19,710 research outputs found
Optimal hedging of Derivatives with transaction costs
We investigate the optimal strategy over a finite time horizon for a
portfolio of stock and bond and a derivative in an multiplicative Markovian
market model with transaction costs (friction). The optimization problem is
solved by a Hamilton-Bellman-Jacobi equation, which by the verification theorem
has well-behaved solutions if certain conditions on a potential are satisfied.
In the case at hand, these conditions simply imply arbitrage-free
("Black-Scholes") pricing of the derivative. While pricing is hence not changed
by friction allow a portfolio to fluctuate around a delta hedge. In the limit
of weak friction, we determine the optimal control to essentially be of two
parts: a strong control, which tries to bring the stock-and-derivative
portfolio towards a Black-Scholes delta hedge; and a weak control, which moves
the portfolio by adding or subtracting a Black-Scholes hedge. For simplicity we
assume growth-optimal investment criteria and quadratic friction.Comment: Revised version, expanded introduction and references 17 pages,
submitted to International Journal of Theoretical and Applied Finance (IJTAF
Indifference Pricing and Hedging in a Multiple-Priors Model with Trading Constraints
This paper considers utility indifference valuation of derivatives under
model uncertainty and trading constraints, where the utility is formulated as
an additive stochastic differential utility of both intertemporal consumption
and terminal wealth, and the uncertain prospects are ranked according to a
multiple-priors model of Chen and Epstein (2002). The price is determined by
two optimal stochastic control problems (mixed with optimal stopping time in
the case of American option) of forward-backward stochastic differential
equations. By means of backward stochastic differential equation and partial
differential equation methods, we show that both bid and ask prices are closely
related to the Black-Scholes risk-neutral price with modified dividend rates.
The two prices will actually coincide with each other if there is no trading
constraint or the model uncertainty disappears. Finally, two applications to
European option and American option are discussed.Comment: 28 pages in Science China Mathematics, 201
On exact null controllability of Black-Scholes equation
summary:In this paper we discuss the exact null controllability of linear as well as nonlinear Black–Scholes equation when both the stock volatility and risk-free interest rate influence the stock price but they are not known with certainty while the control is distributed over a subdomain. The proof of the linear problem relies on a Carleman estimate and observability inequality for its own dual problem and that of the nonlinear one relies on the infinite dimensional Kakutani fixed point theorem with topology
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
Pricing Asian Options with Floating Exercise Price Equal to the Arithmetic Mean: An Optimal Stochastic Control Approach
En esta investigación se caracteriza la prima de una opción de venta asiática de tipo europeo con precio de ejercicio flotante igual a la media aritmética, mediante el análisis de solución a un problema de control óptimo estocástico en tiempo continuo, que modela el proceso de toma de decisiones de consumo-inversión de un consumidor racional en un horizonte finito de planeación con fecha final estocástica. La prima obtenida es una ecuación diferencial parcial de segundo orden que corresponde a la ecuación de Black-Scholes-Merton, con la diferencia de que esta se establece con fundamentos de racionalidad económica. Asimismo, se resuelve analÃticamente la ecuación diferencial parcial de Hamilton-Jacobi-Bellman que optimiza el problema de control óptimo estocástico planteado.This research characterizes the premium of a European-type +Asian put option with floating exercise price equal to the arithmetic mean, through the solution analysis to a stochastic optimal control problem in continuous time, that models the decision-making process of consumption-investment of a rational consumer in a finite horizon of planning with stochastic terminal date. The premium obtained is a partial differential equation of second order corresponding to the Black-Scholes-Merton equation, with the difference that this is established with fundamentals of economic rationality. Also, the Hamilton-Jacobi-Bellman partial differential equation is solved, which optimizes the stochastic optimal control problem
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