Results 1  10
of
16
The Russian Option: Reduced Regret
, 1993
"... this paper the value of the option (i.e. the supremum in (1.2)) will be found exactly, and in particular it will be shown that the maximum in (1.2) is finite if and only if r ? ¯ : (1.4) Assuming (1.4), an explicit formula is given for both the maximal expected present value and the optimal stopping ..."
Abstract

Cited by 36 (2 self)
 Add to MetaCart
this paper the value of the option (i.e. the supremum in (1.2)) will be found exactly, and in particular it will be shown that the maximum in (1.2) is finite if and only if r ? ¯ : (1.4) Assuming (1.4), an explicit formula is given for both the maximal expected present value and the optimal stopping rule in (2.4), which is not a fixed time rule but depends heavily on the observed values of X t and S t . We call the financial option described above a "Russian option" for two reasons. First, this name serves to (facetiously) differentiate it from American and European options, which have been extensively studied in financial economics, especially with the new interest in market economics in Russia. Second, our solution of the stopping problem (1.2) is derived by the socalled principle of smooth fit, first enunciated by the great Russian mathematician, A. N. Kolmogorov, cf. [4, 5]. The Russian option is characterized by "reduced regret" because the owner is paid the maximum stock price up to the time of exercise and hence feels less remorse at not having exercised at the maximum. For purposes of comparison and to emphasize the mathematical nature of the contribution here, we conclude the paper by analyzing an optimal stopping problem for the Russian option based on Bachelier's (1900) original linear model of stock price fluctuations, X
On the optimal stopping problem for onedimensional diffusions, 2002. Working Paper (http://www.stat.columbia.edu/ ˜ik/DAYKAR.pdf
"... A new characterization of excessive functions for arbitrary one–dimensional regular diffusion processes is provided, using the notion of concavity. It is shown that excessivity is equivalent to concavity in some suitable generalized sense. This permits a characterization of the value function of the ..."
Abstract

Cited by 35 (2 self)
 Add to MetaCart
A new characterization of excessive functions for arbitrary one–dimensional regular diffusion processes is provided, using the notion of concavity. It is shown that excessivity is equivalent to concavity in some suitable generalized sense. This permits a characterization of the value function of the optimal stopping problem as “the smallest nonnegative concave majorant of the reward function ” and allows us to generalize results of Dynkin and Yushkevich for standard Brownian motion. Moreover, we show how to reduce the discounted optimal stopping problems for an arbitrary diffusion process to an undiscounted optimal stopping problem for standard Brownian motion. The concavity of the value functions also leads to conclusions about their smoothness, thanks to the properties of concave functions. One is thus led to a new perspective and new facts about the principle of smooth–fit in the context of optimal stopping. The results are illustrated in detail on a number of non–trivial, concrete optimal stopping problems, both old and new.
Risk vs. ProfitPotential; A Model for Corporate Strategy
 J. Econ. Dynam. Control
, 1996
"... A firm whose net earnings are uncertain, and that is subject to the risk of bankruptcy, must choose between paying dividends and retaining earnings in a liquid reserve. Also, different operating strategies imply different combinations of expected return and variance. We model the firm's cash reserve ..."
Abstract

Cited by 27 (0 self)
 Add to MetaCart
A firm whose net earnings are uncertain, and that is subject to the risk of bankruptcy, must choose between paying dividends and retaining earnings in a liquid reserve. Also, different operating strategies imply different combinations of expected return and variance. We model the firm's cash reserve as the difference between the cumulative net earnings and the cumulative dividends. The first is a diffusion (additive), whose drift/volatility pair is chosen dynamically from a finite set, A. The second is an arbitrary nondecreasing process, chosen by the firm. The firm's strategy must be nonclairvoyant. The firm is bankrupt at the first time, T , at which the cash reserve falls to zero (T may be infinite), and the firm's objective is to maximize the expected total discounted dividends from 0 to T , given an initial reserve, x; denote this maximum by V (x). We calculate V explicitly, as a function of the set A and the discount rate. The optimal policy has the form: (1) pay no dividends if ...
On the American option problem
 Math. Finance
, 2005
"... We show how the changeofvariable formula with local time on curves derived recently in [17] can be used to prove that the optimal stopping boundary for the American put option can be characterized as the unique solution of a nonlinear integral equation arising from the early exercise premium repre ..."
Abstract

Cited by 17 (7 self)
 Add to MetaCart
We show how the changeofvariable formula with local time on curves derived recently in [17] can be used to prove that the optimal stopping boundary for the American put option can be characterized as the unique solution of a nonlinear integral equation arising from the early exercise premium representation. This settles the question raised in [15] (dating back to [13]). 1.
Optimal stopping of linear diffusions with random discounting. Working paper, http://www.princeton.edu/ sdayanik/papers/additive.pdf
, 2003
"... Abstract. We propose a new solution method for optimal stopping problems for linear diffusions with random discounting. First, we extend the class of excessive functions for general diffusions and show that they are essentially concave. Then we use the new characterization of excessive functions to ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
Abstract. We propose a new solution method for optimal stopping problems for linear diffusions with random discounting. First, we extend the class of excessive functions for general diffusions and show that they are essentially concave. Then we use the new characterization of excessive functions to show that optimal stopping problems for linear diffusions discounted with respect to a continuous additive functional, recently studied by Beibel and Lerche [Theory Probab. Appl., 45(4):547–557, 2001], can be reduced to an undiscounted optimal stopping problem for standard Brownian motion. The latter problem can be solved essentially by inspection. The necessary and sufficient conditions for the existence of an optimal stopping rule are proved when the reward function is continuous. A proof of the smooth–fit principle is also provided. The results are illustrated on examples. 1.
Principle of smooth fit and diffusions with angles
 Stochastics
, 2007
"... We show that there exists a regular diffusion process X and a differentiable gain function G such that the value function V of the optimal stopping problem V (x) = sup ExG(Xτ) τ fails to satisfy the smooth fit condition V ′(b) = G ′(b) at the optimal stopping point b. On the other hand, if the sca ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
We show that there exists a regular diffusion process X and a differentiable gain function G such that the value function V of the optimal stopping problem V (x) = sup ExG(Xτ) τ fails to satisfy the smooth fit condition V ′(b) = G ′(b) at the optimal stopping point b. On the other hand, if the scale function S of X is differentiable at b, then the smooth fit condition V ′(b) = G ′(b) holds (whenever X is regular and G is differentiable at b). We give an example showing that the latter can happen even when d + G/dS < d + V/dS < d − V/dS < d − G/dS at b. 1.
Optimal Bond Trading with Personal Taxes: Implications for Bond Prices and Estimated Tax Brackets and Yield Curves
 Business, University of Chicago
, 1983
"... Earlier versions of the paper were presented at the annual AFA meeting in Washington, D.C. and at workshops at the University of ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
Earlier versions of the paper were presented at the annual AFA meeting in Washington, D.C. and at workshops at the University of
On nonlinear integral equations arising in problems of optimal stopping
 Proc. Functional Anal. VII (Dubrovnik 2001), Various
, 2002
"... Let B = (Bt)0 t 1 be a standard Brownian motion started at zero, let 0 be given and fixed, and let G: [0; 1]2IR! IR be a measurable function. Consider the optimal stopping problem: ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
Let B = (Bt)0 t 1 be a standard Brownian motion started at zero, let 0 be given and fixed, and let G: [0; 1]2IR! IR be a measurable function. Consider the optimal stopping problem:
The Wiener Disorder Problem with Finite Horizon
"... The Wiener disorder problem seeks to determine a stopping time which is as close as possible to the (unknown) time of ’disorder ’ when the drift of an observed Wiener process changes from one value to another. In this paper we present a solution of the Wiener disorder problem when the horizon is fin ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
The Wiener disorder problem seeks to determine a stopping time which is as close as possible to the (unknown) time of ’disorder ’ when the drift of an observed Wiener process changes from one value to another. In this paper we present a solution of the Wiener disorder problem when the horizon is finite. The method of proof is based on reducing the initial problem to a parabolic freeboundary problem where the continuation region is determined by a continuous curved boundary. By means of the changeofvariable formula containing the local time of a diffusion process on curves we show that the optimal boundary can be characterized as a unique solution of the nonlinear integral equation. 1.
MODELING SHORTEST PATH GAMES WITH PETRI NETS: A LYAPUNOV BASED THEORY
"... In this paper we introduce a new modeling paradigm for shortest path games representation with Petri nets. Whereas previous works have restricted attention to tracking the net using Bellman’s equation as a utility function, this work uses a Lyapunovlike function. In this sense, we change the tradit ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
In this paper we introduce a new modeling paradigm for shortest path games representation with Petri nets. Whereas previous works have restricted attention to tracking the net using Bellman’s equation as a utility function, this work uses a Lyapunovlike function. In this sense, we change the traditional cost function by a trajectorytracking function which is also an optimal costtotarget function. This makes a significant difference in the conceptualization of the problem domain, allowing the replacement of the Nash equilibrium point by the Lyapunov equilibrium point in game theory. We show that the Lyapunov equilibrium point coincides with the Nash equilibrium point. As a consequence, all properties of equilibrium and stability are preserved in game theory. This is the most important contribution of this work. The potential of this approach remains in its formal proof simplicity for the existence of an equilibrium point.