May 17 - 19, 2012
We discuss the pricing of carbon emission allowances in the European Union Emission Trading Scheme (EU ETS) context. We study the EU ETS price, according to a given sector's players aggregation : the electricity producers.
We model the indifference price with a stochastic control problem for a producer that can dynamically switch between coal, gas or hydro power plants, and/or buy/sell emission allowances. We discuss the computation of the indifference price sensitivities.
Keywords of the presentation: multi-agent model, bifurcation, system instability, systemic risk
We build a multi-agent economic model as a dynamical system on a compact set, and show that the market instability is closely related to the leverage. The higher the leverage the greater the reaction of market participants to changes in their wealth. This gives rise to a bifurcation mechanism, and eventually a strong dynamical instability in capital markets which is commonly referred as financial crisis.
We divide an economy into finitely many aggregates called economic “agents,” and build a deterministic dynamical system of their wealth. We introduce a market instability indicator whose size is proportional to the instability level of the financial market, and provide conditions under which a local financial crisis propagates throughout the system to become a global one, creating systemic risk.
We extend the one-economy model to multiple economies to build a global multi-agent model, and provide a quantitative definition of “contagion.” We compare three financial crises, the 1997-98 Asian-Russian crisis, 2007-2009+ US subprime crisis, and the current eurozone sovereign credit crisis, as case studies of our research.
Keywords of the presentation: electricity markets, fuels, emissions, spread options, forward prices, structural model
Spread option pricing is of utmost importance in all energy markets, and in electricity in particular due to its application to power plant valuation and risk management. However, many practitioners still rely on either complicated, intractable production cost models or convenient but overly simplified approaches like Margrabe’s formula. In addition to the highly non-Gaussian nature of log power prices, such reduced-form models fail to capture the crucial state-dependent correlation structure between electricity, fuel (coal, natural gas, etc.) and emissions prices. We propose an intuitive and tractable structural model that incorporates important but subtle effects such as merit order changes, and price feedback through emissions markets. We discuss the availability of closed-form forward and option prices in certain cases, and highlight the implications for physical asset valuation, as compared with other common approaches.
Keywords of the presentation: Dynamic Utility, SDE, SPDE, Yields Curve
A large debate is open for several years within mathematical nance
about the criterion to optimise, in particular for long term policy. From
the perspective of public decision, such strategy must be time consistent.
Moreover the use of adaptative criterion is necessary to integrate some major
variation in the environment. A typical example is the forward utilities
introduced by M. Musiela and T. Zariphopoulou in 2003, for which there is
no-prespecied trading horizon.
First we characterize utility random elds by showing that the associated
marginal utility is a monotonic solution of SDE with random coecients;
its inverse, the marginal conjuguate utility is solution of a SPDE driven
by the adjoint elliptic operator. When forward utilities satisfy a property
of consistency with a given incomplete nancial market, as in the classical
case, dynamic utilities and its conjugate may be characterized in terms of
Hamilton Jacobi Bellman SPEs as value functions of control problems. More
interesting is the splitting property of the marginal utility in terms of optimal
processes, leading to an explicit solution given by the composition of the
optimal conjugate process with the inverse of the optimal wealth.
Then, it is possible to generate time consistency yield curves by indif-
ference pricing. In the controversy on the discount rate used in nancing
long term projects, such a criterion leads to a time consistency yield curve
depending of the wealth of the economy.
Keywords of the presentation: stress testing, extremes, systemic risk
This paper develops methods for selecting and analyzing stress scenarios. We begin by focusing on reverse stress testing --- finding the most likely scenarios leading to losses exceeding a given threshold. We approach this problem using a nonparametric empirical likelihood estimator (in the sense of Owen (2001)) of the conditional mean of the underlying market factors given large losses. We then scale the confidence regions for the conditional mean by a factor that depends on the tails of the market factors to estimate the most likely loss scenarios. We provide rigorous justification for the confidence regions and the scaling procedure in three models of the joint distribution of market factors with qualitatively different tail behavior: multivariate normal (light-tailed), multivariate Laplace (exponentially tailed), and multivariate-t (regularly varying). The key to this analysis (and the differences across the three cases) lies in the asymptotics of the conditional variances and covariances in extremes. These results also lead to asymptotics for marginal expected shortfall and the corresponding variance, conditional on extreme losses; we combine these results with empirical likelihood significance tests of systemic risk rankings based on marginal expected shortfall. We further apply these ideas to compare macro stress scenarios and to propose a scenario sampling method for exploring regions of large losses. This is joint work with Chulmin Kang and Wanmo Kang.
Keywords of the presentation: credit, stochastic time-change, Bayesian methods, MCMC
We introduce stochastic time change to default intensity models of credit risk as a parsimonious way to account for stochastic volatility in credit spreads. We derive two series solutions for the survival probability function, and show that both methods are applicable when the intensity follows the widely-used basic affine process. This leads to straightforward and efficient solutions to bond prices and CDS spreads. We then estimate the time-changed model on panels of CDS spreads (across maturity and observation time) using Bayesian MCMC methods. We find strong evidence of stochastic time change.
Co-authors are: Ovidiu Costin, Min Huang, and Pawel Szerszen.
The Keen model consists of a dynamical system describing the interactions between wages, employment rate, and debt in a closed economy. It exhibits one equilibrium where all variables remain finite and another where wages and employment collapse to zero while debt explodes to infinity, both of which are locally stable for typical parameter values. The introduction of a variable representing Ponzi speculation has the effect of destabilizing the first equilibrium, corresponding to a mathematical formulation of Minsky's famous "financial instability hypothesis". We propose a stochastic extension of this system by modelling a financial index through a price process with a jump component whose intensity depends on the magnitude of the Ponzi variable. The corresponding compensator that needs to be added to the drift of the index can then be interpreted as an asset price bubble. In addition, we postulate that downward jumps in the index increase the cost of borrowing across the economy, providing a feedback effect in the original system. (This is joint work with Bernardo Costa Lima)
Keywords of the presentation: Diffusions with rank-dependent interaction, collision local times, reflected Brownian motions in polyhedral domains, invariant measures, Bessel processes.
We introduce and study ergodic diffusion processes interacting through their ranks. These interactions give rise to invariant measures which are in broad agreement with stability properties observed in large equity markets over long time-periods. The models we develop assign growth rates and variances that depend on both the name (identity) and the rank (according to capitalization) of each individual asset. Such models are able realistically to capture critical features of the observed stability of capital distribution over the past century, all the while being simple enough to allow for rather detailed analytical study.
The methodologies used in this study touch upon the question of triple points for systems of interacting diffusions; in particular, some choices of parameters may permit triple (or higher-order) collisions to occur. We show, however, that such multiple collisions have no effect on any of the stability properties of the resulting
system. This is accomplished through a detailed analysis of collision local times.
The models have connections with the analysis of Queueing Networks in heavy traffic, and with competing particle systems in Statistical Mechanics (e.g., Sherrington-Kirkpatrick model for spin-glasses). Their hydrodynamic-limit behavior is governed by generalized porous medium equations with convection, whereas limits of a different kind display phase transitions and are governed by Poisson-Dirichlet laws.
Keywords of the presentation: exhaustible resources, stochastic games
We study the stochastic effect of resource
exploration in dynamic Cournot models of non-renewable resources, such as oil. We
consider a stochastic game between an exhaustible producer and a "green"
producer that has access to an inexhaustible but relatively expensive
source, such as solar power. Through exploration, the exhaustible producer can replenish reserves with new oil discoveries taking place according to a jump
process with intensity given by the exploration effort. This leads to
a study of systems of nonlinear first order delay ODEs. We derive
asymptotic expansions for the case of a small exploration success rate
and study the analogue of the Hotelling theorem in our game setting.
Time permitting, we will also present extensions, such as fixed exploration start-up costs and possibility for the green
producer to lower her production costs through technology R&D.
This is joint work with Ronnie Sircar (Princeton).
Keywords of the presentation: Systemic risk, Control of epidemics, Financial networks
Distress propagation in financial systems may be modeled by epidemics on a random graph in which nodes represent financial institutions and edges the exposures between them. Cascade dynamics may be reduced to the evolution of a multi‐dimensional Markov chain that corresponds to a sequential discovery of exposures and determines at any time the size of contagion. The end of contagion becomes a stopping time with respect to the history of the Markov chain. We study the optimal intervention strategy by a lender of last resort with objective to minimize the size of contagion under budget constraints. Our results show that, in the case of non‐anticipative information, the optimal strategy strongly depends on the proportion of banks that use short-term financial instruments for funding.
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Keywords of the presentation: utility, optimal portfolios, mean-variance optimization
Recently a new concept of the so called forward utility has been proposed, studied and used in the context of portfolio optimization. We recall the fundamental results of this approach in the case when the forward utility is monotone. Then we construct an analogous multi-period mean-variance optimization framework, derive the optimal portfolios and compare the outcomes. It turns out that the utility and mean-variance based optimal portfolios coincide when one chooses the appropriate levels of the risk tolerance.
Keywords of the presentation: particle methods, power plants valuation, stochastic control, variance reduction
In presence of market prices, energy utilities have to evaluate and hedge the
impact of energy prices fluctuations on their Profit and Loss. This requires, in particular,
to develop efficient computational methods to price, optimize and hedge physical assets
(such as thermal power plants). Mathematically, the problem of pricing a thermal asset
can be stated in terms of a stochastic control problem and can be reduced in a
very specific case to the thoroughly studied optimal stopping problem or American option
pricing. We present some original variance reduction techniques for American option pricing
and thermal power plants valuation.
We consider models which generalize the Volatility-Stabilized Markets introduced in Fernholz and Karatzas (2005) in the context of Stochastic Portfolio Theory. We show how to construct a weak solution of the underlying system of stochastic differential equations, express the solution in terms of time changed squared-Bessel processes, and argue that this solution is unique in distribution. Moreover, we discuss sufficient conditions for existence of strong solution, and show that strong relative arbitrage opportunities exist in these markets.
Keywords of the presentation: Equilibrium, production, prices, interest rates
This paper sets up and analyses a continuous-time equilibrium
model with firms, households and a bank. The model allows us to
study the inter-relation of production, consumption, levels of
working, interest rates, debt, inflation and
wage levels.
Keywords of the presentation: optimal investment and consumption, transaction costs, shadow prices
We revisit the optimal investment and consumption
model of Davis and Norman (1990) and Shreve and Soner (1994),
following a shadow-price approach similar to that of Kallsen and
Muhle-Karbe (2010). Making use of the completeness of the model
without transaction costs, we reformulate and reduce the
Hamilton-Jacobi-Bellman equation for this singular stochastic
control problem to a non-standard free-boundary problem for a
first-order ODE with an integral constraint. Having shown that the
free boundary problem has a smooth solution, we use it to construct
the solution of the original optimal investment/consumption problem
in a self-contained manner and without any recourse to the dynamic
programming principle. Furthermore, we provide an explicit
characterization of model parameters for which the value function is
finite. The presentation is based on joint work with Jin Hyuk Choi and Gordan Zitkovic.
Keywords of the presentation: Commodities markets, Financial Mathematics
Tang and Xiong (2009) discuss the financialization of commodities markets as a
result of increased index investing activity in the past decade. They find empirical
evidence of increased exposure of commodities prices to shocks to other asset
classes. We build a feedback model to try and capture some of these effects in which
traditional economic demand for a commodity, oil say, is perturbed by the influence
of portfolio optimizers. The analysis reveals correlation effects proportional to the
long or short positions of the investors, along with a lowering of volatility.
Keywords of the presentation: Bonds, Business cycle, Commodities, Currencies, Futures market, Inflation
Economists have traditionally viewed futures prices as fully informative about future economic activity and asset prices. We argue that open interest could be more informative than futures prices in the presence of hedging demand and limited risk absorption capacity in futures markets. We find that movements in open interest are highly pro-cyclical, correlated with both macroeconomic activity and movements in asset prices. Movements in commodity market interest predict commodity returns, bond returns, and movements in the short rate even after controlling for other known predictors. To a lesser degree, movements in open interest predict returns in currency, bond, and stock markets.