Minicourses
Jérôme Detemple: Dynamic asset allocation
Dynamic asset allocation models have attracted intense interest from
academics and practitioners in recent years. These lectures will focus on
recent developments in the field. A classic portfolio formula, based on
Malliavin calculus, will be reviewed and implementation aspects discussed.
An alternative decomposition formula, emphasizing the role of long term
bonds, will also be presented. Various applications of the formulas will be
examined. Aspects pertaining to the structure of optimal portfolios and the
role of dynamic hedging terms will be clarified. (slides)
References
[1] Cox and Huang, "Optimal consumption and portfolio policies when asset prices follow a diffusion process," Journal of Economic Theory, 49, 1989: 33-83.
[2] Cox and Huang. "A variational problem arising in financial economics," Journal of Mathematical Economics, 20, 1991: 465-487.
[3] Detemple, Garcia and Rindisbacher, "A Monte-Carlo method for optimal portfolios," Journal of Finance, 58, 2003: 401-446.
[4] Detemple, Garcia and Rindisbacher, "Intertemporal asset allocation: a comparison of methods," Journal of Banking and Finance, 29, 2005b: 2821-2848.
[5] Karatzas, Lehoczky and Shreve, "Optimal portfolio and consumption decisions for a "small investor" on a finite horizon," SIAM J. Control and Optimization, 25, 1987: 1557-1586.
[6] Karatzas and Shreve, Methods of Mathematical Finance, Springer-Verlag, 1999.
[7] Markowitz, "Portfolio selection," Journal of Finance, 7, 1952: 77-91.
[8] Merton, "Lifetime portfolio selection under uncertainty: the continuous-time case," Review of Economics and Statistics, 51, 1969: 247-257.
[9] Merton, "Optimum consumption and portfolio rules in a continuous time model," Journal of Economic Theory, 3, 1971: 273-413.
[10] Nualart, The Malliavin Calculus and Related Topics, Springer Verlag, New York, 1995.
[11] Ocone and Karatzas, "A generalized Clark representation formula, with application to optimal portfolios," Stochastics and Stochastics Reports, 34, 1991: 187-220.
The minicourse especially relies on [1, 3, 5, 9, 11].
Georg Pflug: Risk functionals for multi-period decision problems
We introduce the setting for (time-discrete) multi-period convex risk functionals (definitions, properties such as time-consistency, dual representations, characterizations) as generalizations of single-period functionals. Unlike as in the one-period situation, the functionals are defined on pairs of stochastic processes and filtrations, to which the processes are adapted. We give a new notion of version-independence (law-invariance) and a concept of equivalence.
The pairs consisting of filtrations and processes can be endowed with a (weak) metric and
continuity as well as Lipschitz properties of risk functionals may be studied. This allows the quantification of the approximation error when a sophisticated model is replaced by a simpler one.
Applications include examples from longer term financial optimization problems such as those arising in pension fund management and electricity portfolio planning. (slides 1, slides 2, slides 3)
Special invited lectures
Piotr Karasinski:
Mindless fitting?
We are required to mark-to-market non-plain, exotic, products
consistently with the market-observed prices of liquid vanilla
products.
Thus for each exotic we must have a one-to-one mapping between
vanilla prices and the exotic's price. Such mapping is called the
mark-to-market model as it produces mark-to-market price and risk
exposure. Risk management policies (risk limits, desire to
minimise volatility of the mark-to-market P&L) typically compel
traders to hedge exotics with vanillas such that the combined risk
exposure, measured by the mark-to-market model, is close to zero.
In the traditional approach we set the price of an exotic equal to
its' value given by a traditional derivatives valuation model that
assumes a certain stochastic evolution of the relevant risk
factors. To fit vanilla prices practitioners often use (are
forced to use?) over-parametrised models in which risk factor
dynamics can be counter-intuitive. Does this produce a good
model, i.e., does hedging to such model's risk exposure result in
realised replication costs that is close to the initial exotic's
price the model produces? How can we find an answer to this
question?
What are the alternatives? Can we start with a price of an exotic
produced by a standard derivatives valuation model, with risk
factors' dynamics that makes sense (who is to judge?), and
somehow, externally, adjust the price to reflect the difference
between market and model prices of relevant vanilla options?
Would the resulting mapping produce a hedging model that is better
than the one based on the traditional approach? (slides)
Damien Lamberton:
Some option pricing problems in exponential Lévy models
In this lecture, we will discuss two types of path-dependent options within exponential Lévy models:
American options, and lookback options.
In the first part, we will highlight the connection between variational inequalities
and optimal stopping problems, and derive some regularity properties of American option prices.
The second part of the lecture will be devoted to discretization issues for the maximum of
a Lévy process and potential applications to lookback options.
The results which will be presented
are based on joint work with M. Mikou for the first part and with E.H.A. Dia
for the second part. (slides)
References
Damien Lamberton, Mohammed Mikou. The critical price for the American put in an exponential Lévy model. Finance Stoch (2008) 12: 561-581. (pdf)
Martin Schweizer:
New insights into exponential utility indifference valuation
One popular approach to the problem of valuing contingent claims in incomplete markets is to use exponential utility indifference valuation. This has the combined advantages of being economically well-founded and yet mathematically fairly tractable in quite general settings. Especially for executive stock option valuation, this approach is quite popular in the literature.
In a Markovian setting with one traded and one nontraded asset driven by two correlated Brownian motions, several authors have used PDE techniques to obtain an almost explicit valuation formula. This involves a reduction of the nonlinear PDE resulting from the HJB equation to a linear one by a well-chosen power transformation, where the distortion power is given explicitly in terms of the correlation which is assumed constant. But it turns out that this kind of formula can be extended to much more general models, going over Ito processes with stochastic correlation to completely general semimartingales. In addition, going up to that level of generality gives a much better understanding of why such a formula holds.
We shall explain the above ideas and results in detail. This talk is based on joint work with Christoph Frei (ETH Zürich). (slides)
Short lectures
Jiajia Cui:
Longevity risk pricing
Longevity risks, i.e. unexpected improvements in life expectancies, may lead to severe solvency issues for annuity providers. Longevity-linked securities provide the desirable hedging instruments to annuity providers, and in the meanwhile, diversification benefits to their counterparties. But longevity-linked securities are not traded in financial markets due to the pricing difficulty. This paper proposes a new method to price the longevity risk premia in order to tackle the pricing obstacle. Based on the equivalent utility pricing principle, our method obtains the minimum risk premium required by the longevity insurance seller and the maximum acceptable risk premium by the longevity insurance buyer. The proposed methodology satisfies four important requirements for applications in practice: i) suitable for incomplete market pricing, ii) consistent with other financial market risk premia and iii) flexible in handling different payoff structures, basis risk and natural hedging possibilities. The method is applied in pricing various longevity-linked securities (bonds, swaps, caps and floors). We show that the size of the risk premium depends on the payoff structure of the security due to the market incompleteness. Furthermore, we show that the financial strength of the longevity insurance seller and buyer, the availability of the natural hedges and the presence of basis risk may significantly affect the size of longevity risk premium. (slides)
References
Jiajia Cui. Longevity Risk Pricing. (preprint)
Xinzheng Huang:
Generalized beta regression models for random loss-given-default
We propose a new framework for modeling systematic risk in Loss-Given-Default
(LGD) in the context of credit portfolio losses. The class of models is very
flexible and accommodates well skewness and heteroscedastic errors. The
quantities in the models have simple economic interpretation. Inference of
models in this framework can be unified. Moreover, it allows efficient
numerical procedures, such as the normal approximation and the saddlepoint
approximation, to calculate the portfolio loss distribution, Value at Risk
(VaR) and Expected Shortfall (ES). (slides)
References
Xinzheng Huang, Cornelis. W. Oosterlee. Generalized Beta Regression Models for Random Loss-Given-Default (preprint)
Coen Leentvaar:
Multi-asset option pricing using a parallel Fourier-based technique
We present and evaluate a Fourier-based sparse grid method for pricing multi-asset options. This involves computing multi-dimensional integrals efficiently and we accomplish it using the fast Fourier transform. We also propose and evaluate ways to deal with the curse of dimensionality by means of parallel partitioning of the Fourier transform and by incorporating a parallel sparse grid method. The benefit of the Fourier-based method is the absence of the time-integration for European options in contrast to the standard Black-Scholes multi-dimensional PDE. We test the presented Fourier-based method by solving pricing equations for options that are dependent on up to seven underlying assets and we compare the results to results obtained from multi-dimensional PDE methods. (slides)
Denitsa Stefanova:
Dynamic correlation hedging in copula models for portfolio selection
In this paper we address the problem of solving for optimal portfolio
allocation in a dynamic setting, where conditional asset return correlations
are modeled using observable factors, which allows us to isolate the demand
for hedging correlation risk. We are able to analyse separately the impact
of tail dependence through the unconditional distribution of the process for
asset prices and that of the conditional correlation on portfolio holdings.
With those distinct ways of modeling dependence we aim at replicating the
stylised fact of increased dependence during extreme market downturns,
rising market-wide volatility, or worsening macroeconomic conditions. We
find that both correlation hedging demands and intertemporal hedges due to
increased tail dependence have distinct portfolio implications and cannot
act as substitutes to each other. As well, there are substantial economic
costs for disregarding both the dynamics of conditional correlation and the
dependence in the extremes. (slides)
References
Denitsa Stefanova and Redouane Elkamhi. Dynamic Correlation Hedging in Copula Models for Portfolio Selection (preprint).
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