333 research outputs found
Consumption processes and positively homogeneous projection properties
We constructively prove the existence of time-discrete consumption processes
for stochastic money accounts that fulfill a pre-specified positively
homogeneous projection property (PHPP) and let the account always be positive
and exactly zero at the end. One possible example is consumption rates forming
a martingale under the above restrictions. For finite spaces, it is shown that
any strictly positive consumption strategy with restrictions as above possesses
at least one corresponding PHPP and could be constructed from it. We also
consider numeric examples under time-discrete and -continuous account
processes, cases with infinite time horizons and applications to income
drawdown and bonus theory.Comment: 24 pages, 2 figure
Optimal consumption and investment with bounded downside risk for power utility functions
We investigate optimal consumption and investment problems for a
Black-Scholes market under uniform restrictions on Value-at-Risk and Expected
Shortfall. We formulate various utility maximization problems, which can be
solved explicitly. We compare the optimal solutions in form of optimal value,
optimal control and optimal wealth to analogous problems under additional
uniform risk bounds. Our proofs are partly based on solutions to
Hamilton-Jacobi-Bellman equations, and we prove a corresponding verification
theorem. This work was supported by the European Science Foundation through the
AMaMeF programme.Comment: 36 page
Financial rogue waves
The financial rogue waves are reported analytically in the nonlinear option
pricing model due to Ivancevic, which is nonlinear wave alternative of the
Black-Scholes model. These solutions may be used to describe the possible
physical mechanisms for rogue wave phenomenon in financial markets and related
fields.Comment: 4 papges, 2 figures, Final version accepted in Commun. Theor. Phys.,
201
An Optimal Execution Problem with Market Impact
We study an optimal execution problem in a continuous-time market model that
considers market impact. We formulate the problem as a stochastic control
problem and investigate properties of the corresponding value function. We find
that right-continuity at the time origin is associated with the strength of
market impact for large sales, otherwise the value function is continuous.
Moreover, we show the semi-group property (Bellman principle) and characterise
the value function as a viscosity solution of the corresponding
Hamilton-Jacobi-Bellman equation. We introduce some examples where the forms of
the optimal strategies change completely, depending on the amount of the
trader's security holdings and where optimal strategies in the Black-Scholes
type market with nonlinear market impact are not block liquidation but gradual
liquidation, even when the trader is risk-neutral.Comment: 36 pages, 8 figures, a modified version of the article "An optimal
execution problem with market impact" in Finance and Stochastics (2014
Eroding market stability by proliferation of financial instruments
We contrast Arbitrage Pricing Theory (APT), the theoretical basis for the
development of financial instruments, with a dynamical picture of an
interacting market, in a simple setting. The proliferation of financial
instruments apparently provides more means for risk diversification, making the
market more efficient and complete. In the simple market of interacting traders
discussed here, the proliferation of financial instruments erodes systemic
stability and it drives the market to a critical state characterized by large
susceptibility, strong fluctuations and enhanced correlations among risks. This
suggests that the hypothesis of APT may not be compatible with a stable market
dynamics. In this perspective, market stability acquires the properties of a
common good, which suggests that appropriate measures should be introduced in
derivative markets, to preserve stability.Comment: 26 pages, 8 figure
The dynamics of financial stability in complex networks
We address the problem of banking system resilience by applying
off-equilibrium statistical physics to a system of particles, representing the
economic agents, modelled according to the theoretical foundation of the
current banking regulation, the so called Merton-Vasicek model. Economic agents
are attracted to each other to exchange `economic energy', forming a network of
trades. When the capital level of one economic agent drops below a minimum, the
economic agent becomes insolvent. The insolvency of one single economic agent
affects the economic energy of all its neighbours which thus become susceptible
to insolvency, being able to trigger a chain of insolvencies (avalanche). We
show that the distribution of avalanche sizes follows a power-law whose
exponent depends on the minimum capital level. Furthermore, we present evidence
that under an increase in the minimum capital level, large crashes will be
avoided only if one assumes that agents will accept a drop in business levels,
while keeping their trading attitudes and policies unchanged. The alternative
assumption, that agents will try to restore their business levels, may lead to
the unexpected consequence that large crises occur with higher probability
Generalized pricing formulas for stochastic volatility jump diffusion models applied to the exponential Vasicek model
Path integral techniques for the pricing of financial options are mostly
based on models that can be recast in terms of a Fokker-Planck differential
equation and that, consequently, neglect jumps and only describe drift and
diffusion. We present a method to adapt formulas for both the path-integral
propagators and the option prices themselves, so that jump processes are taken
into account in conjunction with the usual drift and diffusion terms. In
particular, we focus on stochastic volatility models, such as the exponential
Vasicek model, and extend the pricing formulas and propagator of this model to
incorporate jump diffusion with a given jump size distribution. This model is
of importance to include non-Gaussian fluctuations beyond the Black-Scholes
model, and moreover yields a lognormal distribution of the volatilities, in
agreement with results from superstatistical analysis. The results obtained in
the present formalism are checked with Monte Carlo simulations.Comment: 9 pages, 2 figures, 1 tabl
A Delayed Black and Scholes Formula I
In this article we develop an explicit formula for pricing European options
when the underlying stock price follows a non-linear stochastic differential
delay equation (sdde). We believe that the proposed model is sufficiently
flexible to fit real market data, and is yet simple enough to allow for a
closed-form representation of the option price. Furthermore, the model
maintains the no-arbitrage property and the completeness of the market. The
derivation of the option-pricing formula is based on an equivalent martingale
measure
Quadratic BSDEs driven by a continuous martingale and application to utility maximization problem
In this paper, we study a class of quadratic Backward Stochastic Differential
Equations (BSDEs) which arises naturally when studying the problem of utility
maximization with portfolio constraints. We first establish existence and
uniqueness results for such BSDEs and then, we give an application to the
utility maximization problem. Three cases of utility functions will be
discussed: the exponential, power and logarithmic ones
Local time and the pricing of time-dependent barrier options
A time-dependent double-barrier option is a derivative security that delivers
the terminal value at expiry if neither of the continuous
time-dependent barriers b_\pm:[0,T]\to \RR_+ have been hit during the time
interval . Using a probabilistic approach we obtain a decomposition of
the barrier option price into the corresponding European option price minus the
barrier premium for a wide class of payoff functions , barrier functions
and linear diffusions . We show that the barrier
premium can be expressed as a sum of integrals along the barriers of
the option's deltas \Delta_\pm:[0,T]\to\RR at the barriers and that the pair
of functions solves a system of Volterra integral
equations of the first kind. We find a semi-analytic solution for this system
in the case of constant double barriers and briefly discus a numerical
algorithm for the time-dependent case.Comment: 32 pages, to appear in Finance and Stochastic
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