68 research outputs found
A theoretical framework for the pricing of contingent claims in the presence of model uncertainty
The aim of this work is to evaluate the cheapest superreplication price of a
general (possibly path-dependent) European contingent claim in a context where
the model is uncertain. This setting is a generalization of the uncertain
volatility model (UVM) introduced in by Avellaneda, Levy and Paras. The
uncertainty is specified by a family of martingale probability measures which
may not be dominated. We obtain a partial characterization result and a full
characterization which extends Avellaneda, Levy and Paras results in the UVM
case.Comment: Published at http://dx.doi.org/10.1214/105051606000000169 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Robust Superhedging with Jumps and Diffusion
We establish a nondominated version of the optional decomposition theorem in
a setting that includes jump processes with nonvanishing diffusion as well as
general continuous processes. This result is used to derive a robust
superhedging duality and the existence of an optimal superhedging strategy for
general contingent claims. We illustrate the main results in the framework of
nonlinear L\'evy processes.Comment: Forthcoming in 'Stochastic Processes and their Applications
The robust superreplication problem: a dynamic approach
In the frictionless discrete time financial market of Bouchard et al.(2015)
we consider a trader who, due to regulatory requirements or internal risk
management reasons, is required to hedge a claim in a risk-conservative
way relative to a family of probability measures . We first
describe the evolution of - the superhedging price at time of
the liability at maturity - via a dynamic programming principle and
show that can be seen as a concave envelope of
evaluated at today's prices. Then we consider an optimal investment problem for
a trader who is rolling over her robust superhedge and phrase this as a robust
maximisation problem, where the expected utility of inter-temporal consumption
is optimised subject to a robust superhedging constraint. This utility
maximisation is carrried out under a new family of measures ,
which no longer have to capture regulatory or institutional risk views but
rather represent trader's subjective views on market dynamics. Under suitable
assumptions on the trader's utility functions, we show that optimal investment
and consumption strategies exist and further specify when, and in what sense,
these may be unique
Hedging with transient price impact for non-covered and covered options
We solve the superhedging problem for European options in a market with
finite liquidity where trading has transient impact on prices, and possibly a
permanent one in addition. Impact is multiplicative to ensure positive asset
prices. Hedges and option prices depend on the physical and cash delivery
specifications of the option settlement. For non-covered options, where impact
at the inception and maturity dates matters, we characterize the superhedging
price as a viscosity solution of a degenerate semilinear pde that can have
gradient constraints. The non-linearity of the pde is governed by the transient
nature of impact through a resilience function. For covered options, the
pricing pde involves gamma constraints but is not affected by transience of
impact. We use stochastic target techniques and geometric dynamic programming
in reduced coordinates
Minimal Supersolutions of Convex BSDEs under Constraints
We study supersolutions of a backward stochastic differential equation, the
control processes of which are constrained to be continuous semimartingales of
the form . The generator may depend on the
decomposition and is assumed to be positive, jointly
convex and lower semicontinuous, and to satisfy a superquadratic growth
condition in and . We prove the existence of a
supersolution that is minimal at time zero and derive stability properties of
the non-linear operator that maps terminal conditions to the time zero value of
this minimal supersolution such as monotone convergence, Fatou's lemma and
-lower semicontinuity. Furthermore, we provide duality results within the
present framework and thereby give conditions for the existence of solutions
under constraints.Comment: 23 page
Mathematics of Quantitative Finance
The workshop on Mathematics of Quantitative Finance, organised at the Mathematisches Forschungsinstitut Oberwolfach from 26 February to 4 March 2017, focused on cutting edge areas of mathematical finance, with an emphasis on the applicability of the new techniques and models presented by the participants
Model-free bounds for multi-asset options using option-implied information and their exact computation
We consider derivatives written on multiple underlyings in a one-period
financial market, and we are interested in the computation of model-free upper
and lower bounds for their arbitrage-free prices. We work in a completely
realistic setting, in that we only assume the knowledge of traded prices for
other single- and multi-asset derivatives, and even allow for the presence of
bid-ask spread in these prices. We provide a fundamental theorem of asset
pricing for this market model, as well as a superhedging duality result, that
allows to transform the abstract maximization problem over probability measures
into a more tractable minimization problem over vectors, subject to certain
constraints. Then, we recast this problem into a linear semi-infinite
optimization problem, and provide two algorithms for its solution. These
algorithms provide upper and lower bounds for the prices that are
-optimal, as well as a characterization of the optimal pricing
measures. Moreover, these algorithms are efficient and allow the computation of
bounds in high-dimensional scenarios (e.g. when ). Numerical experiments
using synthetic data showcase the efficiency of these algorithms, while they
also allow to understand the reduction of model-risk by including additional
information, in the form of known derivative prices
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