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We provide results on the existence and uniqueness of equilibrium in dynamically incomplete financial markets in discrete time. Our framework allows for heterogeneous agents, unspanned random endowments and convex trading constraints. In the special case where all agents have preferences of the same type and all random endowments are replicable by trading in the financial market we show that a one-fund theorem holds and give an explicit expression for the equilibrium pricing kernel. If the underlying noise is generated by finitely many Bernoulli random walks, the equilibrium dynamics can be described by a system of coupled backward stochastic difference equations, which in the continuous-time limit becomes a multi-dimensional backward stochastic differential equation. If the market is complete in equilibrium, the system of equations decouples, but if not, one needs to keep track of the prices and continuation values of all agents to solve it. As an example we simulate option prices in the presence of stochastic volatility, demand pressure and short-selling constraints.
We consider a class of generalized capital asset pricing models in continuous time with a finite number of agents and tradable securities. The securities may not be sufficient to span all sources of uncertainty. If the agents have exponential utility functions and the individual endowments are spanned by the securities, an equilibrium exists and the agents’ optimal trading strategies are constant. Affine processes, and the theory of information-based asset pricing are used to model the endogenous asset price dynamics and the terminal payoff. The derived semi-explicit pricing formulae are applied to numerically analyze the impact of the agents’ risk aversion on the implied volatility of simultaneously-traded European-style options.
In the paradigm of VON N EUMANN AND M ORGENSTERN, a representation of affine pref-
erences in terms of an expected utility can be obtained under the assumption of weak continu-
ity. Since the weak topology is coarse, this requirement is a priori far from being negligible.
In this work, we replace the assumption of weak continuity by monotonicity. More precisely,
on the space of lotteries on an interval of the real line, it is shown that any affine preference
order which is monotone with respect to the first stochastic order admits a representation in
terms of an expected utility for some nondecreasing utility function. As a consequence, any
affine preference order on the subset of lotteries with compact support, which is monotone
with respect to the second stochastic order, can be represented in terms of an expected util-
ity for some nondecreasing concave utility function. We also provide such representations
for affine preference orders on the subset of those lotteries which fulfill some integrability
conditions. The subtleties of the weak topology are illustrated by some examples.
We consider backward stochastic differential equations with drivers of quadratic growth (qgBSDE). We prove several statements concerning path regularity and stochastic smooth-
ness of the solution processes of the qgBSDE, in particular we prove an extension of Zhang's path regularity theorem to the quadratic growth setting. We give explicit convergence rates for the difference between the solution of a qgBSDE and its truncation, filling an important gap in numerics for qgBSDE. We give an alternative proof of second order Malliavin differentiability for BSDE with drivers that are Lipschitz continuous (and differentiable), and
then derive the same result for qgBSDE.
We consider the problem of numerical approximation for forward-backward stochastic
differential equations with drivers of quadratic growth (qgFBSDE). To illustrate the significance
of qgFBSDE, we discuss a problem of cross hedging of an insurance related financial
derivative using correlated assets. For the convergence of numerical approximation schemes for
such systems of stochastic equations, path regularity of the solution processes is instrumental.
We present a method based on the truncation of the driver, and explicitly exhibit error estimates
as functions of the truncation height. We discuss a reduction method to FBSDE with globally
Lipschitz continuous drivers, by using the Cole-Hopf exponential transformation. We finally
illustrate our numerical approximation methods by giving simulations for prices and optimal
hedges of simple insurance derivatives.
The LIBOR market model is very popular for pricing inter-
est rate derivatives, but is known to have several pitfalls. In addition, if
the model is driven by a jump process, then the complexity of the drift
term is growing exponentially fast (as a function of the tenor length). In
this work, we consider a Levy-driven LIBOR model and aim at developing accurate and efficient log-Levy approximations for the dynamics of
the rates. The approximations are based on truncation of the drift term
and Picard approximation of suitable processes. Numerical experiments
for FRAs, caps and swaptions show that the approximations perform
very well. In addition, we also consider the log-Levy approximation of
annuities, which offers good approximations for high volatility regimes.
Optimal dual martingales, their analysis and application to new algorithms for Bermudan products
(2012)
In this paper we introduce and study the concept of optimal and surely
optimal dual martingales in the context of dual valuation of Bermudan
options, and outline the development of new algorithms in this context.
We provide a characterization theorem, a theorem which gives conditions
for a martingale to be surely optimal, and a stability theorem concerning martingales which are near to be surely optimal in a sense. Guided
by these results we develop a framework of backward algorithms for constructing such a martingale. In turn this martingale may then be utilized
for computing an upper bound of the Bermudan product. The methodology is pure dual in the sense that it doesn't require certain (input)
approximations to the Snell envelope.
In an Ito-Levy environment we outline a particular regression based
backward algorithm which allows for computing dual upper bounds with-
out nested Monte Carlo simulation. Moreover, as a by-product this algorithm also provides approximations to the continuation values of the
product, which in turn determine a stopping policy. Hence, we may obtain lower bounds at the same time.
In a first numerical study we demonstrate a backward dual regression algorithm in a Wiener environment that is easy to implement and
is regarding accuracy comparable with the method of Belomestny et. al.
(2009).
With an emphasis on generators with quadratic growth in the control variable we consider
measure solutions of BSDE, a solution concept corresponding to the notion of risk neutral
measure in mathematical finance. In terms of measure solutions, solving a BSDE reduces
to martingale representation with respect to an underlying filtration. Measure solutions
related to measures equivalent to the historical one provide classical solutions. We derive
the existence of measure solutions in scenarios in which the generating functions are just
continuous, of at most linear growth in the control variable (corresponding to generators of
at most quadratic growth in the usual sense), and with a random bound in the time parameter
whose stochastic integral is a BMO martingale. Our main tools include a stability property
of sequences of measure solutions, for which a limiting solution is obtained by means of the
weak convergence of measures.