9th Winter School on Mathematical Finance

Special Topics:
numerical methods and illiquid markets

The Winter School is supported financially by AMaMeF, NWO, Thomas Stieltjes Institute, MRI and CentER

Training will be provided by a number of renowned international lecturers at the conference center "De Werelt" in the heart of the beautiful forest of the Veluwe.

Special price CANdiensten - EURO 1.700,- VAT included

minicourses

• Peter Forsyth, (University of Waterloo, Waterloo)
  Numerical methods for Hamilton-Jacobi-Bellman equations in Finance
• Alexander Schied, (Mannheim University, Mannheim)
  Market impact models and optimal execution

special invited lectures

• Pauline Barrieu (London School of Economics, London)
  Robust asset allocation under model uncertainty
• Mark Davis (Imperial College, London)
  Risk-sensitive asset management with jump-diffusion price processes
• Peter Tankov (Ecole Polytechnique, Palaiseau )
  Discrete hedging in exponential Lévy models

short lectures

• Dion Bongaerts (Universiteit van Amsterdam)
  Corporate bond liquidity and the credit spread puzzle
• Alexander van Haastrecht (Universiteit van Amsterdam)
  Valuation of guaranteed annuity options using a stochastic volatility model for equity prices
• Vincent Leijdekker(Universiteit van Amsterdam, ABN AMRO)
  Sample-path large deviations in credit risk
• Mitja Stadje (Eurandom, Eindhoven)
  Extending time-consistent risk measures from discrete time to continuous time: a convergence approach

schedule

minicourses

Peter Forsyth
Numerical methods for Hamilton-Jacobi-Bellman equations in Finance

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: Guaranteed Minimum Withdrawal Benefit (GMWB) Variable Annuity
Lecture 5: Gas Storage

Alexander Schied
Market impact models and optimal execution

We consider mathematical problems arising in illiquid markets or, more specifically, when trading asset positions that are large enough to move the underlying asset price. In dealing with this situation, we first need a suitable modeling framework. We will thus discuss several model classes proposed in the literature. It turns out that requiring that these models are free of arbitrage opportunities in the usual sense may not be enough, and we illustrate this fact by several examples. We therefore discuss additional requirements that should be met by any viable market impact model. The problem of model viability is actually closely related to finding strategies that unwind or acquire large asset positions in an optimal way. This problem is known as the optimal execution problem. We give an overview over some of the results that are known to date. We conclude by discussing the situation in which there are several market participants: a seller who needs to unwind a large asset position and a group of informed arbitrageurs trying to make a profit out of this.

special invited lectures

Pauline Barrieu
Robust asset allocation under model uncertainty

In the present paper, we propose a robust decision methodology, when there is some ambiguity concerning the potential future scenarii about decision variables, such as financial asset dynamics. The decision maker considers several prior models for those scenarii and displays an ambiguity aversion against them. We have developed a two step ambiguity robust methodology, that offers the advantage to be more tractable and easier to implement than the various approaches proposed in the literature. This methodology decomposes the ambiguity adjustment into a model specific ambiguity adjustment as well as a relative ambiguity adjustment for each of the considered models. The optimal solutions inferred by each prior are transformed through a generic absolute ambiguity function. Then, the transformed solutions are together mixed through a measure that reflects the relative ambiguity aversion of the decision maker for the different priors considered. This methodology is then illustrated through empirical study of asset allocation. Based on a joint work with Sandrine Tobelem.

Mark Davis
Risk-sensitive asset management with jump-diffusion price processes

Risk-sensitive asset management (RSAM) is in some sense 'dynamic Markowitz' - a trade-off between risk and return in a fully dynamic setting. In earlier work we studied a problem in which asset prices follow jump-diffusions where the mean rates of return are functions of an exogenous 'factor' process X_t which was supposed to be a diffusion (i.e. continuous paths). We showed that RSAM reduces to a stochastic control problem for a diffusion process, for which the Bellman equation has a classical solution. Here we consider the general case where X_t also has jumps. Then the associated stochastic control problem is one of controlled jump-diffusions. We show that the value function is a viscosity solution of the associated Bellman PIDE and derive smoothness properties of the solution. (Joint work with Sebastien Lleo)

Peter Tankov
Discrete hedging in exponential Lévy models

Most authors who studied the problem of option hedging in incomplete markets, focused on finding the strategies which minimize the residual hedging error. However, the resulting strategies are usually unrealistic because they require continuous trading. In practice, the portfolios are readjusted at discrete dates, which leads to a 'hedging error of the second type', due to the difference between the optimal portfolio and its discretized version. In this talk, we review some recent results on the structure of this discretization error in the context of exponential Lévy models. We discuss the rates of convergence of the error to zero when the number of trading dates increases, under different strategies and pay-off profiles, and show how this convergence can be improved with a suitable choice of readjustment dates.

short lectures

Dion Bongaerts
Corporate bond liquidity and the credit spread puzzle

This paper explores the role of expected liquidity and liquidity risk in the pricing of corporate bonds. In particular, we investigate to which extent liquidity can help to explain the credit spread puzzle. For our analysis, we use the TRACE data on an intra-day frequency. Since these data do not contain bid-ask-spreads, we estimate them latently using a Gibbs sampler in a similar spirit as in Hasbrouck (forthcoming). We find a significant premium on expected liquidity, but little evidence for liquidity risk. The liquidity premium goes a long way in explaining the credit spread puzzle.

Alexander van Haastrecht
Valuation of guaranteed annuity options using a stochastic volatility model for equity prices

Guaranteed Annuity Options are options providing the right to convert a policyholder's accumulated funds to a life annuity at a fixed rate when the policy matures. These options were a common feature in UK retirement savings contracts issued in the 1970's and 1980's when interest rates were high, but caused problems for insurers as the interest rates began to fall in the 1990's. Currently, these options are frequently sold in the U.S. and Japan as part of variable annuity products. The last decade the literature on pricing and risk management of these options evolved. Until now, for pricing these options generally a geometric Brownian motion for equity prices is assumed. However, given the long maturities of the insurance contracts a stochastic volatility model for equity prices would be more suitable. In this paper closed form expressions are derived for prices of guaranteed annuity options assuming stochastic volatility for equity prices and either a 1-factor or 2-factor Gaussian interest rate model. The results indicate that the impact of ignoring stochastic volatility can be significant.

Vincent Leijdekker
Sample-path large deviations in credit risk

The event of large losses plays an important role in a portfolio credit risk. As these large losses are typically rare, and portfolios usually consist of a large number of positions, large deviation theory is the natural tool to analyze the tail asymptotics of the probabilities involved. We first derive a sample-path large deviation principle (LDP) for the portfolio’s loss process, which enables the computation of the logarithmic decay rate of the probabilities of interest. In addition, we derive exact asymptotic results for a number of specific rare-event probabilities, such as the probability of the loss process exceeding some given function.

Mitja Stadje
Extending time-consistent risk measures from discrete time to continuous time: a convergence approach

The main aim of this talk is to present an approach to the transition from risk measures in discrete time to their counterparts in continuous time. After a general introduction to risk assessment in mathematical finance, it is shown that a large class of risk measures in continuous time can be obtained as limits of time-consistent risk measures in a discrete setting. The discrete-time risk measures are constructed from properly rescaled ('tilted') one-period risk measures, using a d-dimensional random walk converging to a Brownian Motion. Under suitable conditions (covering the classical one-period risk measures) we obtain convergence of the discrete risk measures to the solution of a backward stochastic differential equation, defining a risk measure in continuous time, whose driver can then be viewed as the continuous-time analogue of the discrete driver characterizing the one-period risk. We derive the limiting drivers for the semi-deviation risk measure, Average Value at Risk, and the Gini risk measure in closed form.

dates

18 January 2010, 3 days

price

Special price CANdiensten (including VAT) EURO 1.700
The registration fee includes accommodation (single room) for the nights of January 18 and 19, all meals starting with lunch on Monday up to and including lunch on Wednesday, and tea and coffee during breaks.

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