scholarly journals Optimal Harvesting When the Exchange Rate Is a Semimartingale

2011 ◽  
Vol 2011 ◽  
pp. 1-19
Author(s):  
E. R. Offen ◽  
E. M. Lungu
Keyword(s):  
Exchange Rate ◽  
General Solution ◽  
Value Function ◽  
Sufficient Conditions ◽  
Optimal Harvesting ◽  
Singular Stochastic Control ◽  
Total Income ◽  
Black Scholes ◽  
The Exchange Rate ◽  
The Value Function

We consider harvesting in the Black-Scholes Quanto Market when the exchange rate is being modeled by the process Et=E0exp⁡{Xt}, where Xt is a semimartingale, and we ask the following question: What harvesting strategy γ* and the value function Φ maximize the expected total income of an investment? We formulate a singular stochastic control problem and give sufficient conditions for the existence of an optimal strategy. We found that, if the value function is not too sensitive to changes in the prices of the investments, the problem reduces to that of Lungu and Øksendal. However, the general solution of this problem still remains elusive.

scholarly journals Optimal Foreign Exchange Rate Intervention in Lévy Markets

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Masimba Aspinas Mutakaya ◽  
Eriyoti Chikodza ◽  
Edward T. Chiyaka
Keyword(s):  
Exchange Rate ◽  
Central Bank ◽  
Impulse Control ◽  
Value Function ◽  
Quasivariational Inequalities ◽  
Expected Number ◽  
A Value ◽  
The Exchange Rate ◽  
Intervention Costs ◽  
Exchange Rate Intervention

This paper considers an exchange rate problem in Lévy markets, where the Central Bank has to intervene. We assume that, in the absence of control, the exchange rate evolves according to Brownian motion with a jump component. The Central Bank is allowed to intervene in order to keep the exchange rate as close as possible to a prespecified target value. The interventions by the Central Bank are associated with costs. We present the situation as an impulse control problem, where the objective of the bank is to minimize the intervention costs. In particular, the paper extends the model by Huang, 2009, to incorporate a jump component. We formulate and prove an optimal verification theorem for the impulse control. We then propose an impulse control and construct a value function and then verify that they solve the quasivariational inequalities. Our results suggest that if the expected number of jumps is high the Central Bank will intervene more frequently and with large intervention amounts hence the intervention costs will be high.

scholarly journals Projection of the two-dimensional Black-Scholes equation for options with underlying stock and strike prices in two different currencies

2021 ◽  
Vol 68 (1 Jan-Feb) ◽  
Author(s):  
Guillermo Chacón-Acosta ◽  
Rubén O. Salas
Keyword(s):  
Exchange Rate ◽  
Interest Rates ◽  
Stock Price ◽  
Option Price ◽  
One Dimensional ◽  
Several Variables ◽  
Exercise Price ◽  
Black Scholes ◽  
Reduction Methods ◽  
The Exchange Rate

The two-variable Black-Scholes equation is used to study the option exercise price of two different currencies. Due to the complexity of dealing with several variables, reduction methods have been implemented to deal with these problems. This paper proposes an alternative reduction by using the so-called Zwanzig projection method to one-dimension, successfully developed to study the diffusion in confined systems. In this case, the option price depends on the stock price and the exchange rate between currencies. We assume that the exchange rate between currencies will depend on the stock price through some model that bounds such dependence, which somehow influences the final option price.As a result, we find a projected one-dimensional Black-Scholes equation similar to the so-called Fick-Jacobs equation for diffusion on channels. This equation is an effective Black-Scholes equation with two different interest rates, whose solution gives rise to a modified Black-Scholes formula. The properties of this solution are shown and were graphically compared with previously found solutions, showing that the corresponding difference is bounded.

scholarly journals Optimal dividends with partial information and stopping of a degenerate reflecting diffusion

2019 ◽  
Vol 24 (1) ◽  
pp. 71-123 ◽  
Author(s):  
Tiziano De Angelis
Keyword(s):  
Stochastic Control ◽  
Value Function ◽  
Partial Information ◽  
Point Of View ◽  
Two Dimensional ◽  
Singular Stochastic Control ◽  
State Dependent ◽  
Optimal Dividend ◽  
Analytical Point ◽  
The Value Function

Abstract We study the optimal dividend problem for a firm’s manager who has partial information on the profitability of the firm. The problem is formulated as one of singular stochastic control with partial information on the drift of the underlying process and with absorption. In the Markovian formulation, we have a two-dimensional degenerate diffusion whose first component is singularly controlled. Moreover, the process is absorbed when its first component hits zero. The free boundary problem (FBP) associated to the value function of the control problem is challenging from the analytical point of view due to the interplay of degeneracy and absorption. We find a probabilistic way to show that the value function of the dividend problem is a smooth solution of the FBP and to construct an optimal dividend strategy. Our approach establishes a new link between multidimensional singular stochastic control problems with absorption and problems of optimal stopping with ‘creation’. One key feature of the stopping problem is that creation occurs at a state-dependent rate of the ‘local time’ of an auxiliary two-dimensional reflecting diffusion.

scholarly journals Differentiability of the Efficient Frontier when Commitment to Risk Sharing is Limited

2006 ◽  
Vol 6 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Thorsten V. Koeppl
Keyword(s):  
Risk Sharing ◽  
Value Function ◽  
Sufficient Conditions ◽  
Efficient Frontier ◽  
Limited Commitment ◽  
History Dependence ◽  
Efficient Allocations ◽  
The Value Function

This paper shows that the value function describing efficient risk sharing with limited commitment is not necessarily differentiable everywhere. We link differentiability of the value function to history dependence of efficient allocations and provide sufficient conditions for both properties.

scholarly journals On properties of one functional used in software constructions for solving differential games

2021 ◽  
Vol 31 (4) ◽  
pp. 668-696
Author(s):  
A.G. Chentsov
Keyword(s):  
Value Function ◽  
Sufficient Conditions ◽  
Game Problem ◽  
Limit Function ◽  
Phase Constraints ◽  
Nonlinear Differential Game ◽  
Target Set ◽  
Iteration Number ◽  
Game Position ◽  
The Value Function

Nonlinear differential game (DG) is investigated; relaxations of the game problem of guidance are investigated also. The variant of the program iterations method realized in the space of position functions and delivering in limit the value function of the minimax-maximin DG for special functionals of a trajectory is considered. For every game position, this limit function realizes the least size of the target set neighborhood for which, under proportional weakening of phase constraints, the player interested in a guidance yet guarantees its realization. Properties of above-mentioned functionals and limit function are investigated. In particular, sufficient conditions for realization of values of given function under fulfilment of finite iteration number are obtained.

The value function for time-related decisions

2011 ◽  
Author(s):  
Anouk Festjens ◽  
Siegfried Dewitte ◽  
Enrico Diecidue ◽  
Sabrina Bruyneel
Keyword(s):  
Value Function ◽  
The Value Function
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