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.
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.