scholarly journals Perpetual American Defaultable Options in Models with Random Dividends and Partial Information

Risks ◽  
2018 ◽  
Vol 6 (4) ◽  
pp. 127 ◽  
Author(s):  
Pavel V. Gapeev ◽  
Hessah Al Motairi
Keyword(s):  
Free Boundary Problems ◽  
Partial Information ◽  
Asset Price ◽  
Risky Asset ◽  
Boundary Problems ◽  
Variable Formula ◽  
Closed Form Solutions ◽  
Price Process ◽  
Pricing Problems ◽  
Change Of Variable

We present closed-form solutions to the perpetual American dividend-paying put and call option pricing problems in two extensions of the Black–Merton–Scholes model with random dividends under full and partial information. We assume that the dividend rate of the underlying asset price changes its value at a certain random time which has an exponential distribution and is independent of the standard Brownian motion driving the price of the underlying risky asset. In the full information version of the model, it is assumed that this time is observable to the option holder, while in the partial information version of the model, it is assumed that this time is unobservable to the option holder. The optimal exercise times are shown to be the first times at which the underlying risky asset price process hits certain constant levels. The proof is based on the solutions of the associated free-boundary problems and the applications of the change-of-variable formula.

scholarly journals Perpetual American Cancellable Standard Options in Models with Last Passage Times

Algorithms ◽  
2020 ◽  
Vol 14 (1) ◽  
pp. 3
Author(s):  
Pavel V. Gapeev ◽  
Libo Li ◽  
Zhuoshu Wu
Keyword(s):  
Optimal Stopping ◽  
Free Boundary Problems ◽  
Asset Price ◽  
Stopping Times ◽  
Hitting Times ◽  
Explicit Solutions ◽  
Boundary Problems ◽  
Price Process ◽  
Optimal Stopping Problems ◽  
Passage Times

We derive explicit solutions to the perpetual American cancellable standard put and call options in an extension of the Black–Merton–Scholes model. It is assumed that the contracts are cancelled at the last hitting times for the underlying asset price process of some constant upper or lower levels which are not stopping times with respect to the observable filtration. We show that the optimal exercise times are the first times at which the asset price reaches some lower or upper constant levels. The proof is based on the reduction of the original optimal stopping problems to the associated free-boundary problems and the solution of the latter problems by means of the smooth-fit conditions.

scholarly journals Discounted Optimal Stopping Problems for Maxima of Geometric Brownian Motions With Switching Payoffs

2021 ◽  
Vol 53 (1) ◽  
pp. 189-219
Author(s):  
Pavel V. Gapeev ◽  
Peter M. Kort ◽  
Maria N. Lavrutich
Keyword(s):  
Optimal Stopping ◽  
Free Boundary Problems ◽  
Sunk Costs ◽  
Boundary Problems ◽  
Closed Form Solutions ◽  
Lookback Options ◽  
Normal Reflection ◽  
Running Maximum ◽  
Optimal Stopping Problems ◽  
Geometric Brownian Motions

AbstractWe present closed-form solutions to some discounted optimal stopping problems for the running maximum of a geometric Brownian motion with payoffs switching according to the dynamics of a continuous-time Markov chain with two states. The proof is based on the reduction of the original problems to the equivalent free-boundary problems and the solution of the latter problems by means of the smooth-fit and normal-reflection conditions. We show that the optimal stopping boundaries are determined as the maximal solutions of the associated two-dimensional systems of first-order nonlinear ordinary differential equations. The obtained results are related to the valuation of real switching lookback options with fixed and floating sunk costs in the Black–Merton–Scholes model.

RISK-NEUTRAL MEASURES AND PRICING FOR A PURE JUMP PRICE PROCESS

2009 ◽  
Vol 24 (1) ◽  
pp. 47-76 ◽  
Author(s):  
Anna Gerardi ◽  
Paola Tardelli
Keyword(s):  
Point Processes ◽  
Marked Point ◽  
Asset Price ◽  
Risky Asset ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Martingale Measure ◽  
Price Process ◽  
Risk Neutral Measures

This article considers the asset price movements in a financial market when risky asset prices are modeled by marked point processes. Their dynamics depend on an underlying event arrivals process, modeled again by a marked point process. Taking into account the presence of catastrophic events, the possibility of common jump times between the risky asset price process and the arrivals process is allowed. By setting and solving a suitable control problem, the characterization of the minimal entropy martingale measure is obtained. In a particular case, a pricing problem is also discussed.

scholarly journals Partially informed investors: hedging in an incomplete market with default

2015 ◽  
Vol 52 (03) ◽  
pp. 718-735 ◽  
Author(s):  
P. Tardelli
Keyword(s):  
Point Processes ◽  
Value Function ◽  
Partial Information ◽  
Marked Point ◽  
Asset Price ◽  
Risky Asset ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Control Problems ◽  
The Value Function

In a defaultable market, an investor trades having only partial information about the behavior of the market. Taking into account the intraday stock movements, the risky asset prices are modelled by marked point processes. Their dynamics depend on an unobservable process, representing the amount of news reaching the market. This is a marked point process, which may have common jump times with the risky asset price processes. The problem of hedging a defaultable claim is studied. In order to discuss all these topics, in this paper we examine stochastic control problems using backward stochastic differential equations (BSDEs) and filtering techniques. The goal of this paper is to construct a sequence of functions converging to the value function, each of these is the unique solution of a suitable BSDE.

scholarly journals Partially informed investors: hedging in an incomplete market with default

2015 ◽  
Vol 52 (3) ◽  
pp. 718-735
Author(s):  
P. Tardelli
Keyword(s):  
Point Processes ◽  
Value Function ◽  
Partial Information ◽  
Marked Point ◽  
Asset Price ◽  
Risky Asset ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Control Problems ◽  
The Value Function

In a defaultable market, an investor trades having only partial information about the behavior of the market. Taking into account the intraday stock movements, the risky asset prices are modelled by marked point processes. Their dynamics depend on an unobservable process, representing the amount of news reaching the market. This is a marked point process, which may have common jump times with the risky asset price processes. The problem of hedging a defaultable claim is studied. In order to discuss all these topics, in this paper we examine stochastic control problems using backward stochastic differential equations (BSDEs) and filtering techniques. The goal of this paper is to construct a sequence of functions converging to the value function, each of these is the unique solution of a suitable BSDE.

scholarly journals Stochastic sensitivity of bull and bear states

2021 ◽  
Author(s):  
Jochen Jungeilges ◽  
Elena Maklakova ◽  
Tatyana Perevalova
Keyword(s):  
Additive Noise ◽  
Asset Price ◽  
Market Model ◽  
Asset Market ◽  
Stochastic Version ◽  
Market Participants ◽  
Price Process ◽  
Stochastic Sensitivity ◽  
Stable Equilibria ◽  
Parametric Noise

AbstractWe study the price dynamics generated by a stochastic version of a Day–Huang type asset market model with heterogenous, interacting market participants. To facilitate the analysis, we introduce a methodology that allows us to assess the consequences of changes in uncertainty on the dynamics of an asset price process close to stable equilibria. In particular, we focus on noise-induced transitions between bull and bear states of the market under additive as well as parametric noise. Our results are obtained by combining the stochastic sensitivity function (SSF) approach, a mixture of analytical and numerical techniques, due to Mil’shtein and Ryashko (1995) with concepts and techniques from the study of non-smooth 1D maps. We find that the stochastic sensitivity of the respective bull and bear equilibria in the presence of additive noise is higher than under parametric noise. Thus, recurrent transitions are likely to be observed already for relatively low intensities of additive noise.

Sign in / Sign up

Export Citation Format

Share Document