Minicourses

Eckhard Platen: A benchmark approach to investing, pricing and hedging

This mini-course introduces into the benchmark approach, which provides a general framework for financial market modelling. It allows for a unified treatment of portfolio optimization, derivative pricing and hedging, financial planning, insurance and risk management. It extends beyond the classical asset pricing theories, with significant differences emerging for portfolio optimization and long dated contracts. The Law of the Minimal Price will be presented for valuation. A Diversification Theorem allows forming an extremely well performing proxy for the numeraire portfolio. The richer modelling framework of the benchmark approach leads to the derivation of tractable, realistic models under the real world probability measure. It will be explained how the approach differs from the classical portfolio optimization approach, the standard risk neutral approach and the classical insurance approach. Examples on long term and extreme maturity derivatives will demonstrate the important fact that a range of contracts can be less expensively priced and hedged than suggested by classical theory. (slides of the course)

Topics:

• Best performing portfolio as benchmark
• Portfolio optimization
• Various approaches to asset pricing
• Benchmarked risk minimization
• The affine nature of diversified wealth dynamics

References

1. Platen, E. and Heath, D.: A Benchmark Approach to Quantitative Finance. Springer 2006/2010.
   
2. Platen, E. and Bruti-Liberati, N. : Numerical Solution of Stochastic Differential Equations with Jumps in Finance (Chapter 3). Springer 2010.
   
3. Baldeaux, J. and Platen, E.: Functionals of Multidimensional Diffusions with Applications to Finance. Springer 2013.
4. E. Platen and R. Rendek: Approximating the numérare portfolio by naive diversification (paper)
5. Ke Du and Eckhard Platen: Benchmarked Risk Minimization (paper)
6. Constantinos Kardaras, Jan Obloj, and Eckhard Platen: The numéraire property and long-term growth optimality for drawdown-constrained investments (paper)

Special invited lectures

David Hobson: Gambling in contests

In a recent paper in the Journal of Economic Theory Seel and Strack introduced a gambling contest. Each agent observes and independent copy of a diffusion and chooses when to stop it (based solely on the information from their own process). The winner of the contest is the agent whose stopped process has the highest value.
In this talk we rederive the form of the optimal strategy for the agents, and discuss several extensions. (Joint work with Han Feng)

References

Han Feng and David Hobson. Gambling in Contests with Regret, report

Agnès Sulem: Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps

In the Brownian case, the links between dynamic risk measures and Backward Stochastic Differential Equations (BSDEs) have been widely studied. We consider here the case with jumps. We first study the properties of BSDEs driven by a Brownian motion and a Poisson random measure and then give some properties of dynamic risk measures induced by BSDEs with jumps. We provide a representation property of such dynamic risk measures in the convex case as well as some results on a robust optimization problem in the case of model ambiguity. We then study the optimal stopping problem for dynamic risk measures represented by Backward Stochastic Differential Equations (BSDEs) with jumps and its relation with reflected BSDEs (RBSDEs) in the case when the obstacle process is a RCLL adapted process. In a Markovian framework, we show that the solution of the reflected BSDE corresponds to the unique viscosity solution of a Partial Integro-diferential Differential Variational Inequality. We further investigate robust optimal stopping problems related to the case with model ambiguity and their links with mixed control/ optimal stopping game problems. Finally we turn to the study of double barrier reflected BSDEs with jumps and their links with generalized Dynkin games.

References

1. Roxana Dumitrescu, Marie-Claire Quenez, Agnès Sulem. Double barrier reflected BSDEs with jumps and generalized Dynkin games, arXiv 1310.2764
2. Roxana Dumitrescu, Marie-Claire Quenez, Agnès Sulem. Reflected backward stochastic differential equations with jumps and partial integro-differential variational inequalities, INRIA research report 8213
3. M.C. Quenez and A. Sulem. BSDEs with jumps, optimization and applications to dynamic risk measures, Stochastic Processes and Applications, Volume 123, Issue 8, August 2013, Pages 3328-3357. - DOI 10.1016/j.spa.2013.02.016
4. M.-C. Quenez and A. Sulem. Reflected BSDEs and robust optimal stopping for dynamic risk measure with jumps. Research Report RR-8211 INRIA Rocquencourt, 2013.

Short lectures

Servaas van Bilsen: Optimal consumption and investment during retirement

The paper explores how a retired individual without a bequest motive should optimally spend and invest a given amount of financial wealth during the rest of his life. The individual faces various sources of risk (stock market, interest rate and inflation risk) and can insure longevity risk. We consider both complete and incomplete financial markets. The desired consumption path is framed in terms of the desired growth of median consumption and the desired standard variation of either the level of consumption, the growth rate of consumption or a mix of both. We show that this model includes various preference models such as constant relative risk aversion utility and habit formation. (slides)

Zhenzhen Fan: Contagion asymmetry and the equity foreign and home biases

We propose an affine jump-diffusion model which is able to generate asymmetric contagion effects both over time and across different equity markets. We solve in closed-form the portfolio optimization problem in this market. We find that our model is capable of reproducing the empirical pattern that dominant equity markets, which transmit their risk more easily to other markets, have a larger share in the market portfolio than implied by classical portfolio choice models, giving rise to foreign bias. Assuming ambiguity averse preferences, we show that reasonable ambiguity levels generate the observed portfolio of country representative investors, exhibiting both foreign and home biases.

Jan de Kort: Optimal investment under uncertain lifetime with stochastic mortality and stochastic interest rates

This talk concerns the the optimal asset allocation of an investor in the presence of stochastic interest rates and a nonnegative stochastic mortality rate. A complete market setting is assumed where wealth can be invested in a zero-coupon bond, a q-forward contract and the money-market account. A q-forward contract, which pays the difference between realized and expected mortality of a given population, can be thought of as an insurance against changes in survival probabilities. The investor, whose lifetime is uncertain, derives utility from accumulated wealth at his/her retirement date and utility from intermediate consumption. The mortality of the agent is modelled by a doubly stochastic Poisson process; the mortality rate and the short rate are assumed to follow independent Cox-Ingersoll-Ross processes. It will be shown that this problem has a closed-form solution. Conditions for existence of an optimal solution will be provided in terms of the model parameters. (slides)

Yanbin Shen: Algorithmic counterparty credit exposure for multi-asset bermudan options

The efficient quantification of counterparty credit risk of high dimensional exotic options is an important and challenging problem both in academics and in the industry. In this paper, an advanced method based on the LongStaff-Schwartz regression idea, which we call Stochastic Grid Bundling Method (SGBM), is applied to the computation of counterparty credit exposure profiles of multi-asset Bermudan options. We reduce the dimensionality of the problem by regressing the next time step option value along a set of functions of the coordinates of the underlying stock price process, instead of regressing along the multi dimensional stock price process itself. Then, we partition the state space in bundles and for every bundle we compute the regression, resulting in a significant improvement of the accuracy. If we chose these functions in a smart way, the calculation problem reduces to conditional moment calculation of the coordinates of the stock price. This enables us to compute exposure profiles where the underlying is from a large set of Markov processes, for which closed form formulas or analytical approximations of the conditional moment exist. We analyse the accuracy of the SGBM in the one dimensional case by benchmarking. Apart from benchmarking against the European options, we also benchmark against Bermudan options, that we price by a combination of Monte Carlo and COS Method (MCCOS). The error analysis for the one-dimensional case shows that the results produced by the SGBM method are very accurate. Finally, we apply the established method to several important examples of multi-asset payoffs. For the Bermudan option we are able to analyse the connection between the exposure profiles and the exercise intensity, where we compute these under both real world measure and risk neutral pricing measure. The efficient calculation of expected exposure (EE) for multi-asset options can be further applied to the computation of the credit value adjustment (CVA). (slides)

Poster presentations

Illia Simonov (University of Leoben): Infinity copulas
Dependence between two random variables is very well described by copulas introduced by Sklar. One can extend this dependence to n random variables. But what happens if we have infinity random variables? We will introduce and construct infinity copulas and try to give some simulation techniques.