799 research outputs found
Valuation of Employee Stock Options (ESOs) by means of Mean-Variance Hedging
We consider the problem of ESO valuation in continuous time. In particular,
we consider models that assume that an appropriate random time serves as a
proxy for anything that causes the ESO's holder to exercise the option early,
namely, reflects the ESO holder's job termination risk as well as early
exercise behaviour. In this context, we study the problem of ESO valuation by
means of mean-variance hedging. Our analysis is based on dynamic programming
and uses PDE techniques. We also express the ESO's value that we derive as the
expected discounted payoff that the ESO yields with respect to an equivalent
martingale measure, which does not coincide with the minimal martingale measure
or the variance-optimal measure. Furthermore, we present a numerical study that
illustrates aspects or our theoretical results
A level-set approach for stochastic optimal control problems under controlled-loss constraints
We study a family of optimal control problems under a set of controlled-loss
constraints holding at different deterministic dates. The characterization of
the associated value function by a Hamilton-Jacobi-Bellman equation usually
calls for additional strong assumptions on the dynamics of the processes
involved and the set of constraints. To treat this problem in absence of those
assumptions, we first convert it into a state-constrained stochastic target
problem and then apply a level-set approach. With this approach, the state
constraints can be managed through an exact penalization technique
A Simulation Approach to Optimal Stopping Under Partial Information
We study the numerical solution of nonlinear partially observed optimal
stopping problems. The system state is taken to be a multi-dimensional
diffusion and drives the drift of the observation process, which is another
multi-dimensional diffusion with correlated noise. Such models where the
controller is not fully aware of her environment are of interest in applied
probability and financial mathematics. We propose a new approximate numerical
algorithm based on the particle filtering and regression Monte Carlo methods.
The algorithm maintains a continuous state-space and yields an integrated
approach to the filtering and control sub-problems. Our approach is entirely
simulation-based and therefore allows for a robust implementation with respect
to model specification. We carry out the error analysis of our scheme and
illustrate with several computational examples. An extension to discretely
observed stochastic volatility models is also considered
Optimal Electoral Timing: Exercise Wisely and You May Live Longer
In many democratic countries, the timing of elections is flexible. We explore this potentially valuable option using insights from option pricing in finance. The paper offers three main contributions on this problem. First, we derive a rationally-based mean-reverting political support process for the parties, assuming that politically heterogeneous voters continuously learn over time about evolving party fortunes. We solve for the long-run density for this process and derive the polling process from it by adding polling noise. Second, we explore optimal timing using the political support process. The incumbent sees its poll support, and must call an election within five years of the last election to maximize its expected total time in office. This resembles the optimal exercise rule for an American financial option. This option is recursive, and the waiting and stopping values subtly interact. We prove the existence of the optimal exercise rule in this setting, and show that the expected longevity is a convex-thenconcave function of the political support. Our model is tractable enough that we can analytically derive how the exercise rule responds to parametric shifts. We calibrate our model to the Labour-Tory rivalry in the U.K., with polling data from 1943-2005 and the 16 elections after 1945. Excluding three elections essentially forced by weak governments, our maximizing story quite well explains when the elections were called, and beats simple linear regressions. We also measure the value of election options, finding that over the long run they should more than double the expected time in power of a fixed term electoral cycle.American option, European option, Brownian motion, Electoral timing
Optimal stopping under probability distortion
We formulate an optimal stopping problem for a geometric Brownian motion
where the probability scale is distorted by a general nonlinear function. The
problem is inherently time inconsistent due to the Choquet integration
involved. We develop a new approach, based on a reformulation of the problem
where one optimally chooses the probability distribution or quantile function
of the stopped state. An optimal stopping time can then be recovered from the
obtained distribution/quantile function, either in a straightforward way for
several important cases or in general via the Skorokhod embedding. This
approach enables us to solve the problem in a fairly general manner with
different shapes of the payoff and probability distortion functions. We also
discuss economical interpretations of the results. In particular, we justify
several liquidation strategies widely adopted in stock trading, including those
of "buy and hold", "cut loss or take profit", "cut loss and let profit run" and
"sell on a percentage of historical high".Comment: Published in at http://dx.doi.org/10.1214/11-AAP838 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Robust Strategies for Optimal Order Execution in the Almgren-Chriss Framework
Assuming geometric Brownian motion as unaffected price process ,
Gatheral & Schied (2011) derived a strategy for optimal order execution that
reacts in a sensible manner on market changes but can still be computed in
closed form. Here we will investigate the robustness of this strategy with
respect to misspecification of the law of . We prove the surprising result
that the strategy remains optimal whenever is a square-integrable
martingale. We then analyze the optimization criterion of Gatheral & Schied
(2011) in the case in which is any square-integrable semimartingale and
we give a closed-form solution to this problem. As a corollary, we find an
explicit solution to the problem of minimizing the expected liquidation costs
when the unaffected price process is a square-integrable semimartingale. The
solutions to our problems are found by stochastically solving a finite-fuel
control problem without assumptions of Markovianity
General Stopping Behaviors of Naive and Non-Committed Sophisticated Agents, with Application to Probability Distortion
We consider the problem of stopping a diffusion process with a payoff
functional that renders the problem time-inconsistent. We study stopping
decisions of naive agents who reoptimize continuously in time, as well as
equilibrium strategies of sophisticated agents who anticipate but lack control
over their future selves' behaviors. When the state process is one dimensional
and the payoff functional satisfies some regularity conditions, we prove that
any equilibrium can be obtained as a fixed point of an operator. This operator
represents strategic reasoning that takes the future selves' behaviors into
account. We then apply the general results to the case when the agents distort
probability and the diffusion process is a geometric Brownian motion. The
problem is inherently time-inconsistent as the level of distortion of a same
event changes over time. We show how the strategic reasoning may turn a naive
agent into a sophisticated one. Moreover, we derive stopping strategies of the
two types of agent for various parameter specifications of the problem,
illustrating rich behaviors beyond the extreme ones such as "never-stopping" or
"never-starting"
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