Minicourses
Rüdiger Kiesel: Modelling Energy Markets
Within the last few years the markets for commodities, in particular energy-related commodities, has changed substantially. New regulations and products have resulted in a spectacular growth in spot and derivative trading. In particular, electricity markets have changed fundamentally over the last couple of years. Due to deregulation energy companies are now allowed to trade not only the commodity electricity, but also various derivatives on electricity on several Energy Exchanges (such as the EEX).
During this Mini-Course we discuss basic principles of commodity markets and outline the stylized facts of electricity price processes. Then we introduce spot and forward models for commodities (with a focus on electricity). In addition, special derivatives for the electricity markets are analysed. In commodities markets the market risk premium, defined as the difference between forward prices and spot forecasts, is an important indicator of the behaviour of buyers and sellers and their views on the market spanning between short-term and long-term horizons. We show that under certain assumptions it is possible to derive explicit solutions that link levels of risk aversion and market power with market prices of risk and the market risk premium.
During the last part of the course we provide a short introduction in the theoretical properties of emission permit price dynamics regarding the effect of banking, linking of different emissions trading schemes and safety-valve mechanisms. Explicit models for the price process are constructed and calibrated to historical data on the permit prices and emissions in the European Union. We show that permit prices in emissions trading schemes without inter-phase banking resemble Digital options and are inherently prone to price jumps and high volatility. Then we discuss so-called hybrid schemes and show that they can be decomposed into ordinary cap-and-trade schemes with plain-vanilla options on permits.
(Slides of lecture 1, lecture 2, lecture 3, lecture 4, lecture 5).
Bernt Øksendal: Malliavin calculus for Lévy processes and applications to finance
The Malliavin calculus was originally introduced by Paul Malliavin in 1978 as a tool to study smoothness of densities of solutions of stochastic differential equations. This was a rather restricted scope of applications and the theory was difficult, so for more than 10 years this was a topic of interest to only a limited group of experts.
However, when Ocone in 1994 showed that Malliavin calculus could be used to obtain an explicit version of the Itô representation theorem (now known as the Clark-Ocone formula), and when subsequently Karatzas and Ocone applied this to finance, the interest in this area exploded. At the same time simpler presentations of the theory were developed. Soon even bank employees started studying Malliavin calculus!
The Malliavin calculus was first introduced for Brownian motion, but it has later been extended to Lévy processes. At the same time new applications of the theory have been discovered.
In this course we give a simple introduction to Malliavin calculus for Lévy processes and we give examples of applications to finance. Here is an outline of the course:
Lecture 1: Introduction to Malliavin calculus for Brownian motion (i):
The Malliavin derivative, or Hida-Malliavin derivative, as a stochastic gradient
Lecture 2: Malliavin calculus for Brownian motion (ii):
The Malliavin derivative by means of chaos expansion. Properties of the Malliavin derivative, including the chain rule, the duality theorem and the fundamental theorem of stochastic calculus.
Lecture 3: Malliavin calculus for Brownian motion (iii):
Applications:
(a)The Clark-Ocone formula and applications to hedging
(b) Sensitivity results and application to efficient numerical computation of the "greeks" in finance.
Lecture 4: Malliavin calculus for Lévy processes (i).
Introduction to stochastic calculus for Lévy processes. The Malliavin derivative by means of chaos expansion. Properties of the Malliavin derivative. The Clark-Ocone formula revisited.
Hedging in incomplete markets.
Lecture 5: Malliavin calculus for Lévy processes (ii).
Applications, for example the following:
(a) Minimal variance portfolio in incomplete markets
(b) Optimal portfolio with partial information
The presentation is based on parts of my book joint with Giulia Di Nunno and Frank Proske, entitled Malliavin Calculus for Lévy Processes and Applications to Finance,
Universitext, Springer 2009. Specifically, the core of the course is based on
Chapter 9,
Chapter 10, and
Chapter 12. Additional parts from the book:
front matter,
Chapter 1,
Chapter 2,
Chapter 3,
back matter.
Special invited lectures
Hansjoerg Albrecher:
Solvency modelling with dependent risks
This talk is a survey on the effects of dependence of risks
on the solvency of a portfolio of insurance policies. Exact and
asymptotic results for ruin probabilities
will be discussed and general techniques will be presented that make
models with dependence tractable for the analysis. Related quantities
like the time to ruin and the deficit of ruin under dependence will also
be treated. Some of the results have applications for the pricing of
path-dependent options in financial markets, including lookback and
barrier options.
Gilles Pagès:
Dual quantization methods and application to Finance
Regular quantization tree methods are based on the mapping of a discrete time Feller Markov chain at each time step on a grid following a nearest neighbour projection. When these grids are optimized with respect to the marginal distributions of the chain, the resulting backward dynamic programming computation method (the so-called quantization tree) has shown its efficiency in the solving of many
non-linear problems arising in Finance like mult-asset American option pricing and hedging, stochastic control like swing options, or non linear filtering (see e.g. [2]).
We recently developed (see [3,4,5]) a new approach to quantization, called {\em dual quantization}, see [4,5], based on the mapping of the Markovian dynamics at each time step onto the vertices of a Delaunay triangulation spanned by a grid (the Delaunay triangulation is the dual geometrical object of the Voronoi tessellation involved in regular vector quantization). The aim of this talk is to present this new approach as well as its first applications to finance.
For a static Rd-valued Lp-integrable random vector, say X, we prove the existence of an optimal dual quantizer of size at most N (see [4]) and, under a slightly stronger moment assumption, we establish the sharp asymptotics for the resulting optimal dual quantization error as N goes to infinity in the expected N-1/d-scale (see [5]). The constant in this asymptotics depends on the distribution of X (through its density) and on a universal constant Cp,d (made explicit in one dimension). This result is the exact counterpart for dual quantization of Zador's Theorem (see [1]). New simulation based stochastic optimization procedures have been derived to produce optimal dual quantization grids for any simulatable distribution.
However, by contrast with optimal regular quantization, dual quantization grids all share a stationarity property, regardless of their optimality. As a first consequence, this induces a significant improvement of the performences of dual quantization trees, especially for smaller grid sizes. But the most striking consequence is the resulting robustness and flexibility of such trees with respect to the possible parameter variations when dealing with a parametrized underlying dynamics. So, it should become an efficient tool for perform calibrations or to solve multi-dimensional stochastic control problems when the underlying dynamics depends on the control.
First applications to credit derivatives (see [3]) and to multi-asset option pricing will be presented as well as first connections with the finite element methods.
This presentation is based on a several joint works with Benedikt Wilbertz References
[1] S. Graf , H. Luschgy, Foundations of Quantization for Probability Distributions, Lecture Notes in Mathematics, 1730, Berlin, 2000.
[2] G. Pagès, J. Printems, Optimal quantization for finance: from random vectors to stochastic processes, in Mathematical Modeling and Numerical Methods in Finance (special volume) (A. Bensoussan, Q. Zhang guest eds.), coll. Handbook of Numerical Analysis (P.G. Ciarlet Editor), North Holland, 595-649, 2009.
[3] G. Pagès, B. Wilbertz. Dual Quantization for random walks with application to credit derivatives, pre-pub. LPMA 1322, arXiv: 0910.5655, 2009, in revision for J. of Comput. Fin.
[4] G. Pagès, B. Wilbertz, Intrinsic stationarity for vector quantization:
Foundation of dual quantization, in progress, 2010.
[5] G. Pagès, B. Wilbertz, Sharp rate for the dual quantization problem, in progress, 2010.
Johan Tysk:
Boundary behaviour of densities for non-negative diffusions
It is well-known that the transition density of a diffusion process
solves the corresponding Kolmogorov forward equation. If the
state space has finite boundary points, then naturally one also needs
to specify appropriate boundary conditions when solving this equation.
However, many processes occurring in finance have degenerate diffusion
coefficients, and for these processes the density may explode at the
boundary. We describe a simple symmetry relation for the density that
transforms the forward equation into a backward equation, the boundary
condition of which is much more straightforward to handle. This relation
allows us to derive new results on the precise asymptotics of the density
at boundary points where the diffusion degenerates. This is joint work
with Erik Ekström. (slides)
Short lectures
Ove Göttsche:
Option pricing and the cost of risk
The problem of pricing and hedging options is well understood in the context of the Black-Scholes model. In this model, a perfect hedge is always possible, meaning, there exists a dynamic strategy such that trading in the underlying asset replicates the payoff of the option. However, the possibility of a perfect hedge is restricted to certain models and restrictive assumptions. In more realistic models a perfect hedge is not possible and thus an option bears a residual risk that cannot be hedged away completely. Therefore, pricing an option consist of two parts: the cost of a hedging strategy that reduces the risk, and a premium to cover the residual risk.
We assume that a trader wishes to minimize the price of a given option. To avoid that she chooses hedging strategies which are too risky, the trader is punished when taking excessive risks. To do so, we introduce an extra capital reserve bank account, which earns a smaller rate of return then a standard deposit bank account. The reserve account should always contain a minimal amount of money, which depends on the residual risk that the trader's portfolio is exposed to. The residual risk is measure by a convex risk measure and the problem leads to a convex optimization problem. This problem can be solved via convex duality methods. We prove the existence of the optimal hedging strategy and give an example. (slides)
References
[1] Barrieu, P. and El Karoui, N. (2005),
Inf-convolution of risk measures and optimal risk transfer.
Finance and Stochastics 9, 269-298.
[2] Rockafellar, R.T. (1997), Convex Analysis. Princeton
University Press, Princeton.
Michiel Janssen:
Portfolio optimisation with a value at risk constraint in the presence of unhedgeable risks
This paper addresses the portfolio optimisation problem that most European insurance companies will face after the introduction of Solvency II (the new regulatory framework in Europe to be introduced in 2012). Solvency II will limit the total Value at Risk of an insurance company. In this paper therefore I derive the optimal portfolio of hedgeable risks when also unhedgeable risks are present and the sum of both risks is constrained by a Value at Risk constraint.
This paper extends the current literature on portfolio optimisation by including both a Value at Risk constraint and Unhedgeable risks where in the current literature maximally only of these two is included. To obtain flexibility with respect to assumptions regarding the probability functions of both the hedgeable and unhedgeable risks, the state price density and the utility function used, the problem is optimised numerically. An example shows the importance of a correct specification of the characteristics of the hedgeable risk. The results also show that the optimal portfolio is much less skewed than the optimal portfolio that is obtained when only hedgeable risks are present.
Keywords: Portfolio optimisation, Value at Risk, Unhegdeable risks, Solvency II (preprint)
Roel Mehlkopf:
Intergenerational risk sharing and long-run labor income risk
The inability of future generations to share risk with current ones causes financial markets to be incomplete and thus inefficient. By using its financial reserves efficiently, a pension fund is able to transfer current equity risk to future generations, thereby alleviating the 'biological' trading constraint that is faced in financial markets. This paper examines how comovements in stock and labor
markets affect the gains from intergenerational risk sharing. If stock and labor markets move together in the long-run, the human wealth of unborn generations becomes highly correlated with stock returns, which reduces their risk appetite. I show that shifting risk into
the future is not optimal anymore once the long-run dynamics of labor income are taken into account. The risk bearing capacity of a pension fund is dramatically decreased if it is unattractive for risk to be transferred to future generations. The results in this paper provide an economic rationale for a tight solvency regime, that requires pension funds to recover from their losses in a short time-period. (preprint)
Enno Veerman:
The affine transform formula for affine jump-diffusions with a general closed convex state space
Affine jump-diffusions are widely used in finance because of their flexibility and mathematical tractability. The latter is revealed by the so-called affine transform formula, which relates exponential moments to solutions of particular ODEs, called generalized Riccati equations. We establish existence of exponential moments and the validity of the affine transform formula for affine jump-diffusions with a general convex state space. This extends known results for affine jump-diffusions with a canonical state space. The key step is to prove the martingale property of an exponential local martingale, using the well-posedness of the associated martingale problem.
By analytic extension we then obtain the affine transform formula for complex exponentials, in particular for the characteristic function.
Next we apply our results to popular models in the literature, including the multivariate stochastic volatility models where the volatility is modelled as a matrix-valued affine process. (joint work with Peter Spreij)
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