Optimal stopping in Hilbert spaces and pricing of American options

1999 ◽  
Vol 50 (1) ◽  
pp. 135-147 ◽  
Author(s):  
Dariusz G&acedil;tarek *, † , Andrzej
Keyword(s):  
Hilbert Spaces ◽  
Optimal Stopping ◽  
American Options

scholarly journals American options in an imperfect complete market with default

2018 ◽  
Vol 64 ◽  
pp. 93-110 ◽  
Author(s):  
Roxana Dumitrescu ◽  
Marie-Claire Quenez ◽  
Agnès Sulem
Keyword(s):  
Optimal Stopping ◽  
Value Function ◽  
American Options ◽  
American Option ◽  
Stopping Times ◽  
Optimal Stopping Problem ◽  
Nonlinear Expectation ◽  
Portfolio Strategy ◽  
Reflected Bsde ◽  
The Value Function

We study pricing and hedging for American options in an imperfect market model with default, where the imperfections are taken into account via the nonlinearity of the wealth dynamics. The payoff is given by an RCLL adapted process (ξt). We define the seller's price of the American option as the minimum of the initial capitals which allow the seller to build up a superhedging portfolio. We prove that this price coincides with the value function of an optimal stopping problem with a nonlinear expectation 𝓔g (induced by a BSDE), which corresponds to the solution of a nonlinear reflected BSDE with obstacle (ξt). Moreover, we show the existence of a superhedging portfolio strategy. We then consider the buyer's price of the American option, which is defined as the supremum of the initial prices which allow the buyer to select an exercise time τ and a portfolio strategy φ so that he/she is superhedged. We show that the buyer's price is equal to the value function of an optimal stopping problem with a nonlinear expectation, and that it can be characterized via the solution of a reflected BSDE with obstacle (ξt). Under the additional assumption of left upper semicontinuity along stopping times of (ξt), we show the existence of a super-hedge (τ, φ) for the buyer.

Optimal Stopping Theory and American Options

2019 ◽  
pp. 376-396
Author(s):  
Tomas Björk
Keyword(s):  
Dynamic Programming ◽  
Optimal Stopping ◽  
Continuous Time ◽  
American Options ◽  
Programming Approach ◽  
Free Boundary Value Problem ◽  
Snell Envelope ◽  
Dynamic Programming Approach ◽  
Optimal Stopping Theory ◽  
Time Theory

In this chapter we present the dynamic programming approach to optimal stopping problems. We start by presenting the discrete time theory, deriving the relevant Bellman equation. We present the Snell envelope and prove the Snell Envelope Theorem. For Markovian models we explore the connection to alpha-excessive functions. The continuous time theory is presented by deriving the free boundary value problem connected to the stopping problem, and we also derive the associated system of variational inequalities. American options are discussed in some detail.

American Option Valuation under Continuous-Time Markov Chains

2015 ◽  
Vol 47 (2) ◽  
pp. 378-401 ◽  
Author(s):  
B. Eriksson ◽  
M. R. Pistorius
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Optimal Stopping ◽  
Continuous Time ◽  
Markov Models ◽  
American Options ◽  
American Option ◽  
Discrete State ◽  
Continuous Time Markov Chains ◽  
Exponential Lévy Models

This paper is concerned with the solution of the optimal stopping problem associated to the value of American options driven by continuous-time Markov chains. The value-function of an American option in this setting is characterised as the unique solution (in a distributional sense) of a system of variational inequalities. Furthermore, with continuous and smooth fit principles not applicable in this discrete state-space setting, a novel explicit characterisation is provided of the optimal stopping boundary in terms of the generator of the underlying Markov chain. Subsequently, an algorithm is presented for the valuation of American options under Markov chain models. By application to a suitably chosen sequence of Markov chains, the algorithm provides an approximate valuation of an American option under a class of Markov models that includes diffusion models, exponential Lévy models, and stochastic differential equations driven by Lévy processes. Numerical experiments for a range of different models suggest that the approximation algorithm is flexible and accurate. A proof of convergence is also provided.

American Option Valuation under Continuous-Time Markov Chains

2015 ◽  
Vol 47 (02) ◽  
pp. 378-401
Author(s):  
B. Eriksson ◽  
M. R. Pistorius
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Optimal Stopping ◽  
Continuous Time ◽  
Markov Models ◽  
American Options ◽  
American Option ◽  
Discrete State ◽  
Continuous Time Markov Chains ◽  
Exponential Lévy Models

This paper is concerned with the solution of the optimal stopping problem associated to the value of American options driven by continuous-time Markov chains. The value-function of an American option in this setting is characterised as the unique solution (in a distributional sense) of a system of variational inequalities. Furthermore, with continuous and smooth fit principles not applicable in this discrete state-space setting, a novel explicit characterisation is provided of the optimal stopping boundary in terms of the generator of the underlying Markov chain. Subsequently, an algorithm is presented for the valuation of American options under Markov chain models. By application to a suitably chosen sequence of Markov chains, the algorithm provides an approximate valuation of an American option under a class of Markov models that includes diffusion models, exponential Lévy models, and stochastic differential equations driven by Lévy processes. Numerical experiments for a range of different models suggest that the approximation algorithm is flexible and accurate. A proof of convergence is also provided.

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