyi

Title: An approximation of utility maximization in incomplete markets

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.

Speaker: Asaf Cohen

Title: Risk Sensitive Control of the Lifetime Ruin Problem

Abstract:

We study a risk sensitive control version of the lifetime ruin probability problem. We consider a sequence of investments problems in Black-Scholes market that includes a risky asset and a riskless asset. We present a differential game that governs the limit behavior. We solve it explicitly and use it in order to find an asymptotically optimal policy.(joint work with Erhan Bayraktar)

Speaker: Kihun Nam

Title: Multidimensional quadratic BSDEs which are related to stochastic differential game

Abstract:

In general, multidimensional quadratic BSDE do not have a solution. In this talk, I will present the existence and uniqueness result for multidimensional quadratic BSDEs of special structure. This types of BSDEs often appears in stochastic differential games. Using Girsanov transform and results from FBSDE and one dimensional quadratic BSDE, I will show how one can construct the solutions. This is a joint work with Patrick Cheridito.

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 (Worcester Polytechnic Institute).

Speaker: Gu Wang

Title: Consumption in Incomplete Markets

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.

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.

Speaker: Michael O. Okelola

Title: Solving a PDE associated with the pricing of power options with time dependent parameters

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.)

Speaker: Gordon Ritter

Multi-period portfolio choice and Bayesian dynamic models

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.)

Speaker: Triet Pham

Two person zero-sum game under feedback controls and path dependent Bellman-Isaacs equations

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.

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Speaker: Ruihua Liu

Title: A tree method for option pricing in switching models with state dependent switching rates

Abstract:

We present a tree approach for option pricing in switching diffusion mod- els where the rates of switching are assumed to depend on the underlying asset price process. The models generalize many existing models in the literature and in particular, the Markovian regime-switching models. The proposed trees grow linearly as the number of tree steps increases. Con- ditions on the choices of key parameters for the tree design are provided that guarantee the positivity of branch probabilities. Numerical results are provided and compared with results reported in the literature for the Markovian regimeswitching cases. The reported numerical results for the state-dependent switching models are new and can be used for comparison in the future.

Speaker: Lakshithe Wagalath

Title: Institutional investors and the dependence structure of asset returns

Abstract:

We propose a model of a financial market with multiple assets, which takes into account the impact of a large institutional investor rebalancing its positions, so as to maintain a fixed allocation in each asset. We show that feedback effects can lead to significant excess realized correlation between asset returns and modify the principal component structure of the (realized) correlation matrix of returns. Our study naturally links, in a quantitative manner, the properties of the realized correlation matrix – correlation between assets, eigenvectors and eigenvalues – to the sizes and trading volumes of large institutional investors. In particular, we show that even starting with uncorrelated ’fundamentals’, fund rebalancing endogenously generates a correlation matrix of returns with a first eigenvector with positive components, which can be associated to the market, as observed empirically. Finally, we show that feedback effects flatten the differences between assets’ expected returns and tend to align them with the returns of the institutional investor’s portfolio, making this benchmark fund more difficult to beat, not because of its strategy but precisely because of its size and market impact.

Speaker: Mingxin Xu

Title: Forward stopping rule within HJM framework.

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.)

Speaker: Duy Nguyen

Title: Numerical schemes for pricing asian options under state-dependent regime-switching jump-diffusion models

Abstract:

We study the pricing problem of Asian options when the underlying asset price follows a very general state-dependent regime-switching jump-diffusion process via a partial differential equation approach. Under this model, the price of the option can be obtained by solving a highly complex system of coupled two-dimensional parabolic partial integro-differential equations (PIDEs). We prove existence of the solution to this system of PIDEs by the method of upper and lower solutions via constructing a monotonic sequence of approximating solutions whose limit is a strong solution of the PIDE system. We then propose several numerical schemes for solving the system of PIDEs. One of the proposed schemes is built upon the constructive proof, hence its results are provably convergent to the solution of the system of PIDEs. We illustrate the accuracy of the proposed methods by several numerical examples. (Joint with D.M. Dang and G. Sewell).

Speaker: Zhixin Yang

Title: Evaluation of risk based premium of pension benefit guaranty corporation with regime switching

Abstract:

This work studies the defined pension plan supported by pension benefit guaranty corporation (PBGC). Our work generalizes Chen ’s work in (2011) and (2014) by taking the market regime switching into consideration. Both the premature of pension fund and distress termination of sponsor asset are analyzed, a closed-form solution for the risk based premiums are achieved.

Speaker: Paolo Guasoni

Title: Spending and investment for shortfall-averse endowments

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)

Speaker: Christian Keller

Title: Pathwise viscosity solutions of stochastic PDEs

Abstract:

We present a notion of pathwise viscosity solutions for fully nonlinear stochastic PDEs and establish well-posedness for a large class of equations. We operate in the framework of rough path theory. Thus we can study stochastic PDEs in a pathwise manner. This is crucial for proving our main results since we can then circumvent very difficult problems regarding null sets. Moreover, ideas from our previous work on path-dependent PDEs play an important role.

This is joint work with Rainer Buckdahn, Jin Ma, and Jianfeng Zhang.

Speaker: Michael Spector

Title: SABR spreads its wings

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.)