8,112 research outputs found
Problems in Mathematical Finance Related to Transaction Costs and Model Uncertainty.
This thesis is devoted to the study of three problems in mathematical finance which involve either transaction costs or model uncertainty or both.
In Chapter II, we investigate the Fundamental Theorem of Asset Pricing (FTAP) under both transaction costs and model uncertainty, where model uncertainty is described by a family of probability measures, possibly non-dominated. We first show that the recent results on the FTAP and the super-hedging theorem in the context of model uncertainty can be extended to the case where only options available for static hedging (hedging options) are quoted with bid-ask spreads. In this set-up, we need to work with the notion of robust no-arbitrage which turns out to be equivalent to no-arbitrage under the additional assumption that hedging options with non-zero spread are non-redundant. Next, we look at the more difficult case where the market consists of a money market and a dynamically traded stock with bid-ask spread. Under a continuity assumption, we prove using a backward-forward scheme that no-arbitrage is equivalent to the existence of a suitable family of consistent price systems.
In Chapter III, we study the problem where an individual targets at a given consumption rate, invests in a risky financial market, and seeks to minimize the probability of lifetime ruin under drift uncertainty. Using stochastic control, we characterize the value function as the unique classical solution of an associated Hamilton-Jacobi-Bellman (HJB) equation, obtain feedback forms for the optimal investment and drift distortion, and discuss their dependence on various model parameters. In analyzing the HJB equation, we establish the existence and uniqueness of viscosity solution using Perron's method, and then upgrade regularity by working with an equivalent convex problem obtained via the Cole-Hopf transformation.
In Chapter IV, we adapt stochastic Perron's method to the lifetime ruin problem under proportional transaction costs which can be formulated as a singular stochastic control problem. Without relying on the Dynamic Programming Principle, we characterize the value function as the unique viscosity solution of an associated variational inequality. We also provide a complete proof of the comparison principle which is the main assumption of stochastic Perron's method.PhDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/111560/1/yuchong_1.pd
Consistent Price Systems under Model Uncertainty
We develop a version of the fundamental theorem of asset pricing for
discrete-time markets with proportional transaction costs and model
uncertainty. A robust notion of no-arbitrage of the second kind is defined and
shown to be equivalent to the existence of a collection of strictly consistent
price systems.Comment: 19 page
A note on the Fundamental Theorem of Asset Pricing under model uncertainty
We show that the results of ArXiv:1305.6008 on the Fundamental Theorem of
Asset Pricing and the super-hedging theorem can be extended to the case in
which the options available for static hedging (\emph{hedging options}) are
quoted with bid-ask spreads. In this set-up, we need to work with the notion of
\emph{robust no-arbitrage} which turns out to be equivalent to no-arbitrage
under the additional assumption that hedging options with non-zero spread are
\emph{non-redundant}. A key result is the closedness of the set of attainable
claims, which requires a new proof in our setting.Comment: Final version. To appear in Risk
Conditional-Mean Hedging Under Transaction Costs in Gaussian Models
We consider so-called regular invertible Gaussian Volterra processes and
derive a formula for their prediction laws. Examples of such processes include
the fractional Brownian motions and the mixed fractional Brownian motions. As
an application, we consider conditional-mean hedging under transaction costs in
Black-Scholes type pricing models where the Brownian motion is replaced with a
more general regular invertible Gaussian Volterra process.Comment: arXiv admin note: text overlap with arXiv:1706.0153
Utility indifference pricing with market incompleteness
Utility indifference pricing and hedging theory is presented, showing
how it leads to linear or to non-linear pricing rules for contingent
claims. Convex duality is first used to derive probabilistic
representations for exponential utility-based prices, in a general
setting with locally bounded semi-martingale price processes. The
indifference price for a finite number of claims gives a non-linear
pricing rule, which reduces to a linear pricing rule as the number of
claims tends to zero, resulting in the so-called marginal
utility-based price of the claim. Applications to basis risk models
with lognormal price processes, under full and partial information
scenarios are then worked out in detail. In the full information case,
a claim on a non-traded asset is priced and hedged using a correlated
traded asset. The resulting hedge requires knowledge of the drift
parameters of the asset price processes, which are very difficult to
estimate with any precision. This leads naturally to a further
application, a partial information problem, with the drift parameters
assumed to be random variables whose values are revealed to the hedger
in a Bayesian fashion via a filtering algorithm. The indifference
price is given by the solution to a non-linear PDE, reducing to a
linear PDE for the marginal price when the number of claims becomes
infinitesimally small
Coherent Price Systems and Uncertainty-Neutral Valuation
We consider fundamental questions of arbitrage pricing arising when the
uncertainty model is given by a set of possible mutually singular probability
measures. With a single probability model, essential equivalence between the
absence of arbitrage and the existence of an equivalent martingale measure is a
folk theorem, see Harrison and Kreps (1979). We establish a microeconomic
foundation of sublinear price systems and present an extension result. In this
context we introduce a prior dependent notion of marketed spaces and viable
price systems. We associate this extension with a canonically altered concept
of equivalent symmetric martingale measure sets, in a dynamic trading framework
under absence of prior depending arbitrage. We prove the existence of such sets
when volatility uncertainty is modeled by a stochastic differential equation,
driven by Peng's G-Brownian motions
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