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. (slides of the lectures)
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. (slides of the lectures)
References (incomplete list, more in the slides of the lectures)
[1] Alfonsi, Aurélien, Fruth, Antje and Schied, Alexander: Optimal Execution Strategies in Limit Order Books with General Shape Functions. To appear in Quantitative Finance. SSRN preprint 1510104
[2] Alfonsi, Aurélien, Fruth, Antje and Schied, Alexander: Constrained portfolio liquidation in a limit order book. Banach Center Publ. 83, 9-25 (2008)
[3] Alfonsi, Aurélien and Schied, Alexander: Optimal Execution and Absence of Price Manipulations in Limit Order Book Models. SSRN preprint 1499209
[4] Alfonsi, Aurélien, Schied, Alexander and Slynko, Alla: Order Book Resilience, Price Manipulation, and the Positive Portfolio Problem. SSRN preprint 1498514
[5] Almgren, Robert: Optimal execution with nonlinear impact functions and trading-enhanced risk, Applied Mathematical Finance 10 (2003) 1-18.
[6] Almgren, Robert and Chriss, Neil: Optimal execution of portfolio transactions, J. Risk 3 (Winter 2000/2001) 5-39.
[7] Gatheral, Jim: No-Dynamic-Arbitrage and Market Impact. Quantitative Finance, Forthcoming. SSRN preprint 1531466
[9] Obizhaeva, Anna A. and Wang, Jiang: Optimal Trading Strategy and Supply/Demand Dynamics. NBER Working Paper No. W11444. SSRN preprint 752022
[10] Schied, Alexander and Schöneborn, Torsten: Risk Aversion and the Dynamics of Optimal Liquidation Strategies in Illiquid Markets. Finance and Stochastics 13, 181-204 (2009). SSRN perprint 1092028
[11] Schied, Alexander, Schoeneborn, Torsten and Tehranchi, Michael: Optimal Basket Liquidation for CARA Investors is Deterministic. To appear in Applied Mathematical Finance. SSRN preprint 1500277
[12] Schöneborn, Torsten: Trade execution in illiquid markets. Optimal stochastic control and multi-agent equilibria. Ph.D. thesis, TU Berlin, May 2008.
[13] Schöneborn, Torsten and Schied, Alexander: Liquidation in the Face of Adversity: Stealth vs. Sunshine Trading. SSRN preprint 1007014
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. (slides)
References
[1] Pauline M. Barrieu,
Sandrine Tobelem, Robust Decision under Model Uncertainty, SSRN preprint 1318579
[2] Sandrine Tobelem and Pauline Barrieu, Robust asset allocation under model risk, Risk, 01 May 2009.
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 Xt 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 Xt 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. (slides)
References
[1] Mark Davis and Sebastien Lleo,
Jump-diffusion risk-sensitive asset management, preprint arXiv:0905.4740
[2] Mark Davis and Sebastien Lleo,
Risk-sensitive asset management in a jump-diffusion factor model, preprint arXiv:1001.1379
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. (slides)
References
[1] Mats Broden and Peter Tankov, Errors from discrete hedging in exponential Lévy models:
the L2-approach (preprint).
[2] Peter Tankov and Ekaterina Voltchkova, Asymptotic analysis of hedging errors in models
with jumps (preprint)
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. (slides)
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. (slides)
Reference
Alexander van Haastrecht, Richard Plat and Antoon Pelsser, Valuation of Guaranteed Annuity Options using a
Stochastic Volatility Model for Equity Prices (preprint)
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 portfolios 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. (slides)
Reference
V. Leijdekker, M. Mandjes, and P. Spreij, Sample-path large deviations in credit risk
(preprint)
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.
(slides)
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