Nonlinear Expectations, Stochastic Calculus under Knightian Uncertainty, and Related Topics - IMS

Nonlinear Expectations, Stochastic Calculus under Knightian Uncertainty, and Related Topics
(03 Jun - 12 Jul 2013)

Jointly organized with Centre for Quantitative Finance, NUS

~ Abstracts ~

Stochastic maximum principle for Hilbert space valued forward-backward doubly SDES with Poisson jumps
Abdul Rahman Al-Hussein, Qassim University, Saudi Arabia

In this talk we consider the optimal control problem of a control problem in infinite dimensions. This problem is governed by a fully coupled forward-backward doubly stochastic differential equation driven by a cylindrical Wiener process on a separable Hilbert spaces and a Poisson random measure. The control variable is allowed to enter in all coefficients appearing in this system. The maximum principle for optimal control of this stochastic optimal control problem is derived. Further discussions and proofs will be given as well in the talk.

Precise large deviations of aggregate claims in a risk model with regression-type size-dependence
Xiuchun Bi, University of Science and Technology of China, China

In this work, we build a risk model under the framework of web Markov skeleton processes (WMSPs for short), which are a new class of stochastic processes and are found to be useful in modeling insurance risk. We introduce some regression-type dependence structures, including semi-Markov dependence and mirror semi-Markov dependence. For a case of heavy-tailed claims, we obtain precise large deviation formulas of the aggregate claims under some assumptions on the regression-type size-dependence.

Authors: Xiuchun Bi and Shuguang Zhang, Department of Statistics and Finance, University of Science and Technology of China, People?s Republic of China;

Nonlinear stochastic differential games involving a major player and a large number of collectively acting minor players
Rainer Buckdahn, Universite de Bretagne Occidentale, France and Shandong University, China

In the talk we consider a 2-person zero-sum non-linear stochastic differential game, in which the one player is a major one and the other player is formed by N collectively acting minor players, whose dynamics are driven by independent Brownian motions but who intervene with their control in a same manner. This leads to a pay-off/cost functional, defined through a backward SDE, which averages over the minor players.

For the game with the N minor players we consider a weak solution, which makes it possible to study the game by using controls. Under suitable assumptions the saddle-point controls of the game are determined. The main objective on which the talk focuses is the limit behavior of the stochastic differential game and of the saddle-point controls, as the number N of minor players tends to infinity. The limit stochastic differential game -a mean-field game- is discussed and its saddle-point controls are characterised as the limit of the saddle-point controls of the game with N minor players.

The talk is based on a common work by Shige Peng and Juan Li (Shandong University) together with the speaker.

Incomplete information, trend following, and liquidity premium
Yingshan Chen, National University of Singapore

We study the optimal investment policy of an investor who trades in a market that switches stochastically between bull and bear regimes. The investor does not fully observe the state of the market and incurs transaction costs. We characterize the solution to this problem, focusing on two main implications. First, we show that in this framework the investor is mainly a trend follower, buying on the upswings and selling on the downswings. Second, compared to the full information case, we show that incomplete information about the state of the market can significantly amplify the magnitude of the effect of transaction costs on liquidity premia. Overall, trading costs combined with imperfect knowledge about the time-varying investment opportunities have a strong first-order role in asset pricing.

This is a joint work with Min Dai and Luis Goncalves-Pinto.

Free boundary problems and variational inequality in finance
Xinfu Chen, University of Pittsburgh, USA

In this talk, I review the classical Stefan problem and its variational formulation. Then I present connections between free boundary problem and variational inequality. Some examples from mathematical finance are given.

On the ruin time for risk reserve processes with heavy-tailed claims
Tsung-Lin Cheng, National Changhua University of Education, Taiwan

In this paper, we extend Lemma 1 of Chow and Zhang (1986) to the risk reserve models with heavy-tailed claims. In particular, we obtain an upper bound and a lower bound for the expectation of the time to ruin for the risk reserve process. Our results don't require any assumptions on the distributions of the claims. Moreover, both of continuous-time indexed and discrete-time indexed risk reserve processes are discussed separately. Finally, we conduct a simulation for a risk reserve process with Cauchy-distributed claims to illustrate our main results.

Authors: Tsung-Lin Cheng, National Changhua University of Education, Taiwan; Henghsiu Tsai, Academia Sinica, Taiwan

Insider trading and option returns around earnings announcements
Chin-han Chiang, Singapore Management University

This paper studies the relation between insider trading and option returns around earnings announcements. We show that put (call) options listed under stocks sold (purchased) by insiders earn a significant return premium. This return premium remains significant after controlling for systematic risk, volatility risk, and transaction cost. We provide the first piece of empirical evidence of rising volatility which generates the put option return premium, following insider sales. This rise in volatility is not fully anticipated by market investors, given a significant spread between the implied and realized volatility, and thus causes put options to be relatively undervalued. On the other hand, the call option premium is due to significant run-up in the underlying stock price. The option return premium is cross-sectionally correlated with lagged stock returns, stock volatility, firm size, R&D cost, and book-to-market ratio.

Characterization of optimal strategy for multi-asset investment and consumption with transaction costs
Min Dai, National University of Singapore

We consider the optimal consumption and investment with transaction costs on multiple assets, where the prices of risky assets jointly follow a multi-dimensional geometric Brownian motion. We characterize the optimal investment strategy and show that the trading region has the shape that is very much needed for well defining the trading strategy, e.g., the no-trading region has distinct corners. In contrast, the existing literature is restricted to either single risky asset or multiple uncorrelated risky assets. This work is jointly with Xinfu Chen.

Optimal investment and consumption with capital gain taxes
Min Dai, National University of Singapore

We consider optimal investment and consumption models with capital gain taxes. We will consider two models: 1) the model with average tax basis, where we present an asymptotic expansion of the optimal strategy with small interest/tax rate and extend the model to the carry over case. 2) the model with asymmetric long-term/short-term tax rates, where we show that the optimal strategy is different from those in the existing literature. The talk is based on two of our recent papers: Min Dai, Hong Liu, and Yifei Zhong (2012), Xinfu Chen and Min Dai (2013).

New convergence properties related to local theorems
Freddy Delbaen, ETH Zurich, Switzerland

An extension of the weak convergence of probability measures is introduced. It measures in the language of characteristic functions the distance between probability neasures and given limits (Gaussian or other measures). Applications are found in statistics (but less general than the known results), number theory, random matrices, large and moderate deviations.

Time consistent coherent extension of time consistent concave monetary utility functions
Freddy Delbaen, ETH Zurich, Switzerland

I showed that a concave monetary utility function can be made positively homogeneous by extending the probability space. For time consistent utility functions the construction is more complicated. I will present a positive answer and show that time consistent concave monetary utility functions can be made coherent and time consistent by extending the probability space.

Martingale optimal transport and robust hedging in continuous time
Yan Dolinsky, ETH Zurich, Switzerland

The duality between the robust (or equivalently, model independent) hedging of path dependent European options and a martingale optimal transport problem is proved. The financial market is modeled through a risky asset whose price is only assumed to be a continuous function of time. The hedging problem is to construct a minimal super-hedging portfolio that consists of dynamically trading the underlying risky asset and a static position of vanilla options which can be exercised at the given, fixed maturity. The dual is a Monge Kantorovich type martingale transport problem of maximizing the expected value of the option over all martingale measures that has the given marginal at maturity. In addition to duality, a family of simple, piecewise constant super-replication portfolios that asymptotically achieve the minimal super replication cost is constructed. This is a joint work with Mete Soner.

Superhedging under model uncertainty
Samuel Drapeau, Humboldt Universität zu Berlin, Germany

We discuss the superhedging problem for financial markets under model uncertainty based on existence and duality results for minimal supersolutions of backward stochastic differential equations. The talk is based on joint works with Gregor Heyne, Michael Kupper and Reinhard Schmidt.

Cascading defaults and systemic risk of a banking system
Jin-Chuan Duan, National University of Singapore

Systemic risk of a banking system arises from cascading defaults due to interbank linkages. Any large external shock can in principle triggers cascading defaults, but shocks to systematic risk factors, as opposed to banks' idiosyncratic elements, are more likely to drive cascading defaults and hence to cause higher systemic risk. This paper proposes a structural model for a banking system in which bank assets are subject to both systematic and idiosyncratic risks and bank liabilities contain interbank exposures which may or may not be subject to netting. This model allows us to define two useful measures: systemic exposure and systemic fragility. The former characterizes the expected losses due to interbank linkages under some prescribed macro stress scenario, whereas the latter measures the pervasiveness of bank defaults under the same condition. Our model is conducive to examining potential impacts on systemic risk under different banking network configurations. We devise a novel bridge sampling technique specifically for computing these two systemic risk measures, and obtain data and estimates for a network of 15 British banks. Our results are quarterly time series of estimates for systemic exposures and fragilities from before the 2008-09 financial crisis to the end of 2012. Through this empirical analysis, we shed light on the nature of systemic risk and open new ways for controlling such risk.

Obstacle problem of path-dependent PDES
Ibrahim Ekren, Southern California, USA

In this talk, we adapt the definition of viscosity solutions of path-dependent PDEs to the obstacle problem associated to a non-Markovian second order reflected backward stochastic differential equation with data uniformly continuous in (t, w) and generator Lipschitz continuous in (y, z). We prove that our definition is consistent with the classical solutions, and satisfy a stability result. We show that the value functional defined via the second order reflected backward stochastic differential equation is the unique viscosity solution of the variational inequalities.

Model uncertainty and risk aggregation
Paul Embrechts, ETH Zurich, Switzerland

The quantitative regulation of banking and insurance is very much based on specific risk measures. Examples include Value-at-Risk (a quantile based measure) and Expected Shortfall (a conditional excess measure). Besides their statistical estimation, recent applications very much use the axiomatic theory of risk measures to investigate allocation and aggregation properties. In this talk I will present the necessary theory (going back to a question of Kolmogorov) on quantile based risk aggregation when only partial information on the underlying stochastic structure is known. Besides discussing some analytic results for sums of risk positions, I will also present a versatile, so-called Rearrangement Algorithm for the numerical calculation of best and worst bounds in a model uncertainty context. As an example we discuss the calculation of risk capital for operational risk within the Basel 3 framework of banking regulation. In my talk I will also address a recent Consultative Document of the Basel Committee and its consequences for risk management research related to robust forecasting of risk measures.

Optimal trade execution: mean variance or mean quadratic variation?
Peter Forsyth, University of Waterloo, Canada

Algorithmic trade execution has become a standard technique for institutional market players in recent years, particularly in the equity market where electronic trading is most prevalent. A trade execution algorithm typically seeks to execute a trade decision optimally upon receiving inputs from a human trader.

A common form of optimality criterion seeks to strike a balance between minimizing pricing impact and minimizing timing risk. For example, in the case of selling a large number of shares, a fast liquidation will cause the share price to drop, whereas a slow liquidation will expose the seller to timing risk due to the stochastic nature of the share price.

We compare optimal liquidation policies in continuous time in the presence of trading impact using numerical solutions of Hamilton Jacobi Bellman (HJB) partial differential equations (PDE). In particular, we compare the time-consistent mean-quadratic-variation strategy (Almgren and Chriss) with the time-inconsistent (pre-commitment) mean-variance strategy.
The Almgren and Chriss strategy should be viewed as the industry standard.

We show that the two different risk measures lead to very different strategies and liquidation profiles.

In terms of the mean variance efficient frontier, the original Almgren/Chriss strategy is significantly sub-optimal compared to the (pre-commitment) mean-variance strategy.

Numerical methods for Hamilton-Jacobi-Bellman equations in finance
Peter Forsyth, University of Waterloo, Canada

Many problems in finance can be posed as non-linear Hamilton Jacobi Bellman (HJB) Partial Integro Differential Equations (PIDEs). Examples of such problems include: uncertain volatility, dynamic asset allocation for pension plans, optimal operation of natural gas storage facilities, optimal execution of trades, and pricing of variable annuity products (e.g. Guaranteed Minimum Withdrawal Benefit). This course will discuss general numerical methods for solving the HJB PDEs which arise from these types of problems. After an introductory lecture, we will give an example where seemingly reasonable methods do not converge to the correct (viscosity) solution of a nonlinear HJB equation. A set of general guidelines is then established which will ensure convergence of the numerical method to the viscosity solution. Emphasis will be placed on methods which are straightforward to implement. We then illustrate these techniques on some of the problems mentioned above.

Lecture 1: Examples of HJB Equations, Viscosity Solutions

Lecture 2: Sufficient Conditions for Convergence to the Viscosity Solution, Case Study: Uncertain Volatilty

Lecture 3: Pension Plan Asset Allocation, Passport Options

Lecture 4: Guaranteed Minimum Withdrawal Benefit (GMWB) Variable Annuity: Impulse Control Formulation

Lecture 5: Gas Storage

Lecture 6: Continuous Time Mean Variance Asset Allocation

Portfolio optimization and stochastic volatility asymptotics
Jean-Pierre Fouque, University of California at Santa Barbara, USA

We study the Merton problem of portfolio optimization over a finite horizon when volatility is stochastic and fluctuating on different time scales. We develop a perturbation method for the associated nonlinear PDE and we show how to relate market data implied volatility skews to optimal strategies.
Joint work with Ronnie Sircar and Thaleia Zariphopoulou:
http://www.pstat.ucsb.edu/faculty/fouque/PubliFM/merton_SV_asymp_0323.pdf

Risk measures
Marco Frittelli, Università degli Studi di Milano, Italy

The already-fifteen-years-old theory of risk measures is still originating many questions and springing out lots of new problems which trigger the interest of researchers. We will review the main steps in the evolution of this theory and its connections with several areas in financial mathematics as, for example, the pricing in incomplete markets, the theory of optimal risk sharing, decision theory. One reason of the large growth of this theory is ascribed to the robust representation of risk measures, which accommodates for model ambiguity. We will discuss this matter in relation to convex and quasi-convex risk measures. The dynamic and conditional features of risk measures are more delicate issues and we show how to analyze this aspects by embedding the theory in the framework of L_0 Modules. As more recent applications, we will describe risk measures defined on distribution functions, a generalization of the notion of the V@R, based on a convex family of acceptance sets.

The optimal execution strategy of employee stock option
Yi Fu, Shanghai Normal University, China

In this paper, we developed an optimal execution strategy for the employee stock option by means of the fluid model. We show that the value function is the viscosity solutions of the Hamilton-Jacobi-Bellman variational inequality equation and prove the comparison principle of the viscosity solutions. Finally, numerical illustrative examples and numerical solution of optimal selling strategies ate given by the finite difference method.

Authors: Baojun Bian, Department of Mathematics, Tongji University, Shanghai, People?s Republic of China; Yi Fu and Jizhou Zhang, Mathematics and Science College, Shanghai Normal University, People's Republic of China.

Mean variance hedging under defaults risk
Stéphane Goutte, CNRS-University Paris Diderot 7, France

We solve a Mean Variance Hedging problem in an incomplete market where multiple defaults can appear. For this, we use a default-density modeling approach. The global market information is formulated as progressive enlargement of a default-free Brownian filtration and the dependence of default times is modeled by a conditional density hypothesis. We prove the quadratic form of each value process between consecutive defaults times and solve recursively systems of quadratic backward stochastic differential equations. Moreover, we obtain an explicit formula of the optimal trading strategy. We illustrate our results with some specific cases.

Keywords: Mean variance hedging; default-density modeling; Quadratic backward stochastic differential equation (BSDE); Dynamic programming.

MSC Classification (2010): 60J75, 91B28, 93E20.

Authors: Sébastien CHOUKROUN, Université Paris 7 Diderot, France; Stéphane Goutte, Université du Luxembourg, Luxembourg; Armand Ngoupeyou, Université Paris 7 Diderot, France.

Optimal order placement in a limit order book
Xin Guo, University of California at Berkeley, USA

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 discuss some simple models and strategies to place orders in a limit order book with the objective of minimizing the expected cost. We show that the optimal strategy may depend on key order book statistics and derive a diffusion limit approximation for such analysis. We will discuss some key difference between the diffusion limit modeling for LOB and that for general queues with reneging.

Statistical models of market reactions to influential trades
Mei-Hui Guo, National Sun Yat-Sen University, Taiwan

In the literature, traders are often classified into informed and uninformed and the trades from informed traders have market impacts. We investigate these trades by first establishing a scheme to identify the influential trades from the ordinary trades under certain criteria. The differential properties between these two types of trades are examined via the four transaction states classified by the trade price, trade volume, quotes, and quoted depth. Marginal distribution of the four states and the transition probability between different states are shown to be distinct for informed trades and ordinary liquidity trades. Furthermore, four market reaction factors are introduced and logistic regression models of the influential trades are established based on these four factors. Empirical study on the high frequency transaction data from the NYSE TAQ database show supportive evidence for high correct classification rates of the logistic regression models.

Keywords: High frequency data, Influential trade, Quoted depth, Transition probability, Logistic regression model, Odds ratio.

Joint work with Yi-Ting Guo, Chi-Jeng Wang and Liang-Ching Lin

Martingale problems under non-linear expectations
Xin Guo, University of California at Berkeley, USA

We consider 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 existence of martingale problems. Under this non-linear expectation space, we further develop the stochastic integral and the Ito's type formula, which are consistent with Peng's G-framework. 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.

A price risk measure and momentum strategy
Hwai-Chung Ho, Institute of Statistical Science Academia Sinica, Taiwan

Examining the properties of stock returns has long been a central topic in finance. Most quantitative analyses conducted by academic researchers and practitioners focus only on the return distribution. However, the return distribution itself hardly helps to determine whether the price of a winner stock picked by using the momentum strategy reaches the level where the risk incurred from the falling of prices is imminent. Therefore, we construct an implied price risk index to quantify the downside risk of a stock and use it to manage the tail risk of the momentum strategy. The empirical results demonstrate that our modified strategy can not only achieve significant improvement on the overall performance, but also substantially reduce the drastic losses suffered from the 2008 global recession. We also establish the connection between the implied price risk index and the cross-sectional return differences based on the well-known three factors, the market beta, the firm size and the book-to-market ratio.

Authors: Hwai-Chung Ho and Hongwei Chuang, Institute of Statistical Science, Academia Sinica, Taiwan

Stochastic control problems with singular value functions - analytic solutions and application to optimal portfolio liquidation
Ulrich Horst, Humboldt University Berlin, Germany

We study stochastic optimal control problems with singular terminal values arising in models of optimal portfolio liquidation under market impact when traders can simultaneously trade in a lit and a dark market. For the benchmark Markovian control model we show that the value function can be characterized by a PDE with a singularity at the terminal time and establish existence and uniqueness of classical solutions results for the resulting HJB equation. If the market impact or cost function is not Markovian, then the value function can be described by a singular BSPDE rather than a PDE. We show that the BSPDE has a unique solution (in certain class).
The talk is based on joint work with Paulwin Graewe, Jinnia Qiu, and Eric Sere.

Formal asymptotics for (i) the penalty method (ii) a model for carbon allowance prices
Sam Howison, University of Oxford, UK

I shall describe how formal asymptotic analysis can help in understanding the structure of two nonlinear problems with non-smooth payoffs: (i) numerical solutions for American put options and butterfly spreads using the penalty method, with a focus on the error analysis, and (ii) the behaviour near expiry of a model for carbon allowance prices, with a focus on the impact of the payoff discontinuity.

Mean-variance portfolio selection with uncertain drift and volatility
Ying Hu, Université de Rennes 1, France

This talk is devoted to the study of mean-variance portfolio selection problem with uncertain drift and volatility. We consider the mean-variance portfolio selection problem when the excess drift rate and the volatility may not be known, which causes uncertainty into the model.

A stochastic recursive optimal control problem under the G-expectation framework
Shaolin Ji, Shandong University, China

In this paper, we study a stochastic recursive optimal control problem in which the objective functional is described by the solution of a backward stochastic differential equation driven by G-Brownian motion. Under standard assumptions, we establish the dynamic programming principle and the related Hamilton-Jacobi-Bellman (HJB) equation in the framework of G-expectation. Finally, we show that the value function is the viscosity solution of the obtained HJB equation.

Pricing variance swaps under stochastic volatility with an Ornstein-Uhlenbeck process
Zhaoli Jia, University of Science and Technology of China, China

In this paper, we present a highly efficient approach to price variance swaps under discrete sampling times. We have found a closed-form exact solution for the partial differential equation system based on the Ornstein-Uhlenbeck?s stochastic volatility embedded in the framework proposed by Little and Pant. The key features of our new solution approach include the following: (1) with the newly found analytic solution, all the hedging ratios of a variance swap can also be analytically derived; (2) numerical values can be very efficiently computed from the newly found analytic formula.

Keywords: variance swaps; stochastic volatility; Ornstein-Uhlenbeck process; closed-form exact solution.

On the estimation of integrated volatility with jumps and microstructure noises
Bingyi Jing, Hong Kong University of Science and Technology, Hong Kong

We propose a nonparametric procedure to estimate the integrated volatility of Ito semi-martingale in the presence of jumps and microstructure noise. The estimator is based on a combination of the pre-averaging method and threshold technique, which serve to remove microstructure noise and jumps, respectively. The estimator is shown to work for both finite and infinite activity jumps. Furthermore, asymptotic properties of the proposed estimator, such as consistency and central limit theorem, are established. Simulations results are given to evaluate the performance of the proposed method in comparison with other alternative methods.

Path-dependent partial integro-differential equations
Christian Keller, University of Southern California, USA

We extend the notion of viscosity solutions for path-dependent PDEs, introduced in my joint work with Ekren, Touzi, and Zhang, to path-dependent partial integro-differential equations (PPIDEs). We establish well-posedness for a class of semilinear PPIDEs with uniformly continuous data. Existence follows relatively easily from a probabilistic representation by solutions of non-Markovian Backward SDEs with jumps. Our uniqueness proof relies on optimal stopping theory and the existence of classical solutions to Cauchy-Dirichlet problems involving (state-dependent) partial integro-differential equations that approximate the PPIDE under consideration. The results and their extensions are potentially useful for Finance problems in non-Markovian jump-diffusion models.

The market timing ability of individual investors
Jussi Keppo, National University of Singapore

We document significant persistence in the ability of individual investors to time the stock market. Using data on all trades by individual Finnish investors over more than 14 years, we show that investors who successfully time the market in the first half of the sample are more likely to successfully time in the second half. We further show that investors who time the market around the run-up and crash in 1999 and 2000 are more likely to time the run-up and crash in 2007 and 2008. Our evidence suggests that it is possible to use the trading patterns of these smart investors to anticipate market movements, lending some credibility to the view that market bubbles are identifiable in real time.

Authors: Jussi Keppo, National University of Singapore; Tyler Shumway and Daniel Weagley, University of Michigan

A new 10-30 rule: an investigation of hedge fund performance fees via behavioral finance
Steven Kou, National University of Singapore

A main cause of the recent financial crisis is excessive risk taking due to the limited liability of fund managers and corporations, which means profits are shared but not losses. We investigate hedge fund performance fees via behavioral finance. In particular, we show that in most cases it is possible to improve the satisfaction of regulators, fund managers, and fund investors simultaneously by replacing the traditional 20% performance fee scheme with a new 10-30 first-loss scheme, in which fund managers take 30% performance fee in return for their 10% first-loss capital investment.
This is a joint work with Xuedong He at Columbia University.

First passage times of two-dimensional Brownian motion
Steven Kou, National University of Singapore

First passage times of two-dimensional Brownian motion have been used to study correlated defaults under structural models in finance. However, despite various attempts since 1950's, there are few analytical solutions available. By analytically solving a modified Helmholtz equation in an infinite wedge with non-homogenous boundary conditions, we propose a unified approach to obtain analytical solutions to these problems. We also point out a link between the Laplace transforms of the first passage times and a bivariate exponential distribution which is absolute continuous but does not have memoryless property. This is a joint work with Haowen Zhong.

Prospect theory and trading patterns
Duan Li, The Chinese University of Hong Kong, Hong Kong

Reference dependence, loss aversion, and risk seeking for losses together comprise the preference-based component of prospect theory that sets its value function apart from the standard risk-aversion model. Using an elasticity analysis, we show that this distinctive preference component serves to underpin negative-feedback trading propensities, but cannot manifest itself in behavior directly or holistically at the individual-choice level. We then propose and demonstrate that the market interaction between prospect-theory investors and regular CRRA investors allows this preference component to dominate in equilibrium behavior and hence helps to reestablish the intuitive link between prospect-theory preferences and negative-feedback trading patterns. In the model, the interaction also reconciles the contrarian behavior of prospect-theory investors with asymmetric volatility and short-term return reversal. The results suggest that prospect-theory preferences can lead investors to behave endogenously as contrarian noise traders in the market interaction process.

Stochastic differential games for fully coupled FBSDEs with jumps
Juan Li, Shandong University, China

This paper is concerned with stochastic differential games (SDGs) defined through fully coupled forward-backward stochastic differential equations (FBSDEs) which are governed by Brownian motion and Poisson random measure. First we give some basic estimates for fully coupled FBSDEs with jumps under the monotonic condition. We also prove the well-posedness and regularity results for fully coupled FBSDEs with jumps on the small time interval under a Lipschitz condition (where the Lipschitz constants of σ h with respect to z, k are small enough) and a linear growth condition. For SDGs, the upper and the lower value functions are defined by the controlled fully coupled FBSDEs with jumps. Using a new transformation, we prove that the upper and the lower value functions are deterministic. Then, after establishing the dynamic programming principle for the upper and the lower value functions of this SDGs, we prove that the upper and the lower value functions are the viscosity solutions to the associated upper and the lower Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations, respectively. Furthermore, for a special case (when σ h do not depend on y, z, k), under the Isaacs' condition, we get the existence of the value of the game.

It's based on a common work with Qingmeng Wei (School of Mathematics, Shandong University, Jinan 250100, P. R. China).

Building time-dependent commodity and energy derivative models: an additive time change approach
Lingfei Li, The Chinese University of Hong Kong, Hong Kong

We characterize Ornstein-Uhlenbeck processes time changed with additive subordinators as time-inhomogeneous Markov semimartigales, based on which a new class of commodity and energy derivative models with time-dependent and mean-reverting jumps is developed. Analytical solutions are obtained for European and Bermudan futures options via eigenfunction expansions, with American option prices computed efficiently by extrapolating Bermudan option prices. Calibration examples show that these new models are better alternatives than those developed in Li and Linetsky ("Time-changed Ornstein-Uhlenbeck processes and their applications in commodity derivative models", Mathematical Finance, 2012) by being much more parsimonious and faster for option pricing while calibrating very well to implied volatility surfaces. Our method can be applied to many other processes popular in finance to develop time-inhomogeneous Markov models with desirable features and tractability.

Keywords: commodity and energy derivatives, time change, additive subordinators, time-dependent and mean-reverting jumps, eigenfunction expansions.

Volatility swaps and volatility options on discretely sampled realized variance
Andy Guanghua Lian, University of South Australia, Australia

Volatility derivatives such as variance swaps, variance options and volatility swaps are financial products written on discretely sampled realized variance. Actively traded in over-the-counter markets, these products are priced often by the continuously sampled approximation to simplify the computations. This paper presents an analytical approach to efficiently and accurately price discretely sampled volatility derivatives, and then analyzes the effect of continuous sampling approximation. Under the Heston stochastic volatility model, we first obtain an accurate approximation for the characteristic function of the discretely sampled realized variance. This characteristic function is then applied to derive semi-analytical (up to inverse Laplace transform) pricing formulae for variance options, volatility swaps and volatility options. We examine with numerical examples the accuracies of the approach in pricing these volatility derivatives. We also test the effect of discrete sampling in pricing volatility derivatives. For realistic contract specifications and model parameters, we find that continuously sampled variance swaps and options are commonly cheaper than their discretely sampled counterparts. Continuously sampled volatility swaps, surprisingly, are more expensive than their discretely sampled counterparts.

Keyword: Variance swaps, Variance options, Stochastic volatility, Characteristic function

J.E.L. Classification. D81, G13.

Authors: Andy Guanghua Lian, School of Commerce, Division of Business, University of South Australia; Carl Chiarella, Finance Discipline Group, Business School, University of Technology, Sydney, Australia; Petko S. Kalev, School of Commerce, Division of Business, University of South Australia.

Mean-variance hedging on uncertain time horizon in a market with a jump
Thomas Lim, Université d'Evry and ENSIIE, France

In this work, we study the problem of mean-variance hedging with a random horizon T Λ τ, where T is a deterministic constant and τ is a jump time of the underlying asset price process. We first formulate this problem as a stochastic control problem and relate it to a system of BSDEs with a jump. We then provide a verification theorem which gives the optimal strategy for the mean-variance hedging using the solution of the previous system of BSDEs. Finally, we prove that this system of BSDEs admits a solution via a decomposition approach coming from filtration enlargement theory.

Keywords: Mean-variance hedging, Backward SDE, random horizon, jump processes, progressive enlargement of filtration, decomposition in the reference filtration.

AMS subject classification: 91B30, 60G57, 60H10, 93E20.

Authors: Idris Kharroubi, CERAMADE, Université Paris Dauphine; Thomas Lim, Laboratoire d'Analyse et Probabilités, Université d'Evry and ENSIIE; Armand Ngoupeyou, Laboratoire de Probabilités et Modèles Aléatoires, Université Paris 7

Condition number regularized covariance estimation with application to portfolio optimization
Johan Lim, Seoul National University, Korea

Estimation of high-dimensional covariance matrices is known to be a difficult problem, has many applications, and is of current interest to the larger statistics community. In many applications including so-called the large p, small n setting, the estimate of the co-variance matrix is required to be not only invertible, but also well-conditioned. Although many regularization schemes attempt to do this, none of them address the ill-conditioning problem directly. In this paper, we propose a maximum likelihood approach, with the direct goal of obtaining a well-conditioned estimator. No sparsity assumptions on either the covariance matrix or its inverse are imposed, thus making our procedure more widely applicable. We demonstrate that the proposed regularization scheme is computationally efficient, yields a type of Steinian shrinkage estimator, and has a natural Bayesian interpretation. We investigate the theoretical properties of the regularized covariance estimator comprehensively, including its regularization path, and proceed to develop an approach that adaptively determines the level of regularization that is required. Finally, we demonstrate the performance of the regularized estimator in decision-theoretic comparisons and in the financial portfolio optimization setting. The proposed approach has desirable properties, and can serve as a competitive procedure, especially when the sample size is small and when a well-conditioned estimator is required.

This is a joint work with Joong-Ho Won, Seung-Jean Kim, and Bala Rajaratnam.

Model uncertainty in insurance problems
Jin Ma, University of Southern California, USA

In this we are interested in the general issue of \model uncertainly" for a typical insurance model with investments. Apart of the well-known uncertainty on market volatility, there are special uncertainties only occurring in insurance problems. For example, the uncertainty on severity and frequency of the claim processes. We show that these uncertainties can be studied under the general paradigm of G-expectation, developed recently by S. Peng. Many problems, especially those related to ruin probabilities, can be conveniently studied within the G-L_evy processes framework. In particular, we study the ruin problems, both in the _nite time setting and in asymptotics, under model uncertainties. Some related issues for market models under G-framework will be also be discussed.
This is a joint work with Xin Wang.

Doubly reflected second order BSDE's and Dynkin game under uncertainty
Anis Matoussi, Université du Maine, France

We first present results about existence and uniqueness of second-order reflected 2BSDEs to the case of upper obstacles. Then, under some regularity assumptions on one of the barrier, and when the two barriers are completely separated, we provide a complete wellposedness theory for doubly reflected second-order BSDEs. We also show that these objects are related to non-standard optimal stopping games, thus generalizing the connection between DRBSDEs and Dynkin games first proved by Cvitanic? and Karatzas (2006). More precisely, we show that the second order DRBSDEs provide solutions of what we call uncertain Dynkin games and that they also allow us to obtain super and subhedging prices for American game options (also called Israeli options) in financial markets with volatility uncertainty. This talk is based on joint works with Lambert Piozin (Université du Mans), Dylan Possamaï (Université Dauphine) and Cha Zhou (National University of Singapore).

Stochastic dominance via quantile regression with applications to investigate arbitrage opportunity and market efficiency
Pin Ng, Northern Arizona University, USA

Tests for stochastic dominance constructed by translating the inference problem of stochastic dominance into parameter restrictions in quantile regressions are proposed. They are variants of the one-sided Kolmogorov-Smirnoff statistic with a limiting distribution of the standard Brownian Bridge. Simulation results show their superior size and power. They are applied to the NASDAQ 100 and S&P 500 indices to investigate dominance relationship before and after the major turning points. Results show no arbitrage opportunity between the bear and bull market, and markets are inefficient in that risk averters are better off by investing in the bull rather than the bear market.

Keywords: Quantile regression, Stochastic dominance, Brownian bridge, Internet bubble crisis, Sub-prime crisis.

JEL Classification: C01, C12, C31

Authors: Pin Ng, Franke College of Business, Northern Arizona University; Wing-Keung Wong, Department of Economics, Hong Kong Baptist University WLB, Shaw Campus, Hongkong; Zhijie Xiao, Department of Economics, Boston College, USA

On lower and upper bounds for Asian-type options: a unified approach
Alex Novikov, University of Technology, Australia

In a context of dealing with financial risk management problems it is desirable to have accurate bounds for option prices in situations when pricing formulae do not exist in the closed form. An unified approach for obtaining upper and lower bounds for Asian-type options is proposed in the paper. These bounds are applicable in the continuous and discrete-time frameworks and for the case of time-dependent interest rates as well. The numerical examples are provided which illustrate accuracy of the bounds.

Arbitrage and duality in nondominated discrete-time models
Marcel Nutz, Columbia University, USA

We consider a nondominated model of a discrete-time financial market where stocks are traded dynamically and options are available for static hedging. In a general measure-theoretic setting, we show that absence of arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of martingale measures. In the arbitrage-free case, we show that optimal superhedging strategies exist for general contingent claims, and that the minimal superhedging price is given by the supremum over the martingale measures. (Joint work with Bruno Bouchard.)

Martingale problems under nonlinear expectations
Chen Pan, University of Science and Technology of China, China

In this work, we define the martingale problems under nonlinear expectations according to Stroock and Varadhan's classical ones. Then, in the light of Peng's (2005) idea about defining time consistent non-linear expectations, we derive the solution of the martingale problems based on viscosity solution theory for fully nonlinear parabolic PDEs. At last, as an application, we define the weak solutions to a class of G-SDEs, and give corresponding solutions.

Authors: Xin Guo, Department of Industrial Engineering and Operations Research, UC Berkeley, USA; Chen Pan Department of Mathematics, University of Science and Technology of China, People?s Republic of China; Shige Peng, School of Mathematics, Shandong University, People's Republic of China

Dynamic comfort: a common market factor non-Gaussian returns model with dynamic conditional correlations
Marc Paolella, University of Zurich, Switzerland

A new multivariate time series model with various attractive properties is motivated and studied. By extending the CCC model in several ways, it allows for all the primary stylized facts of Financial asset returns, including volatility clustering, non-normality (excess kurtosis and asymmetry), and also dynamics in the dependency between assets over time. A fast EM-algorithm is developed for estimation. The predictive conditional distribution is a (possibly special case of a) multivariate generalized hyperbolic, so that sums of marginals (as required for portfolios) are tractable. Each element of the vector return at time t is endowed with a common univariate shock, interpretable as a common market factor, and this stochastic process has a predictable component. This leads to the new model being a hybrid of GARCH and stochastic volatility, but without the estimation problems associated with the latter, and being applicable in the multivariate setting for potentially large numbers of assets. Formulae associated with portfolio optimization, risk measures and option pricing based on the predictive density are developed. In-sample fit and out-of-sample conditional density forecasting exercises using daily returns on the 30 DJIA stocks confirm the superiority of the model to numerous competing ones. Extensions to the DCC and CCC Markov switching models are discussed, as well as extension to the stable Paretian and tempered stable distributional setting.

Authors: Marc S. Paolella and Pawel Polak, Department of Banking and Finance, University of Zurich, Switzerland & Swiss Finance Institute

Mean or median: an implication for BIS trading book capital requirements
Xianhua Peng, The Hong Kong University of Science and Technology, Hong Kong

We give the precise definition of a new risk measure called median shortfall that captures the tail risk in an alternative way to that of expected shortfall. We compare the properties of median shortfall with those of some other risk measures that capture tail risks such as tail conditional median and expected shortfall. We show that (i) the median shortfall can capture tail risk; (ii) the median shortfall is more robust than the expected shortfall with respect to small changes in the data; (iii) the estimation error of the median shortfall is much smaller than that of the expected shortfall. We argue that the risk measure used for setting trading book capital requirements should be robust; hence, it is better to use the median shortfall than to use the expected shortfall in setting capital requirement in the Basel accords. This is a joint work with Steven Kou.

G-expectation weighted Sobolev spaces, backward SDE and path dependent PDE
Shige Peng, Shandong University, China

We introduce a new notion of G-expectation-weighted Sobolev spaces, or in short, G-Sobolev spaces, and prove that a backward SDEs driven by G-Brownian motion are in fact path dependent PDEs in the corresponding G-Sobolev spaces. For the linear case of G corresponding the classical Wiener probability space, we have established a 1-1 correspondence between BSDE and such new type of quasi linear PDE in the corresponding P-Sobolev space. When G is nonlinear, we also provide such 1-1 correspondence between a fully nonlinear PDE in the corresponding G-Sobolev space and BSDE driven by G-Brownian. Consequently, the existence and uniqueness of such type of fully nonlinear path-dependence PDE in G-Sobolev space have been obtained via a recent results of BSDE driven by G-Brownian motion.

Location, location, location: the econometrics of asset pricing with spatial interaction
Xianhua Peng, The Hong Kong University of Science and Technology, Hong Kong

It is common knowledge that spatial interaction is important in modeling real estate assets, as house prices are significantly affected by the neighborhood prices. Although spatial econometrics have been applied to empirical studies of housing markets, there is little theoretical work that studies the risk and return of real estate assets. In this paper, we attempt to fill this gap by proposing a spatial capital asset pricing model (S-CAPM) and a spatial arbitrage pricing theory (S-APT), which extend the classical asset pricing models by incorporating spatial interaction among asset returns. Furthermore, we study asymptotic properties of the estimators and test statistics needed for implementing the models. An empirical study of the futures contracts on the S&P/Case-Shiller Home Price Indices shows that the spatial interaction is statistically significant.

This is a joint work with Steven Kou and Haowen Zhong.

Backward SDEs with partially nonpositive jumps and Hamilton-Jacobi-Bellman IPDE
Huyên Pham, Université Paris Diderot, France

We aim to provide a Feynman-Kac type representation for Hamilton-Jacobi-Bellman equation, in terms of Forward Backward Stochastic Di_erential Equation (FBSDE) with a simulatable forward process. For this purpose, we introduce a class of BSDE where the jumps component of the solution is subject to a partial nonpositive con- straint. Existence and approximation of a unique minimal solution is proved by a penalization method under mild assumptions. We then show how minimal solution to this BSDE class provides a new probabilistic representation for fully nonlinear integro-partial di_erential equations (IPDEs) of Hamilton-Jacobi-Bellman (HJB) type, when considering a regime switching forward SDE in a Markovian framework. This includes in particular equations in _nance arising from option pricing under model uncertainty. In contrast with the recent theory of G-expectation and 2BSDEs, our representation is formulated under a single probability measure, thus avoiding quasi-sure analysis and nondominated measures. We also introduce a direct numerical scheme for BSDEs with constrained jumps, which gives an original probabilistic algorithm for solving HJB equations. Finally, we present some extensions of our approach for dealing with ergodic HJB equations, and Bellman-Isaacs equation in stochastic di_erential games. This talk is based on joint works with I. Kharroubi (Paris Dauphine), Andrea Cosso, Marco Fuhrman (Politechnico Milano), and S_ebastien Choukroun, Nicolas Langren_e (Paris Diderot).

Financial bubbles and the possibility of real time detection
Philip Protter, Columbia University, USA

Due to their occasional spectacular consequences, mathematical models for financial bubbles have been developed over the last 10 years. We will survey many of these developments, explaining how the more interesting models lead to the close analysis of strict local martingales, and we will explain how models of bubbles are more reasonable in incomplete market settings than they are in complete market settings. We will close by exhibiting a method under which one can detect (but without certainty) whether or not a given stock price is undergoing bubble pricing.

The generation of simple linear regression model
Manachai Rodchuen, Chiang Mai University, Thailand

A simple linear regression model is the relationship between two variables. The value of one variable can estimate or predict based on the relationship with the other variable. For specified problem in linear regression model such as outlier, multicollinearity and missing data, the solution cannot be proof using statistical theory. In recently, the computer simulations are used as a tool to provide the answer of the problems in practical. This article presents a method to generate the population values under the simple linear regression model in order to satisfy the specified correlation coefficient.

Keywords: simulation; regression; coefficient; correlation

Authors: Manachai Rodchuen and Puttpong Bookkamana, Department of Statistics, Faculty of Science, Chiang Mai University, Chiang Mai, Thailand

Why are quadratic normal volatility models analytically tractable?
Johannes Ruf, University of Oxford, UK

We discuss the class of Quadratic Normal Volatility (QNV) models, which have drawn much attention in the financial industry due to their analytic tractability and exibility. We characterize these models as the ones that can be obtained from stopped Brownian motion by a simple transformation and a change of measure that only depends on the terminal value of the stopped Brownian motion. This explains the existence of explicit analytic formulas for option prices within QNV models in the academic literature. Furthermore, via a different transformation, we connect a certain class of QNV models to the dynamics of geometric Brownian motion and discuss changes of numeraires if the numeraire is modelled as a QNV process.

Keywords: Local volatility, Pricing, Foreign Exchange, Riccati equation, Change of numeraire, Local martingale, Semi-static hedging, Hyperination

Joint work with Peter Carr and Travis Fisher.

Comparative and qualitative robustness for law-invariant risk measures
Alexander Schied, University of Mannheim, Germany

When estimating the risk of a P&L from historical data or Monte Carlo simulation, the robustness of the estimate is important. We argue here that Hampel's classical notion of qualitative robustness is not suitable for risk measurement and we propose and analyze a refined notion of robustness that applies to tail-dependent law-invariant convex risk measures on Orlicz space. This concept of robustness captures the tradeoff between robustness and sensitivity and can be quantified by an index of qualitative robustness. By means of this index, we can compare various risk measures, such as distortion risk measures, in regard to their degree of robustness. Our analysis also yields results that are of independent interest such as continuity properties and consistency of estimators for risk measures, or a Skorohod representation theorem for {\psi}-weak convergence. This is joint work with Volker Krätschmer and Henryk Zähle.

On a stochastic Fubini theorem
Martin Schweizer, ETH Zurich, Switzerland

We present a new stochastic Fubini theorem in a setting where we integrate measure-valued stochastic processes with respect to a d-dimensional martingale. To that end, we develop a notion of measure-valued stochastic integrals. As an application, we show how one can handle a class of quite general stochastic Volterra semimartingales. The talk is based on joint work with Tahir Choulli (University of Alberta, Edmonton, Canada).

Second order BSDE's
Mete Soner, ETH Zurich, Switzerland

As well known standard backward stochastic differential equations (BSDE) have many applications in finance. It also has natural connections semi-linear parabolic, partial differential equations (PDE). In this talk, I discuss the extension of this theory that allows to include fully nonlinear PDE's, which is called 2BSDE's, which was initiated in collaboration with Nizar Touzi and Jianfeng Zhang. A particular case of this is the deep theory of G-expectations that was recently introduced by Peng. In this case, the nonlinear of the corresponding PDE is Pucci maximal operator and this connection to the PDE is essentially used to construct solutions. In the general theory of 2BSDE's a probabilistic approach is used to construct the unique solution. I will explain the general theory and the related results. The connections to risk measures and to robust hedging will also be outlined.

Processes with stationary and independent increments under G-expectation
Yongsheng Song, Chinese Academy of Sciences, China

Our purpose is to investigate properties for processes with stationary and independent increments under G-expectation. As applications, we give a martingale characterization to G-Brownian motion and present a decomposition for generalized G-Brownian motion.

Conditional heteroskedasticity of return range processes
Yan Sun, Utah State University, USA

Assets prices often show a significant variation within a small period of time. To take into account such short-term variations, along with the volatility over time, we propose an interval-valued GARCH (I-GARCH) model for analyzing the return range data. We present the statistical properties in the framework of random sets. We obtain sufficient conditions under which the I-GARCH model is covariance stationary, and give an explicit formula for the auto-correlation function. In addition, we propose a conditional least squares estimate (CLSE) for estimating the model parameters. Simulation studies support our theorems. Furthermore, we apply our I-GARCH model to analyze the daily log return range data of several Dow Jones component stocks and derive interesting and promising results. Especially, our model successfully captures a subtle type of high volatility that is typically underestimated by the classical (point-valued) GARCH models. We elaborate on the impact of these findings to financial practice.

Authors: Yan Sun, Department of Mathematics & Statistics, Utah State University, U.S.A; Jennifer Loveland, Department of Mathematics & Statistics, Utah State University, U.S.A

Calibration of stochastic volatility models: a Tikhonov regularization approach
Ling Tang, National University of Singapore

We aim to calibrate stochastic volatility models from option prices. We develop a Tikhonov regularization approach to recover the risk neutral drift and volatility terms of stochastic volatility. Numerical results and empirical studies are presented. In contrast to existing literature, we do not assume that the model has special structure.

This is a joint work with Min Dai and Xingye Yue.

Model-free hedging and martingale optimal transport
Nizar Touzi, Ecole Polytechnique, France

In the simplest one-period model, the dual formulation of the robust superhedging cost differs from the standard optimal transport problem by the presence of a martingale constraint on the set of coupling measures. The one-dimensional Brenier theorem has a natural extension. However, in the present martingale version, the optimal coupling measure is concentrated on a pair of graphs which can be obtained in explicit form. These explicit extremal probability measures are also characterized as the unique left and right monotone martingale transference plans, and induce an optimal solution of the kantorovitch dual, which coincides with our original robust hedging problem. By iterating the above construction over n steps, we define a Markov process whose distribution is optimal for the n-periods martingale transport problem corresponding to a convenient class of cost functions. Similarly, the optimal solution of the corresponding robust hedging problem is deduced in explicit form. Finally, by sending the time step to zero, this leads to a continuous-time version of the Brenier theorem in the present martingale context, thus providing a new remarkable example of Peacock, i.e. Processus Croissant pour l'Ordre Convexe. Here again, the corresponding robust hedging strategy is obtained in explicit form, and the superhedging property holds in the sense of the Follmer stochastic calculus without probability.

A new approach to Poisson approximations
Loc Hung Tran, University of Finance and Marketing (UFM), Vietnam

The main purpose of this note is to present a new approach to Poisson Approximations. Some bounds in Poisson Approximations in term of classical Le Cam's inequalities for various wise-row triangular arrays of wide class of discrete independent random variables are established via Trotter-Renyi distance based on Trotter-Renyi operator method. Some analogous results related to random sums in Poisson Approximations are considered, too.

Keywords: Poisson approximation, Random summand, Le Cam's inequality, Trotter's operator, Renyi's operator, Poisson-binomial random variables, Geometric random variables, Negative binomial random variables

Mathematics Subject Classification 2010: 60F05, 60G50, 41A36.

Authors: Tran Loc Hung, University of Finance and Marketing; Le Truong Giang, Can Tho University, Vietnam

Estimating and backtesting distortion risk measures
Hideatsu Tsukahara, Seijo University, Japan

We have shown in our previous work that for a wide class of distortion functions, it is possible to construct an estimator for distortion risk measures (DRMs) with reasonable accuracy based on weakly dependent data. In this presentation, we first show that the estimator always has a negative bias and illustrate a bootstrap-based method for bias correction. The method will be shown to possess consistency under a certain regularity condition.

For a Monte Carlo simulation study, we consider a stochastic volatility (SV) model with inverse gamma AR(1) volatility process. Simulation results for estimating value-at-risk, expected shortfall and proportional odds risk measure under various values of the parameters show that the normal approximation, our asymptotic variance estimation and bias correction methods are working to a reasonable extent.

As a next step in financial risk management, we need to evaluate the accuracy of the model and/or estimation procedure for risk measurement. To this end, a simple backtesting procedure will be proposed for DRMs which can be made theoretically rigorous with i.i.d. data. We can also implement the conditional approach by McNeil and Frey with GARCH-type observations.

BSDE, path-dependent PDE and nonlinear Feynman-KAC formula
Falei Wang, Shandong University, China

In this paper, we introduce a type of path-dependent quasilinear (parabolic) partial differential equations in which the (continuous) paths on an interval [0; t] becomes the basic variables in the place of classical variables (t, x). This new type of PDE are formulated through a classical backward stochastic differential equations in which the terminal values and the generators are allowed to be general function of Brownian paths. In this way we have established a new type of nonlinear Feynman-Kac formula for a general non-Markovian BSDE. Some main properties of regularities for this new PDE is obtained.

Tests for the equality of multiple sharpe ratios
John Alexander Wright, The Chinese University of Hong Kong, Hong Kong

In this talk we discuss tests for the equality of multiple Sharpe ratios. First we extend the multivariate Sharpe ratio statistic of Leung and Wong (2008) for the case when excess returns are independently and identically distributed (IID). We then provide a test that holds under the much more general assumption that the excess returns are stationary and ergodic, making use of the generalised method of moments and heteroskedasticity and autocorrelation consistent estimation of covariance matrices. As such, this test relies on the asymptotic distribution of the estimators concerned. In contrast, the method for pairwise comparison of Sharpe ratios put forward in Ledoit and Wolf (2008) is a bootstrap approach and we extend it to work for many Sharpe ratios. Comparisons are made between the two approaches, using simulated IID returns, time-series returns and real-world data in the form of iShares returns. We conclude that the bootstrap approach of Ledoit and Wolf (2008) is, in general, more accurate but it can be impractically time-consuming.

Authors: Wei Rosa Huang, John Alexander Wright, and Sheung Chi Philip Yam, Department of Statistics, The Chinese University of Hong Kong, Shatin, Hong Kong

Asset pricing under probability distortions
Jianming Xia, Chinese Academy of Sciences, China

We provide conditions on a one-period-two-date pure exchange economy with rank-dependent utility agents under which Arrow-Debreu equilibria exist. When such an equilibrium exists, we show that the state-price density is a weighted marginal rate of intertemporal substitution of a representative agent, while the weight is expressed through the di fferential of the probability distortion function. Based on the result we reach several findings, including (1) that asset prices depend upon agents' subjective beliefs regarding overall consumption growth, (2) that an uncorrelated security's entire probability distribution and its interdependence with the other part of the economy should be priced, and (3) that there is a direction of thinking about the equity premium and risk-free rate puzzles. (Based on a joint work with Xun Yu Zhou).

Firms' credit rating dynamics in the presence of unobserved market structural changes
Haipeng Xing, State University of New York, at Stony Brook, USA

Many efforts have been made in the past decades to identify and measure new significant sources of credit risk. We find in this paper that credit market structural change, a new source of joint credit risk, can not be captured by observable or unobservable firm-specific and macroeconomic risk factors. To demonstrate this, we proposes for firms' credit rating transition intensities a modulated semi-Markov model in which the model parameters may experience sharp shifts caused by credit market structural changes. We then develop a semiparametric estimation theory for the inference of time varying model parameters, baseline transition intensities, and probabilities of market structural changes. Based on the analysis of U.S. public firms between 1986 and 2008, we show strong evidence of the effect of market structural changes in the firm's rating transitions after the inclusion of observable firm-specific variables (distance to default and trailing return), macroeconomic covariates, and frailty effects.

Implicit incentives of mutual fund flows and liquidity premia
Jing Xu, National University of Singapore

We study the optimal investment policy of an open-end equity fund manager who needs to deal with periodic and tradable money flows into and out of her fund. The fund manager invests in a benchmark portfolio and in an alternative risky asset. We assume that trading on the alternative asset incurs transaction costs, while trading on the benchmark is costless. We assume a positive and convex sensitivity of fund flows to relative past performance, which gives the manager implicit incentives to gamble to finish ahead of the benchmark. We show that such implicit incentives significantly increase the frequency and volume of endogenous trading of the fund manager, while keeping her exogenous trading, which is imposed by the flows themselves, at negligible levels. As a result, if the fund manager is the marginal investor in the alternative asset, transaction costs can have a strong first-order effect on the liquidity premium of that asset.

This is a joint work with Min Dai and Luis F Gonclaves-Pinto.

The dual process and the Monte-Carlo method
Weiqiang Yang, Shandong University, China

This talk will discusses some types of Monte-Carlo methods to solve PDE, which may involving Feynman-Kac formula, BSDE, martingale representation, Brownian bridge, Green function etc. Further, a new kind of Monte-Carlo method induced by the dual process will be introduced.

Time consistent optimal controls for mean-variance portfolio selection problems
Siu-Pang Yung, The University of Hong Kong, Hong Kong

In a mean-variance portfolio selection problem, the portfolio wealth appears as a nonlinear term in the objective function. This makes the dynamic programming principle invalid and causes the traditional optimal selection strategy "time inconsistent", in the sense that the optimal strategy started at the initial time may not be optimal again when we start at a later time. In this talk, we shall describe some of our current results in the search of time consistent optimal strategies for some mean-variance problems.

Financial Stochastic differential equations
Behnam Zarpak, Shahed University, Iran

ARIMA, GARCH and Neural Network are important discrete time models for financial time series data. But in this paper we have considered stochastic differential equation for discrete sample data. One advantage of this model is to work with a huge data. We have used maximum likelihood estimation with an explicit formula in Black-Scholes equation. Finally data analysis have shown with a specify financial data.

Keyword: Stochastic Differential Equation, Maximum Likelihood Estimation, Black-Scholes Differential Equation, Data Analysis.

Backward stochastic partial differential equations and their application to stochastic black-scholes formula
Qi Zhang, Fudan University, China

The backward SPDEs, originated from the study of optimal control theory of SPDEs, can be applied to mathematical finance problems. We demonstrate their application to stochastic Black-Scholes formula, in a general setting to the parameters of the model. This application is based on our recent studies of the solvability to degenerate backward SPDEs without technical assumptions and their connection to forward-backward SDEs. The connection between backward SPDEs and forward-backward SDEs can also be regarded as an extension of Feynman-Kac formula to non-Markovian framework.

Transposition method for BSDEs/BSEEs
Xu Zhang, Chinese Academy of Sciences, China

Stimilated by the classical transposition method in PDEs, we introduced a new notion of solution, i.e., transposition solution to BSDEs/BSEEs. This is something like the generalized function solutions to PDEs. We obtained the well-posedness of linear and semilinear BSDEs/BSEEs in general filtration space, without using the Martingale Representation Theorem and Ito's formula. Some applications are also presented.

Two person zero sum game under weak formulation
Jianfeng Zhang, University of Southern California, USA

In this talk we study a two person zero sum stochastic differential game under weak formulation. Unlike standard literature which uses strategy type of controls, the weak formulation allows us to consider the game with control against control. We shall prove the existence of the game value under natural conditions. Another main feature of the work is that we 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. This is a joint work with Triet Pham (USC).

Study on multiple attribute decision making problems based on D-S evidence theory and hypercube segmentation
Fangwei Zhang, Southeast University, China

The theory of multiple attribute decision making (MADM) is an important branch of modern decision sciences. Uncertain MADM is the extending and development of classical theory in MADM. In this speech, the necessity, the feasibility and the application of combining D-S evidence theory with hypercube segmentation in multiple attribute decision making is studied. The main innovation of our research is that three methods are proposed based on the combination of evidence theory and hypercube segmentation. The first method is proposed for solving the multiple attribute decision making problems with normal random variables; the second method is for uncertain multiple attribute group decision making problems; the third method deals with two-people multiple attribute decision making problems. At last, a brief summary of this research is offered, the application and prospect of our main results is introduced.

Keywords: Multiple attribute decision making; D-S evidence theory; Hypercube segmentation; Numerical simulation.

Viscosity solutions of path dependent PDEs
Jianfeng Zhang, University of Southern California, USA

Path Dependent PDEs (PPDEs, for short) is a convenient tool to characterize the value functions of various types of stochastic control problems in non-Markovian framework. Its typical examples include Backward SDEs (semi-linear PPDEs), Second order backward SDEs (path dependent HJB equations), path dependent Bellman-Isaacs equations, and Backward Stochastic PDEs. PPDEs can rarely have classical solutions. The main goal of this mini-course is to propose a notion of viscosity solutions for PPDEs and establish its wellposedness. Our definition relies heavily on the Functional Ito formula initiated by Dupire. Unlike the viscosity theory of standard PDEs, the main technical difficulty in path dependent case is that the state space is not locally compact. To overcome such difficulty, we replace the point-wise maximization in standard theory with an optimal stopping problem. The course is based on joint works with Nizar Touzi, and Ibrahim Ekren, Christian Keller, Triet Pham.

Lecture 1. Motivation and examples

Lecture 2. Basics of pathwise stochastic analysis

Lecture 3. Viscosity solutions of path dependent heat equations

Lecture 4. Peng's nonlinear expectation

Lecture 5. Viscosity solutions of fully nonlinear PPDEs

Lecture 6. Wellposedness of fully nonlinear PPDEs

Quadratic BSDEs with jumps: a fixed-point approach
Chao Zhou, National University of Singapore

We prove the existence of bounded solutions of quadratic backward SDEs with jumps, using a direct fixed-point approach as in Tevzadze (2008). Under an additional standard assumption, we prove a uniqueness result, thanks to a comparison theorem. Then we study the properties of the corresponding g-expectations, we obtain in particular a non-linear Doob-Meyer decomposition for g-submartingales. We also give applications for dynamic risk measures and compute their inf-convolution, with some explicit examples of optimal risk transfer between two agents.

Second order BSDEs with jumps
Chao Zhou, National University of Singapore

We provide a wellposedness result for second order backward stochastic differential equation with jumps, which generalizes the continuous case considered by Soner, Touzi and Zhang. We also give an application of second order BSDEs to the study of a robust exponential utility maximization problem under model uncertainty. The uncertainty affects both the volatility process and the jump measure compensator. This talk is based on joint works with N. Kazi-Tani and D. Possamai.

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