Financial Mathematics - IMS

Financial Mathematics
(02 Nov - 23 Dec 2009)

Jointly organized with Risk Management Institute, NUS

~ Abstracts ~

Liquidity matters - pure & applied techniques in liquidity management
Ranjan Bhaduri, AlphaMetrix Alternative Investment Advisors, LLC, USA

The problems of some of the statistical due diligence techniques of hedge funds used in industry pertaining to liquidity are examined, and superior techniques are introduced. The entanglement between model risk and liquidity risk is explored. The basics of Managed Accounts, and the interplay between liquidity, transparency, and technology are taught. The importance of tying the quantitative and qualitative together are also highlighted. This workshop covers different liquidity matters such as: liquidity buckets, liquidity duration, liquidity indices, liquid hedge funds, and liquidity derivatives.

The balls in the hat game - an optimal stopping problem and its connections to the value of liquidity
Ranjan Bhaduri, AlphaMetrix Alternative Investment Advisors, LLC, USA

This talk furnishes a game-theoretic illustration of an optimal stopping problem and its connection to the value of liquidity. The solution to the finite game is based on combinatorics, a recursive relation, and elementary probability. Variations of the Balls in the Hat Game are presented, including the respective solutions, and in each case, insights are given to liquidity and portfolio management. The asymptotic behaviour is also explored.

Pricing path dependent options under a flexible jump diffusion model
Ning Cai, The Hong Kong University of Science and Technology, Hong Kong

We provide analytical solutions to pricing problems of a variety of path dependent options under a flexible hyper-exponential jump diffusion model, including Asian options, lookback options, single- or double-barrier options, perpetual American options with fixed or special time-dependent strikes (i.e., stock loans), and occupation-time related options such as step, quantile and corridor options. These solutions can be obtained primarily because we manage to derive analytical solutions to the distributions of the first passage times, the occupation times, and the integral of the underlying asset price process. It is worth mentioning that for Asian option pricing, in a simpler and more robust way, we extend the elegant analytical approach of Geman and Yor (1993) from the Black-Scholes model to a quite general jump diffusion model. Numerical results illustrate that our pricing methods are accurate and efficient.

Pricing and hedging in carbon emissions markets
Umut Cetin, London School of Economics, UK

We propose a model for trading in emission allowances in the EU Emission Trading Scheme (ETS). Exploiting an arbitrage relationship we derive the spot prices of carbon allowances given a forward contract whose price is exogenous to the model. The modeling is done under the assumption of no banking of carbon allowances (which is valid during the Phase I of Kyoto protocol), however, we also discuss how the model can be extended when banking of permits is available. We employ results from filtering theory to derive the spot prices of permits and suggest hedging formulas using a local risk minimization approach. We also consider the effect of intermediate announcements regarding the net position of the ETS zone on the prices and show that the jumps in the prices can be attributed to information release on the net position of the zone. We also provide a brief numerical simulation for the price processes of carbon allowances using our model to show the resemblance to the actual data.

Discrete-time approximation of obliquely reflected BSDEs
Jean-Francois Chassagneux, Universite D'Evry, France

We study the regularity and discrete-time approximation of a class of multi-dimensional obliquely reflected BSDEs introduced by Hu and Tang (2009). These BSDEs are linked to the valuation of "switching problem''. Introducing the notion of discretely obliquely reflected BSDEs, we provide a control of the L2-time regularity of the Z component of the BSDEs. We then use a natural scheme based on oblique projections and prove a control on the error of the algorithm. In the particular case when the driver of the BSDE does not depend on the Z component, we obtain a control of the error of order 1/4. This is joint work with R. Elie (Université Paris Dauphine) and I. Kharroubi (Université Paris 7).

Processes of class Sigma, last passage times and drawdowns
Patrick Cheridito, Princeton University, USA

We propose a general framework to study last passage times, suprema and draw downs of a large class of stochastic processes. A central role in our approach is played by processes of class Sigma. After investigating convergence properties and a family of transformations that leave processes of class Sigma invariant, we provide three general representation results. The first one allows to recover a process of class Sigma from its final value and the last time it visited the origin. In many situations this gives access to the distribution of the last time a stochastic process hit a certain level or was equal to its running maximum. It also leads to a formula recently discovered by Madan, Roynette and Yor expressing put option prices in terms of last passage times. Our second representation result is a stochastic integral representation of certain functionals of processes of class Sigma, and the third one gives a formula for their conditional expectations. From the latter one can deduce the laws of a variety of interesting random variables such as running maxima, drawdowns and maximum drawdowns of suitably stopped processes. As an application we discuss the pricing and hedging of options that depend on the running maximum of an underlying price process and are triggered when the underlying price drops to a given level or alternatively, when the drawdown or relative drawdown of the underlying price attains a given height.

Dynamic CAPM with risk measures
Patrick Cheridito, Princeton University, USA

Dynamic equilibrium prices are derived for options in incomplete markets for agents with mean-risk preferences. Explicit formulas are given for the equilibrium pricing measure in a discrete-time model, and the corresponding option prices are discussed.

Modeling and measuring systemic risk
Rama Cont, Columbia University, New York and CNRS, France

The financial crisis has underlined the crucial role of systemic risk, as well as the lack of a coherent framework for measuring and discussing this important issue.
We propose a quantitative measure - the Systemic Risk Index - for assessing the systemic risk posed by large financial institutions, using a network-based model of the financial system. This indicator combines counterparty contagion effects with the effect of correlated market shocks across financial institutions balance sheets. We discuss some theoretical properties of the Systemic Risk Index and show how it can be estimated in practice using a database of interbank exposures obtained from the Brazilian central bank.
We illustrate the usefulness of our approach by exploring the impact of credit default swaps on systemic risk and discuss whether a clearinghouse for such contracts can help mitigate systemic risk and default contagion. Our results underline the complexity of the issues at hand when discussing systemic risk and plead for an analysis taking into account counterparty network structure.

This talk draws on joint work with E Bastos (Banco Central, Brasil), Andreea MINCA (Universite de Paris VI) and Amal Moussa (Columbia University).

Leverage management in a bull-bear switching market
Min Dai, National University of Singapore

We characterize an investor's optimal trading strategy with finite horizon and transaction costs in an economy that switches stochastically between two market conditions. We find that whether the optimal strategy involves deleveraging depends crucially on the switch intensity from a ``bear" market with a negative risk premium to a "bull" market with a positive risk premium. If the switch intensity is high, leverage may be optimal even in the "bear" market and no deleveraging activity is needed. On the other hand, if this switch intensity is low, the investor's may choose not to leverage at all in either market.

Counterparty risk via Bessel bridge
Mark H. A. Davis, Imperial College, UK

Counterparty risk concerns evaluating potential losses on a financial trade conditioned on one's counterparty defaulting at a specific time in the future. Often, default can be modelled as the hitting time of a barrier by Brownian motion. Generally this barrier is not flat, but we show how to reformulate the problem so that it is flat. Then conditioning on default at a specific time is equivalent to replacing the BM by a BES(3) bridge. We suppose that the factors determining the value of the trade are further BMs, correlated with the first. Then we can calculate the conditional value of the trade in terms of SDEs whose 'inputs' are the Bessel bridge and additional independent BMs. An example relating to interest rate swaps will be presented.

Irrational borrowers and the pricing of residential mortgages
Yongheng Deng, National University of Singapore

Mortgage terminations arise because borrowers exercise options. This paper investigates the nonoptimal and apparently irrational behavior of those borrowers who do not terminate their mortgages even when the exercise value of the option is deeply in the money. We develop an option-based empirical model to analyze this phenomenon -- the behavior of irrational or boundedly rational "woodheads." Of course we do not observe "woodheads" explicitly in any body of data. Instead, we analyze correlates of unobserved heterogeneity within a large sample of mortgage holders. We develop a three-stage maximum likelihood (3SML) estimator using martingale transforms to estimate the competing risks of mortgage prepayment and default, recognizing unobserved heterogeneity which in part generates the behavior of "woodheads." The extended model is clearly superior to alternatives on statistical grounds. We then analyze the economic implications of this more powerful model. We analyze the predictions of the model for the valuation and pricing of mortgage pools and mortgage-backed securities. Based upon an extensive Monte Carlo simulation, we find that the 3SML model yields prices for seasoned mortgage pools that deviate quite substantially from more primitive estimates. The results indicate the empirical importance of heterogeneity and the implications of non-optimizing behavior for the valuation and pricing of mortgages and mortgage-backed securities. This is joint work with JOHN M. QUIGLEY.

Clustered defaults
Jin-Chuan Duan, National University of Singapore

Defaults in a credit portfolio of many obligors or in an economy populated with firms tend to occur in waves. This may simply reflect their sharing of common risk factors and/or manifest their systemic linkages via credit chains. One popular approach to characterizing defaults in a large pool of obligors is the Poisson intensity model coupled with stochastic covariates, or the Cox process for short. A constraining feature of such models is that defaults of different obligors are independent events after conditioning on the covariates, which makes them ill-suited for modeling clustered defaults. Although individual default intensities under such models can be high and correlated via the stochastic covariates, joint default rates will always be zero, because the joint default probabilities are in the order of the length of time squared or higher. In this paper, we develop a hierarchical intensity model with three layers of shocks -- common, group-specific and individual. When a common (or group-specific) shock occurs, all obligors (or group members) face individual default probabilities, determining whether they actually default. The joint default rates under this hierarchical structure can be high, and thus the model better captures clustered defaults. This hierarchical intensity model can be estimated using the maximum likelihood principle. A default signature plot is invented to complement the typical power curve analysis in default prediction. We implement the new model on the US corporate bankruptcy data and find it far superior to the standard intensity model both in terms of the likelihood ratio test and default signature plot.

Extremes from meta distributions and the shape of their sample clouds
Paul Embrechts, Swiss Federal Institute of Technology (ETH) Zurich, Switzerland

Meta distributions are multivariate models constructed and fitted in two stages: marginal fitting followed by dependence (copula) modelling. These models are being used in numerous applications, ranging from biostatistics, reliability to insurance and finance. Perhaps the most notorious example is the so called Li (Gaussian) copula model for CDO pricing. In this talk I will discuss some mathematical properties of this class of models mainly related to the behaviour of extremes. This talk is based on joint work with Natalia Lysenko (ETH Zurich) and Guus Balkema (University of Amsterdam).

Design of cap and trade schemes
Max Fehr, London School of Economics and Political Sciences, UK

Protocol in 2004 become popular environmental instruments. However recent price development of carbon allowances in the EU Emission Trading Scheme and its impact on European electricity prices exhibits the importance of a clear understanding of such Trading Systems. To this end we propose a stochastic equilibrium model for the joint price formation of allowances and products, whose production causes pollution. It turns out that for any cap and trade scheme, designed in the spirit of the EU ETS, the consumers burden exceeds by far the overall reduction costs, giving rise for huge windfall profits. Following this insight we show how to adapt the regulatory framework of emission trading schemes in such a way that windfall profits are reduced.

Analysis of a penalty method for pricing a guaranteed minimum withdrawal benefit (GMWB)
Peter A. Forsyth, University of Waterloo, Canada

The no arbitrage pricing of Guaranteed Minimum Withdrawal Benefits (GMWB) contracts results in a singular stochastic control problem which can be formulated as a Hamilton Jacobi Bellman (HJB) Variational Inequality (VI). Recently, a penalty method has been suggested for solution of this HJB variational inequality (Dai et al, 2008). This method is very simple to implement. In this talk, we present a rigorous proof of convergence of the penalty method to the viscosity solution of the HJB VI. Numerical tests of the penalty method are presented which show the experimental rates of convergence, and a discussion of the choice of the penalty parameter is
also included. A comparison with an impulse control formulation of the same problem, in terms of generality and computational complexity, is also presented.

Numerical methods for Hamilton Jacobi Bellman equations in finance
Peter A. 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: 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
Lecture 3: Pension Plan Asset Allocation, Passport Options
Lecture 4: Gas Storage

Socially efficient discounting under ambiguity aversion
Christian Gollier, University Toulouse I, France

We consider an economy with an ambiguity-averse representative agent who faces uncertain consumption growth. We examine conditions under which ambiguity aversion reduces the socially efficient discount rate. It is shown that ambiguity aversion affects the interest rate in two ways. The first effect is an ambiguity prudence effect, similar to the prudence effect that prevails in the expected utility model. In contrast, it requires decreasing ambiguity aversion in order to be signed. The second effect is that ambiguity also entails pessimism. But this implicit shift in beliefs generally has an ambiguous effect on the interest rate. We provide sufficient conditions under which ambiguity aversion does indeed decrease the socially efficient discount rate. The calibration of the model suggests that the effect of ambiguity aversion on the way we should discount distant cash flows is potentially large.

How to price Asian temperature risk
Wolfgang Härdle, Humboldt-Universität zu Berlin, Germany

Weather influences our daily lives and choices and has an enormous impact on corporate revenues and earnings. Weather derivatives differ from most derivatives in that the underlying cannot be traded and the market for weather derivatives is relatively illiquid. The weather derivative market is therefore incomplete. This paper implements a pricing methodology for weather derivatives that can increase the precision of measuring weather risk. We have applied continuous autoregressive models (CAR) with seasonal variation to model the temperature in Berlin in order to get the explicit nature of non-arbitrage prices for temperature derivatives. We infer the implied market price from Berlin cumulative monthly temperature futures that are traded at the Chicago Mercantile Exchange (CME), which is an important parameter of the associated equivalent martingale measures used to price and hedge weather futures/options in the market. We have studied the market price of risk, not only as a piecewise constant linear function, but also as a time dependent object. In all of the previous cases, we found that the market price of weather risk is different from zero and shows a seasonal structure related to the seasonal variance of the temperature process. With the information extracted we price other exotic options, such as cooling/heating degree day temperatures and non-standard maturity contracts.

Cross hedging with stochastic correlation
Gregor Heyne, Humboldt-Universität zu Berlin, Germany

This talk is concerned with the study of quadratic hedging of contingent claims with basis risk. We extend existing results by allowing for the correlation between the hedging instrument and the underlying of the contingent claim to be random itself. We assume that the correlation process evolves according to a stochastic differ-ential equation with values between the boundaries -1 and 1. We keep the correlation dynamics general and derive a simple integrability condition on the correlation pro-cess that allows to describe and compute the quadratic hedge by means of a simple hedging formula that can be directly implemented. Furthermore we show that the conditions on correlation process are fulfilled by a large class of dynamics. The theory is exemplified by various, explicitly given correlation dynamics.

Modeling risk-neutral allowance price evolution with applications to option pricing
Juri Hinz, National University of Singapore

The climate rescue is on the top of many agendas. In this context, emission trading schemes are considered as promising tools. The regulatory framework of an emission trading scheme introduces a market for emission allowances and creates need for risk management by appropriate financial contracts. We address logical principles underlying the fair valuation of such derivatives. Starting from the equilibrium of a market with risk averse players, we show that the risk-neutral allowance price dynamics can be characterized in terms of a fixed point equation which plays the same role as the central planer optimal control problem for the non-risk averse situation. We show that derivatives valuation is naturally addressed in and can be obtained in this setting.

Backward stochastic equations and equilibrium pricing in incomplete financial markets
Ulrich Horst, Humboldt-Universität zu Berlin, Germany

The problem of equilibrium pricing in dynamically incomplete financial markets is one of the oldest problems in mathematical economics. The problem of equilibrium pricing is well understood for the benchmark case of complete markets where all risk factors can be hedged using the available assets. When markets are in-complete the situation is more involved, and to date no united approach to incomplete markets is available. In this talk we review some recent results on equilibrium pricing in incomplete markets in discrete time when the market participants evaluate their risk exposures using dynamic risk measures. For such market situations we establish
existence and uniqueness of equilibrium results and show that the problem of dynamic equilibrium pricing can be reduced to a recursive sequence of static one-period problems. When the flow of information is generated by independent random walks the equilibrium dynamics can be described by a coupled system of backward stochastic difference equations which renders our approach easily amenable to numerical simulations. We also comment on some of the mathematical challenges that currently prevent us from translating our results from discrete to continuous time, including the lack of existence and differentiability of solutions results for coupled systems of backward stochastic differential equations with non Lipschitz continuous drivers.

Equilibrium pricing in incomplete markets under translation invariant preferences
Ulrich Horst, Humboldt-Universität zu Berlin, Germany

We consider a partial equilibrium framework within to price financial securities in dynamically incomplete markets in discrete time when the agents have translation invariant preference functionals. We show the existence of equilibrium is equivalent to the existence of a representative agent which, in turns, is satisfied if the agents are sensitive to large losses. It turns out that the analysis of equilibrium in dynamic models can be reduced to a recursive sequence of static equilibrium models and when the flow of information is generated by independent random walks the equilibrium dynamics can be described by a coupled system if backward stochastic difference equations. This renders our theoretical analysis readily amenable to numerical simulations.

The talk is based on joint work with P. Cheridito, M. Kupper and Train Pirvu

Games with exhaustible resources
Sam Howison, University of Oxford, UK

Two producers own finite resources of a commodity (for example mineral water). It costs nothing to produce, and there is an alternative technology (say, desalination) which allows production at a nonzero cost. The producers compete in a Cournot market and the goal is to determine their optimal production strategies in a continuous-time Markov-perfect setting, including the option of producing using the alternative technology. The resulting differential game leads to a strongly nonlinear pair of coupled partial differential equations with some unexpected properties. I shall describe the model and our work on it (which is very much in progress, with many open issues). Joint work with Chris Harris (Cambridge) and Ronnie Sircar (Princeton).

Arbitrage models of commodity prices
Rachid Id Brik, ESSEC Business School, France

Arbitrage models of commodity prices are relative pricing devices. They allow market operators to evaluate commodity-linked securities in terms of quoted commodity prices or indices. We present a comprehensive overview of these models and provide the participant with recent advances on modeling issues. Concrete examples of models illustrate the underlying concepts and methods.

Behavioural optimal liquidation ----- a model for disposition and break-even effect
Hanqing Jin, University of Oxford, UK

In this paper, we studied an optimal selling/liquidation problem of a risky asset in a finite time horizon for an agent with some simplified behavioral objective, which is motivated from the observation that people prefer to realize gain earlier and defer loss later. The optimal solution turns out to be consistent with the disposition and break-even phenomenon observed in financial markets, which were studied by some complicated model in literature. We also studied how the optimal solution will change under different parameters, which make the optimal decision not always suggest those special phenomena.

Exercise boundary of the American put option in the black-Scholes model with discrete dividends
Benjamin Jourdain, Ecole Nationale des Ponts et Chaussées, France

We are interested in the American Put option in the Black-Scholes model when a discrete dividend computed as a function D of the pre-dividend stock level is paid at some time s prior to the maturity T of the option. The exercise region at time t < s is an interval (0,c(t)] with boundary c(t). When D is a positive concave function such that x-D(x) is non-negative, we study the behavior of the exercise boundary c(t) on a left-hand neighbourhood of the dividend times.

Risk preferences and their robust representations
Michael Kupper, Humboldt Universität zu Berlin, Germany

To answer the need for a rational assessment of risk, apart many descriptive attempts conducting to ad-hoc measure instruments (mean-variance, quantile, index of riskiness, etc.), two major normative approach were undertaken: in the mid 20th century with the utility theory of von Neumann and Morgenstern on the level of probability distributions, and in the end of the 20th century with the so called sub-additive monetary risk measures introduced by Artzner et al. on the level or random variables. These two paradigms (as well as the descriptive approaches) lead to very different objects which however share the denomination of risk.

Motivated by a recent work on the duality of quasi-convex functionals for bounded random variables by Cerreia-Vioglio et al., we study the concept of risk preference orders in a very general setting and characterize risk only structurally by the concept of diversification without (deliberately) paying attention to the nature of the objects. After a general study of the global properties of the risk preference orders and the related quasi-convex monotone functionals, we provide a robust representation in a dual sense. This generality allows us to interpret the risk under different lights depending on the specification of the setting, and in particular to cover all the previous concepts of risk measurement. We discuss then several examples in different settings. It is joint work with Samuel Drapeau.

Optimal stopping with multiple priors and variational expectations in continuous time
Frank Riedel, Bielefeld University, Germany

We develop a theory of optimal stopping for variational preferences (convex risk measures). We construct a Snell envelope along the classical lines; for multiple priors (kappa--ambiguity, coherent risk measures), we identify the worst case prior. We show how to extend the classical Hamilton--Jacobi--Bellman equation to our case. We then apply our results to a number of examples in economics and finance.

Arbitrage models of commodity prices
Andrea Roncoroni, ESSEC Business School, France

Arbitrage models of commodity prices are relative pricing devices. They allow market operators to evaluate commodity-linked securities in terms of quoted commodity prices or indices. We present a comprehensive overview of these models and provide the participant with recent advances on modeling issues. Concrete examples of models illustrate the underlying concepts and methods.

Order book resilience, price manipulations, and the positive portfolio problem
Alexander Schied, Mannheim University, Germany

We consider a class of limit order book models, extending the block-shaped, exponential-resilience model introduced by Obizhaeva and Wang (2005). An important question is the viability of these models. Examples show that the requirement of the absence of price manipulation strategies as introduced by Huberman & Stanzl (2004) is not sufficient to guarantee that the model is indeed well-behaved, as these examples admit price manipulation strategies in a weak sense.
For block-shaped limit order book models we can characterize those resilience functions for which there are no price manipulation strategies in both the usual and the weak sense. We also discuss the case of nonlinear price impact and exponential resilience. Here it turns out that the optimal strategy is extremely robust with respect to variations of the shape of the limit order book. The talk is based on joint work with A. Alfonsi and A. Slynko.

Large deviations controls for long-term investment
Jun Sekine, Kyoto University, Japan

A new representation and an extension for large deviations control problems are introduced in complete market setting. Linear model examples are discussed in detail, and the following observations will be focused:
(1) important roles of algebraic Riccati equations, similar to those in linear robust control problem,
(2) "partial" breakdown phenomenon when investor's attitude is too risk-averse,
(3) prior-distribution-free expression for partial information problem.

Flexibility premium in marketable permits
Luca Taschini, London School of Economics, UK

We study the market for emission permits in the presence of reversible abatement measures characterized by delay in implementation. We assume that the new operating profits follow a one-dimensional geometric Brownian motion and that the company is risk-neutral. The optimal "investment" and "disinvestment" policy for reversible abatement options is evaluated under both instantaneous and Parisian criteria, nesting the previous instantaneous models. By taking the difference between these two values at their respective optima, we derive an analytic solution of the premium for flexibility embedded in marketable permits. This extends the findings in the literature on marketable permits. Numerical results are presented to illustrate the likely magnitude of the premium and how it is affected by uncertainty and delays in
implementation.

Optimal calibration of the LIBOR market model
Lixin Wu, Hong Kong University of Science and Technology, Hong Kong

We claim to have developed the optimal methodology for the non-parametric calibration of market model to the prices of at-the-money (ATM) caps/floors and swaptions, as well as to the historic correlations of the LIBOR rates. We take the approach of divide-andconquer: first fit the model to historic correlations, then to the implied Black volatilities of the input options. Regularization is adopted and the calibration is cast into minimization maximization problems and then solved by the method of Lagrange multiplier. During the solution process we have utilized the quadratic functional form of both objective function and constraints: we solve the inner maximization problems with a single matrix eigenvalue decomposition, and solve the outer minimization problem by a Hessian-based descending method. The well-posedness of the Lagrange multiplier problems and the convergence of the descending methods are rigorously justified. Numerical results show that we have achieved very quality calibration. We will also discuss how to incorporate this calibration technology into the calibration of market models with stochastic volatilities.

Risk aversion and portfolio selection
Jianming Xia, National University of Singapore

The comparative statics of the optimal portfolios across individuals is carried out for a continuous-time market model, where the risky assets price process follows a joint geometric Brownian motion with time-dependent and deterministic coefficients. It turns out that the indirect utility functions inherit the order of risk aversion (in the Arrow-Pratt sense) from the von-Neumann-Morgenstern utility functions, and therefore, a more risk-averse agent would invest less wealth (in absolute value) in the risky assets.

Optimal stopping with prospect preference
Zuoquan Xu, University of Oxford, UK

Prospect theory, featuring S-shaped utility (value) function and probability distortion, proposed by Kahneman and Tversky (1979) has been widely accepted as a successful supplement and extension of traditional expected utility theory. In this paper, we study general optimal stopping with prospect preference problems. All the three key features in prospect theory are considered in our problems. In particular, probability distortion is considered in such problems for the first time. Probability distortion has destroyed the time-consistency structure of the problems which leads to the failure of using dynamic programming and stochastic control approach to solve the problems. With the help of distribution/quantile formulation and Skorokhod embedding theorem, we propose a three-step procedure to solve the problems. The optimal stopping times turn out to be highly depending on the shapes of the utility function and probability distortion function. In particular, we have re-found and generalized the results that have been obtained without considering probability distortion in the literature. The main contribution of this paper is tackling the time-inconsistency arising from the probability distortion in the optimal stopping problems.

Callable stock loans and beyond
Sheung Chi Phillip Yam, The Hong Kong Polytechnic University, Hong Kong

A stock loan is a loan in which the borrower, who owns one share of a stock, obtains a loan from the lender with the stock as a collateral. In their work, Xia and Zhou (2007) provided the ¯rst quantitative analysis of stock loans under the Black-Scholes framework and determined the fair price charged by the lender for providing such a service. In this talk, I shall consider the pricing issue of stock loans with a callable feature that lender can call back the loan at any time before maturity; upon calling the loan, lender has the right to enforce the borrower either to immediately redeem the stock by paying back the loan at a reduced amount or surrender his share of stock. Financial products with such a feature are commonly traded under the name: Callable REPO. Explicit solution together with range of loan-to-value ratio for marketable REPOs will be illustrated in infinite time horizon setting; while for the finite time counterpart, a couple of integral equations characterizing the two exercising boundaries will be shown. On the other hand, in a recent work of Kunita and Seko (2007), they attempted to identify the exercising region of game call options (with ±-penalty) with ¯finite time to maturity. In this talk, I shall also illustrate a complete solution to the same problem which is in contrast to their expected results; indeed, by applying similar method, we had shown the non-trivial nature of the pair of exercising boundaries of the corresponding optimal stopping game (Dynkin's game). This is joint work with S. P. Yung and W. Zhou.

Risk measures with comonotonic subadditivity or convexity and respecting stochastic orders
Jia-An Yan, Chinese Academy of Sciences, China

This paper proposes some new classes of risk measures, which are not only comonotonic subadditive or convex, but also respect the (first) stochastic dominance or stop-loss order. We give their representations in terms of Choquet integrals w.r.t. distorted probabilities, and show that if the physical probability is atomless then a comonotonic subadditive (resp. convex) risk measure respecting stop-loss order is in fact a law-invariant coherent (resp. convex) risk measure. This is joint work with Yongsheng Song.

Time-inconsistent optimal control problems
Jiongmin Yong, University of Central Florida, USA

Common sense tells us that people are changing their minds/objectives frequently. Also, the environment (in the broad sense) is changing as time goes by. When these issues are taken into account in studying optimal controls, one will lead to some time-inconsistent optimal control problems. In this talk, we will present a general framework on time-inconsistent optimal control problems. It is shown by some simple examples that naive optimal controls are not time-consistent. Using some sophisticated approach by means of hierarchical differential games, one can obtain time-consistent solutions to the time-inconsistent problems in certain sense.

Finding quantiles
Xunyu Zhou, University of Oxford, UK

Existing portfolio choice models in continuous time typically reduce to finding optimal terminal cash flows which are random variables. While it works for expected utility maximisation, it generally fails to work for models with non-expected utility criteria, such as the goal-achieving model, Yaari's dual model, Lopes' SP/A model, behavioural model under prospect theory, models with coherent risk measures, as well as optimal stopping with probability distortions. This talk reviews the latest development in solving these non-classical models by changing decision variables - from random variables to their quantile functions.

Singular stochastic control with partial information of Itô-Lévy processes and associated optimal stopping
Bernt Øksendal, University of Oslo, Norway

In the first part of this talk we study general singular control problems of Itô-Lévy processes, in which the controller has only partial information to her disposal and the system is not necessarily Markovian. We present two different approaches to obtain solution methods for such problems:

(i) by using Malliavin calculus, leading to generalized variational inequalities for partial information singular control (of possibly non-Markovian systems)

(ii) by introducing a singular control version of the Hamiltonian and using backward stochastic differential equations (BSDEs) to obtain a partial information maximum principle for such problems.

We show that the two methods are related, and we find a connection between them.

In the second part of the paper we study the relation between the generalized variational inequalities found in (i) above and general reflected backward stochastic differential equations (RBSDEs) for Itô-Lévy processes. These are again shown to be equivalent to general optimal stopping problems for such processes. Combining this we get a connection between singular control and optimal stopping in this general setting.

In the third part we illustrate the results by studying some examples.

The presentation is based on recent joint work with Agnès Sulem, INRIA, Paris.

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