MITACS-PIMS-UBC Summer School in Risk Management and Risk Sharing

GRADUATE COURSE 1

 

Instructor: Bruno Bouchard (University Paris-Dauphine) Title: Portfolio management under risk constraint.

 

Course Material and Topics:

The aim of this lecture is to discuss classical and recent techniques used for the hedging and pricing risks.

 

We will start with the classical dual approach for financial markets, which allows to rewrite super-hedging problems in terms of optimal control problems in standard form. Based on this, we shall then consider hedging and pricing problems under utility or risk minimization criteria. This approach will turn out to be powerful whenever linear (or essentially linear) problems are considered, but not adapted to more general settings with non-linear dynamics (e.g. large investor models, high frequency trading with market impact features, mixed finance/insurance issues).

 

In the second part of this lecture, we will develop on a new approach for risk control problems based on a stochastic target formulation. We will see how flexible this approach is and how it allows to characterize very easily super-hedging prices in term of suitable Hamilton-Jacobi-Bellman type partial differential equations (PDEs). We will then see how quantile hedging and expected loss pricing problems can be embeded into this framework, for a very large class of financial models.

 

The third part will be dedicated to optimal management problems under risk constraints. Based on the results of the previous part, we shall see how they can naturally be embeded into optimal control problems with state constraints. Here the state constraint formulation is somehow unusual as it will be given in terms of a corresponding stochastic target problem associated to the risk constraint.

 

Many practical problems will be discussed as examples of applications.

 

The course will cover the following topics The classical dual approach: Dual formulation for super-hedging. Utility maximization and risk minimization. Indifference pricing/hedging. Risk minimization and quantile type pricing and hedging. Stochastic targets and risk controlled hedging problems: Problem formulation and examples. Direct dynamic programming. Viscosity solutions. PDE characterization. Examples, comparison and verification arguments. Optimal management under risk control : State constraint formulation. Dynamic programming. PDE characterization (Hamilton-Jacobi-Bellman equation with constraints). Examples, comparison and verification arguments

 

Learning Objectives:

 

1. Mathematics: apart from stochastic calculus tools, the students will learn how to discuss optimal control problems both by using convex duality and PDE (partial differential equation) approaches. In particular, a notion of weak solutions of PDEs, namely viscosity solutions, will be introduced. They will also become familiar with the recent literature on stochastic target problems.

 

2. Economics and finance: the students will learn recent and powerful techniques for the control of risks and optimal portfolio management under risk constraints.

 

Evaluation:

The instructional format for the course will consist of 23 lectures. To ensure regular progress of the students, there will be small classes and one homework per week, counting towards 50% of the final mark. There will be a final examination, counting for the remaining 50%

 

Rationale for Course:

Optimal management under risk constraints problems have a long history. In the most simple models, they can be handled by using either the so-called dual approach or through standard stochastic optimal control techniques. Both approaches are based on the dual formulation for super-hedging prices which are valid under strong linear assumptions on the underlying models. In particular, they do not allow to handle recent models in which traders have an impact on the price dynamics. They neither allow to discuss more general situations coming from insurance or mixed finance/insurance considerations. In these lectures, we shall see how very general situations can be handled via the theory of stochastic target problems. This new tool allows a systematic and powerful analysis of growing practical issues.

 

Textbooks and References:

Bouchard B., R. Elie and C. Imbert, Optimal Control under Stochastic Target Constraints, to appear in SIAM Journal on Control and Optimization.

 

Bouchard B., R. Elie and N. Touzi, Stochastic target problems with controlled loss, SIAM Journal on Control and Optimization, 2009, 48 (5), 3123-3150.

 

Follmer H. and P. Leukert, Quantile Hedging. Finance and Stochastics, , 1999, 3 (3), 251-273.

 

Follmer H. et P. Leukert, Efficient hedging : cost versus shortfall risk, Finance and Stochastics, 2000, 4, 117-146.

 

Follmer H. and A. Schied, Stochastic finance : An introduction in discrete time, 2004, De Gruyter editions.

 

GRADUATE COURSE 2

Instructor: Ivar Ekeland (UBC)

Title: Large Asymmetry of information and risk sharing in finance

 

Course Material and Topics:

Economic theory classifies asymmetry of information into two types: adverse selection and moral hazard. Adverse selection occurs when parties to a transaction have private information which they do not want to share (typically, an insurer would like to know more about his prospective clients than they would be willing to tell). Moral hazard occurs when parties to a transaction can undertake actions which the other parties will not detect, but which will affect the outcome (typically, an insurer will worry that too complete a coverage will induce reckless behaviour).

 

The course will cover the following topics:

 

1. Asymmetry of information: Adverse selection (hidden information), moral hazard (hidden action). The principal-agent problem. Incentive-compatibility constraints. First-best contracts versus second-best contracts.
2. Adverse selection: Expressing the incentive-compatibility constraint in a static (one-shot) framework. The case of one-dimensional types and the Spence-Mirrlees condition. The case of multidimensional types: generalized convexity $u$-convex functions, $u$-conjugates and their properties. The principal's problem as an optimization problem with convexity constraints. The Rochet-Chone problem. Selling risk over the counter to investors with different degrees of risk aversion. The case of an informed principal. Numerical algorithms for solving adverse selection problems.
3. 
   Moral hazard: Limited liability in finance: a dynamic model with continuous time and infrequent but costly accidents. The martingale representation theorem. Expressing the incentive-compatibility constraint. Description of the optimal (second-best) contract. Extension of the method to other situations in corporate finance.

 

Learning Objectives:

1. Mathematics: the students will learn fundamental techniques of optimization ($u\$-convex analysis, functional analysis calculus of variations), and stochastic analysis (the martingale representation theorem)


2. Economics and finance: the students will learn fundamental techniques for handling asymmetry of information, and apply them in a financial context.

The course will train them to detect and manage an important aspect of risk management, namely that part of the risk which arises from human behaviour: the propensity of employees and managers to look after their own interests rather than that of the shareholders.

 

Evaluation:

The instructional format for the course will consist of 23 lectures. To ensure regular progress of the students, there will be one homework per week, counting towards 50% of the final mark. There will be a final examination, counting for the remaining 50%

 

Rationale for Course:

The recent ongoing economic crisis, which started as a financial crisis, has highlighted the importance of informational asymmetry: generally speaking, banks took more risk than investors and shareholders would have wanted, and that they themselves realized. Risk management has now moved to center stage in the practice and theory of finance. It is crucial that graduates in mathematical finance, whether they seek a job in industry, or want to do research in the field, are familiarized with the various aspects of risk, and learn to take it into account when pricing and designing assets. This course covers one of the most important aspects of risk management, namely the one that arises from human behaviour

 

Textbooks and References:

1. Biais, Mariotti, Rochet and Villeneuve, Large risks, limited liability and dynamic moral hazard, Econometrica 2009
2. Carlier, Ekeland, and Touzi, Optimal derivatives design for mean-variance agents under adverse selection
   Mathematics and Financial Economics, 1, 1 (April 2007), pp. 57-80.
3. Sannikov, A continous-time version of the principal-agent problem, Review of Economic Studies Oksendal, "Stochastic differential equations", Springer.

 

 

Optimal High Frequency Trading

 

Charles-Albert Lehalle, CA Chevreux, groupe Calyon Mathieu Rosenbaum, Ecole Polytechnique Paris

 

 

The financial crisis and the new regulations (Reg NMS in the US and MiFID in Europe) have conduced financial institutions to look more carefully at their execution costs (brokerage), trading (hedging) and arbitrage (high frequency arbitrage) activities. Indeed, the always higher complexity of the orders execution process on electronic markets implies to optimize the trading strategies :

 

-In time : On the one hand, buying or selling a lot of shares too quickly has a so called market impact, which can be an important cost for the trader. On the other hand, if the trader is too slow, he will face the market risk.

 

-In place: the trader has to choose an (some) optimal electronic platform(s) where he will be able too find liquidity, even though most platform hide most of the liquidity information (explicitly like dark pools, or implicitly, when high frequency players interacts within the pool).

 

Since the end of the 90's some mathematical works have provided a clear mathematical framework enabling to solve the typical trader's problem of buying or selling a large quantity of shares over a given time period, leading to optimal strategies.

 

In the high frequency context, designing such strategies in practice requires an accurate statistical estimation of some relevant market parameters. This is an intricate problem since usual statistical methods collapse when dealing with high frequency data.

 

The goal of this course is to explain both the optimal trading strategies and the associated statistical procedures, in an applied context.