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- chance constraints (4)
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Metastability in reversible diffusion processes I. Sharp asymptotics for capcities and exit times
(2004)
We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form +rF ( )r on R or subsets of , where F is a smooth function with finitely many local minima. In analogy to previous work in discrete Markov chains, we show that metastable exit times from the attractive domains of the minima of F can be related, up to multiplicative errors that tend to one as # 0, to the capacities of suitably constructed sets. We show that this capacities can be computed, again up to multiplicative errors that tend to one, in terms of local characteristics of F at the starting minimum and the relevant saddle points. As a result, we are able to give the first rigorous proof of the classical Eyring-Kramers formula in dimension larger than 1. The estimates on capacities make use of their variational representation and monotonicity properties of Dirichlet forms. The methods developed here are extensions of our earlier work on discrete Markov chains to continuous diffusion processes.
We continue the analysis of the problem of metastability for reversible diffusion processes,
initiated in [BEGK3], with a precise analysis of the low-lying spectrum of the generator.
Recall that we are considering processes with generators of the form 1+rF()r on Rd or subsets
of Rd , where F is a smooth function with finitely many local minima. Here we consider only
the generic situation where the depths of all local minima are different. We show that in general
the exponentially small part of the spectrum is given, up to multiplicative errors tending to one, by
the eigenvalues of the classical capacity matrix of the array of capacitors made of balls of radius
centered at the positions of the local minima of F. We also get very precise uniform control on the
corresponding eigenfunctions. Moreover, these eigenvalues can be identified with the same precision
with the inverse mean metastable exit times from each minimum. In [BEGK3] it was proven
that these mean times are given, again up to multiplicative errors that tend to one, by the classical
Eyring–Kramers formula.
In this paper the numerical approximation of solutions of Itô stochastic differential
equations is considered, in particular for equations with a small parameter ? in the noise coex-
cient. We construct stochastic linear multi-step methods and develop the fundamental numerical
analysis concerning their mean-square consistency, numerical stability in the mean-square sense and
mean-square convergence. For the special case of two-step Maruyama schemes we derive conditions
guaranteeing their mean-square consistency. Further, for the small noise case we obtain expansions
of the local error in terms of the stepsize and the small parameter ?. Simulation results using several
explicit and implicit stochastic linear k-step schemes, k = 1; 2, illustrate the theoretical findings.
In Kolodko & Schoenmakers (2004) and Bender & Schoenmakers (2004) a policy iteration was introduced which allows to achieve tight lower approximations of the price for early exercise options via a nested Monte-Carlo simulation in a Markovian setting. In this paper we enhance the algorithm by a scenario selection method. It is demonstrated by numerical examples that the scenario selection can significantly reduce the number of actually performed inner simulations, and thus can heavily speed up the method (up to factor 10 in some examples). Moreover, it is shown that the modified algorithm retains the desirable properties of the original one such as the monotone improvement property, termination after a finite number of iteration steps, and numerical stability.
An important issue for solving multistage stochastic programs consists in the approximate representation of the (multivariate) stochastic input process in the form of a scenario tree. In this paper, forward and backward approaches are developed for generating scenario trees out of an initial fan of individual scenarios. Both approaches are motivated by the recent stability result in [15] for optimal values of multistage stochastic programs. They are based on upper bounds for the two relevant ingredients of the stability estimate, namely, the probabilistic and the filtration distance, respectively. These bounds allow to control the process of recursive scenario reduction [13] and branching. Numerical experience is reported for constructing multivariate scenario trees in electricity portfolio management.
The paper provides a structural analysis of the feasible set defined by linear probabilistic constraints. Emphasis is laid on single (individual) probabilistic constraints. A classical convexity result by Van de Panne/Popp and Kataoka is extended to a broader class of distributions and to more general functions of the decision vector. The range of probability levels for which convexity can be expected is exactly identified. Apart from convexity, also nontriviality and compactness of the
feasible set are precisely characterized at the same time. The relation between feasible sets with negative and with nonnegative right-hand side is revealed. Finally, an existence result is formulated for the more difficult case of joint probabilistic constraints.
We propose a valuation method for callable structures in a multi-factor Libor model which are path-dependent in the sense that, after calling, one receives a sequence of cash-flows in the future, instead of a well specified cash-flow at the calling date. The method is based on a Monte Carlo procedure for standard Bermudans recently developed in Kolodko & Schoenmakers (2004), and is applied to the cancelable snowball interest rate swap. The proposed procedure is quite generic, straightforward to implement, and can be easily adapted to other related path-dependent products.
We show that pricing a big class of relevant options by hedging
and no-arbitrage can be extended beyond semimartingale models. To
this end we construct a subclass of self-financing portfolios that
contains hedges for these options, but does not contain arbitrage
opportunities, even if the stock price process is a
non-semimartingale of some special type.
Moreover, we show that the option prices depend
essentially only on a path property of the stock price process,
viz. on the quadratic variation. As a consequence, we can
incorporate many stylized facts to a pricing model without
changing the option prices.
In this paper we lay the foundation for a numerical algorithm to
simulate high-dimensional coupled FBSDEs under weak coupling or
monotonicity conditions. In particular we prove convergence of a
time discretization and a Markovian iteration. The iteration
differs from standard Picard iterations for FBSDEs in that the
dimension of the underlying Markovian process does not increase
with the number of iterations. This feature seems to be
indispensable for an efficient iterative scheme from a numerical
point of view. We finally suggest a fully explicit numerical
algorithm and present some numerical examples with up to
10-dimensional state space.
Discrete approximations to chance constrained and mixed-integer two-stage stochastic programs require moderately sized scenario
sets. The relevant distances of (multivariate) probability
distributions for deriving quantitative stability results for such stochastic programs are $\mathcal{B}$-discrepancies, where the class $\mathcal{B}$ of Borel sets depends on their structural properties.
Hence, the optimal scenario reduction problem for such models is stated with respect to $\mathcal{B}$-discrepancies. In this paper,
upper and lower bounds, and some explicit solutions for optimal scenario reduction problems are derived. In addition, we develop
heuristic algorithms for determining nearly optimally reduced probability measures, discuss the case of the cell discrepancy (or
Kolmogorov metric) in some detail and provide some numerical experience.
We analyse stability aspects of linear multistage stochastic programs with polyhedral risk measures in the objective. In particular, we consider sensitivity of the optimal value with respect perturbations of the underlying stochastic input process. An existing stability result for multistage stochastic programs with expectation objective is carried forward to the case of polyhedral risk-averse objectives. Beside Lr-distances these results also involve filtration distances of the perturbations of the stochastic process. We discuss additional requirements for the
polyhedral risk measures such that the problem dependent filtration distances can be bounded by problem independent ones. Stability and such bounds are the basis for scenario tree approximation techniques used in practical problem solving.
We study a two-species interacting particle model on a subset of $\Z$
with open boundaries. The two species are injected with time
dependent rate on the left, resp.~right boundary.
Particles of different species annihilate when
they try to occupy the same site. This model has been proposed as a
simple model for the dynamics of an ``order book'' on a stock
market. We consider the hydrodynamic scaling limit for the empirical
process and prove a large deviation principle that implies
convergence to the solution of a non-linear parabolic equation.
We study various properties of a dynamic convex risk measure for bounded random variables which describe the discounted terminal values of financial positions. In particular we characterize time-consistency by a joint supermartingale property of the risk measure and its penalty function. Moreover we discuss the limit behavior of the risk measure in terms of asymptotic safety and of asymptotic precision, a property which may be viewed as a non-linear analogue of martingale convergence. These results are illustrated by the entropic dynamic risk measure.
In this paper we develop several regression algorithms for solving general stochastic optimal control problems via Monte Carlo. This type of algorithms is particulary useful for problems with a high-dimensional state space and complex dependence structure of the underlying Markov process with respect to some control. The main idea behind the algorithms is to simulate a set of trajectories under some reference measure and to use the Bellman principle combined with fast methods for approximating conditional expectations and functional optimization. Theoretical properties of the presented algorithms are investigated and the convergence to the optimal solution is proved under mild assumptions. Finally, we present numerical results for the problem of pricing a high-dimensional Bermudan basket option under transaction costs in a financial market with a large investor.
We present two approximation methods for pricing of CMS spread options in Libor market models. Both approaches are based on approximating the underlying swap rates with lognormal processes under suitable measures. The first method is derived straightforwardly from the Libor market model. The second one uses a convexity adjustment technique under a linear swap model assumption. A numerical study demonstrates that both methods provide satisfactory approximations of spread option prices and can be used for calibration of a Libor market model to the CMS spread option market.
The paper focuses on multi-period aspects of risk functionals. It discusses properties,
provides dual representations and offers methods for constructing multiperiod
risk functionals. On the way, existence results and representations for conditional
risk mappings are derived. In particular, conditional, multi-period, and
nested versions of the average value-at-risk are given. Finally, the importance of
polyhedral multi-period risk functionals for their employment in practical dynamic
decision making and risk management is discussed.
In a rather general setting of multivariate stochastic volatility market models we derive global iterative probabilistic schemes for computing the free boundary and its Greeks for a generic class of American derivative models using front-fixing methods. Establishment of convergence is closely linked to a proof of global regularity of the free boundary surface.
Arrow Debreu Prices
(2010)
Arrow Debreu prices are the prices of ‘atomic’ time and state contingent
claims which deliver one unit of a specific consumption good if a specific uncertain
state realizes at a specific future date. For instance, claims on the good
‘ice cream tomorrow’ are split into different commodities depending whether the
weather will be good or bad, so that good-weather and bad-weather ice cream
tomorrow can be traded separately. Such claims were introduced by K.J. Arrow
and G. Debreu in their work on general equilibrium theory under uncertainty,
to allow agents to exchange state and time contingent claims on goods. Thereby
the general equilibrium problem with uncertainly can be reduced to a conventional
one without uncertainty. In finite state financial models, Arrow-Debreu
securities delivering one unit of the numeraire good can be viewed as natural
atomic building blocks for all other state-time contingent financial claims; their
prices determine a unique arbitrage-free price system.
Under market frictions like illiquidity or transaction costs, contingent claims
can incorporate some inevitable intrinsic risk that cannot be completely hedged
away but remains with the holder. In general, they cannot be synthesized by
dynamical trading in liquid assets and hence not be priced by no-arbitrage arguments alone. Still, an agent can determine a valuation with respect to her
preferences towards risk. The utility indifference value for a variation in the
quantity of illiquid assets held by the agent is defined as the compensating variation
of wealth, under which her maximal expected utility remains unchanged.
The Real Multiple Dual
(2009)
In this paper we present a dual representation for the multiple stopping
problem, hence multiple exercise options. As such it is a natural generalization of the
method in Rogers (2002) and Haugh and Kogan (2004) for the standard stopping
problem for American options. We consider this representation as the real dual as it is
solely expressed in terms of an infimum over martingales rather than an infimum over
martingales and stopping times as in Meinshausen and Hambly (2004). For the multiple
dual representation we present three Monte Carlo simulation algorithms which require
only one degree of nesting.
In this paper we consider the optimal stopping problem for general dynamic monetary utility functionals. Sufficient conditions for the Bellman principle and the existence of optimal stopping times are provided. Particular attention is payed to representations which allow for a numerical treatment in real situations. To this aim, generalizations of standard evaluation methods like policy iteration, dual and consumption based approaches are developed in the context of general dynamic monetary utility functionals. As a result, it turns out that the possibility of a particular generalization depends on specific properties of the utility functional under consideration.
Recently, there is a growing trend to offer guarantee products where the investor is allowed to shift her account/investment value between multiple funds. The switching right is granted a finite number per year, i.e. it is American style with multiple exercise possibilities. In consequence, the pricing and the risk management is based on the switching strategy which maximizes the value of the guarantee put option. We analyze the optimal stopping problem in the case of one switching right within different model classes and compare the exact price with the lower price bound implied by the optimal deterministic switching time. We show that, within the class of log-price processes with independent increments, the stopping problem is solved by a deterministic stopping time if (and only if) the price process is in addition continuous. Thus, in a sense, the Black & Scholes model is the only (meaningful) pricing model where the lower price bound gives the exact price. It turns out that even moderate deviations from the Black & Scholes model assumptions give a lower price bound which is really below the exact price. This is illustrated by means of a stylized stochastic volatility model setup.
We propose a class of Markovian agent based models for the time evolution of a share price in an interactive market. The models rely on a microscopic description of a market of buyers and sellers who change their opinion about the stock value in a stochastic way. The actual price is determined in realistic way by matching (clearing) offers until no further transactions can be performed. Some analytic results for a non-interacting model are presented. We also propose basic interaction mechanisms and show in simulations that these already reproduce certain particular features of prices in real stock markets.
We apply theoretical results of S. Peng on supersolutions for BS-DEs
to the problem of finding optimal superhedging strategies in a
Black-Scholes market under constraints. Constraints may be imposed
simultaneously on wealth process and portfolio. They may be nonconvex,
time-dependent, and random. Constraints on the portfolio may
e.g. be formulated in terms of the amount of money invested, the portfolio
proportion, or the number of shares held.
We introduce a systematic approach to the problem of maximizing the robust
utility of the terminal wealth of an admissible strategy in a general complete market
model, where the robust utility functional is defined by a set Q of probability measures.
Our main result shows that this problem can be reduced to determining a “least favorable”
measure Q0 2 Q, which is universal in the sense that it does not depend on the
particular utility function. The robust problem is thus equivalent to a standard utility
maximization problem with respect to the “subjective” probability measure Q0. By using
the Huber-Strassen theorem from robust statistics, it is shown that Q0 always exists if Q
is the core of a 2-alternating upper probability. We also discuss the problem of robust
utility maximization with uncertain drift in a Black-Scholes market and the case of “weak
information” as studied by Baudoin (2002).
We introduce a forward scheme to simulate backward SDEs. Compared
to existing schemes, we avoid high order nestings of conditional
expectations backwards in time. In this way the error, when
approximating the conditional expectation, in dependence of the
time partition is significantly reduced. Besides this generic
result, we present an implementable algorithm and provide an error
analysis for it. Finally, we demonstrate the strength of the new
algorithm by solving some financial problems numerically.
The paper provides a condition for differentiability as well as an equivalent criterion
for Lipschitz continuity of singular normal distributions. Such distributions are of interest,
for instance, in stochastic optimization problems with probabilistic constraints, where
a comparatively small (nondegenerate-) normally distributed random vector induces a large
number of linear inequality constraints (e.g. networks with stochastic demands). The criterion
for Lipschitz continuity is established for the class of quasi-concave distributions which
the singular normal distribution belongs to.
In this paper we carry over the concept of reverse probabilistic representa-
tions developed in Milstein, Schoenmakers, Spokoiny (2004) for diffusion pro-
cesses, to discrete time Markov chains. We outline the construction of reverse
chains in several situations and apply this to processes which are connected
with jump-diffusion models and finite state Markov chains. By combining
forward and reverse representations we then construct transition density esti-
mators for chains which have root-N accuracy in any dimension and consider
some applications.
In a rather general setting of Itô-Lévy processes we study a class of transforms (Fourier for example) of the state variable of a process which are holomorphic in some disc around time zero in the complex plane. We show that such transforms are related to a system of analytic vectors for the generator of the process, and we state conditions which allow for holomorphic extension of these transforms into a strip which contains the positive real axis. Based on these extensions we develop a functional series expansion of these transforms in terms of the constituents of the generator. As application, we show that for multidimensional affine Itô-Lévy processes with state dependent jump part the Fourier transform is holomorphic in a time strip under some stationarity conditions, and give log-affine series representations for the transform.
Modeling several competitive leaders and followers acting in an electricity market
leads to coupled systems of mathematical programs with equilibrium constraints,
called equilibrium problems with equilibrium constraints (EPECs). We consider
a simplified model for competition in electricity markets under uncertainty of demand
in an electricity network
as a (stochastic) multi-leader-follower game. First order necessary conditions are
developed for the corresponding stochastic EPEC based on a result of Outrata.
For applying the general result an explicit representation of the co-derivative of
the normal cone mapping to a polyhedron is derived. Later the
co-derivative formula is used for verifying constraint qualifications and for identifying
$M$-stationary solutions of the stochastic EPEC if the demand is represented by a
finite number of scenarios.
Stability and Sensitivity of Optimization Problems with First Order Stochastic Dominance Constraints
(2007)
We analyze the stability and sensitivity of stochastic optimization problems with stochastic dominance constraints of first order. We consider general perturbations of the underlying probability measures in the space of regular measures equipped with a suitable discrepancy distance. We show that the graph of the feasible set mapping is closed under rather general assumptions. We obtain conditions for the continuity of the optimal value and upper-semicontinuity of the optimal solutions, as well as quantitative stability estimates of Lipschitz type.
Furthermore, we analyze the sensitivity of the optimal value and obtain upper and lower bounds for the directional
derivatives of the optimal value. The estimates are formulated in terms of the dual utility functions associated with the
dominance constraints.
We investigate the convexity of chance constraints with independent random variables. It will be shown, how concavity properties of the mapping related to the decision vector have to be combined with a suitable property of decrease for the marginal densities in order to arrive at convexity of the feasible set for large enough probability levels. It turns out that the required decrease can be verified for most prominent density functions. The results are applied then, to derive convexity of linear chance constraints with normally distributed stochastic coefficients when assuming independence of the rows of the coefficient matrix.
In this paper we introduce efficient Monte Carlo estimators for the valuation
of high-dimensional derivatives and their sensitivities (”Greeks”).
These estimators are based on an analytical, usually approximative representation
of the underlying density. We study approximative densities
obtained by the WKB method. The results are applied in the context of
a Libor market model.
We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the "limit" of the recursively constructed family of polynomials to the solution and error estimates are obtained from a priori estimates for some standard classes of linear partial differential equations, i.e. elliptic and hyperbolic equations. Another variation of the algorithm allows to construct polynomial interpolations which preserve systems of linear partial differential equations at the interpolation points. We show how this can be applied in order to compute higher order terms of WKB-approximations of fundamental solutions of a large class of linear parabolic equations. The error estimates are sensitive to the regularity of the solution. Our method is compatible with recent developments for solution of higher dimensional partial differential equations, i.e. (adaptive) sparse grids, and weighted Monte-Carlo, and has obvious applications to mathematical finance and physics.
In this paper we propose a Libor model with a high-dimensional specially structured system of
driving CIR volatility processes. A stable calibration procedure which takes into account
a given local correlation structure is presented. The calibration algorithm is FFT based, so fast and easy
to implement.
We present a generic non-nested Monte Carlo procedure for computing true upper bounds for Bermudan products, given an approximation of the Snell envelope. The pleonastic ``true'' stresses that, by construction, the estimator is biased above the Snell envelope. The key idea is a regression estimator for the Doob martingale part of the approximative Snell envelope, which preserves the martingale property. The so constructed martingale may be employed for computing dual upper bounds without nested simulation. In general, this martingale can also be used as a control variate for simulation of conditional expectations. In this context, we develop a variance reduced version of the nested primal-dual estimator (Anderson & Broadie (2004)) and nested consumption based (Belomestny & Milstein (2006)) methods . Numerical experiments indicate the efficiency of the non-nested Monte Carlo algorithm and the variance reduced nested one.
We develop a generic method for constructing a weak static minimum
variance hedge for a wide range of derivatives that may involve optimal exercise features or contingent cash flow streams, to provide a hedge along a
sequence of future hedging dates. The optimal hedge is constructed using
a portfolio of preselected hedge instruments which could be derivatives
with different maturities. The hedge portfolio is weakly static in that
it is initiated at time zero, does not involve intermediate re-balancing,
but hedges may be gradually unwound over time. We study the static
hedging of a convertible bond to demonstrate the method by an example
that involves equity and credit risk. We investigate the robustness of the
hedge performance with respect to parameter and model risk by numerical
experiments.
Good-deal bounds have been introduced as a way to obtain valuation bounds
for derivative assets which are tighter than the arbitrage bounds. This is achieved by ruling out not only those prices that violate no-arbitrage restrictions but also
trading opportunities that are `too good'.
We study dynamic good-deal valuation bounds that are derived from bounds on optimal
expected growth rates. This leads naturally to restrictions on the set of pricing measure which are local in time, thereby inducing good dynamic properties for the good-deal valuation bounds.
We study good-deal bounds by duality arguments in a general semimartingale setting.
In a Wiener space setting where asset prices evolve as It\^o-processes,
good-deal bounds are then conveniently described by backward SDEs.
We show how the good-deal bounds arise as the value function for an
optimal control problem, where a dynamic coherent a priori risk measure is minimized by the choice of a suitable hedging strategy.
This demonstrates how the theory of no-good-deal valuations can be associated to an established concept of dynamic hedging in continuous time.
In a strong constant electric field, a dielectric particle immersed in a weakly conducting fluid exhibits spontaneous rotations. This phenomenon is known under the name of the Quincke effect. In the original setup the particle was suspended on a silk thread and performed torsional oscillations of remarkably high amplitude. We derive the governing equations for this experiment, and ascertain
that onset of oscillations from the quiescent state corresponds to the supercritical Hopf bifurcation.
For the case of a soft thread, we characterize the regime of large-scale torsional relaxation oscillations:
explicit estimates are derived for their period and amplitude, effects of bifurcation delay are described. In a stronger electric field, these relaxation oscillations yield to small-scale erratic rotations of the pendulum.