Optimal Stopping with Model Uncertainty and Pricing the American Option

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.

Download Full-text

American Option Valuation under Continuous-Time Markov Chains

Advances in Applied Probability ◽

10.1239/aap/1435236980 ◽

2015 ◽

Vol 47 (2) ◽

pp. 378-401 ◽

Cited By ~ 6

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.

Download Full-text

American Option Valuation under Continuous-Time Markov Chains

Advances in Applied Probability ◽

10.1017/s0001867800007904 ◽

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.

Download Full-text

SUPER-HEDGING AMERICAN OPTIONS WITH SEMI-STATIC TRADING STRATEGIES UNDER MODEL UNCERTAINTY

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024917500364 ◽

2017 ◽

Vol 20 (06) ◽

pp. 1750036 ◽

Cited By ~ 6

Author(s):

ERHAN BAYRAKTAR ◽

ZHOU ZHOU

Keyword(s):

Discrete Time ◽

Model Uncertainty ◽

American Options ◽

American Option ◽

Trading Strategies ◽

Martingale Measure ◽

European Options ◽

Model Based ◽

Price Formula ◽

Dynamic Trading

We consider the super-hedging price of an American option in a discrete-time market in which stocks are available for dynamic trading and European options are available for static trading. We show that the super-hedging price [Formula: see text] is given by the supremum over the prices of the American option under randomized models. That is, [Formula: see text], where [Formula: see text] and the martingale measure [Formula: see text] are chosen such that [Formula: see text] and [Formula: see text] prices the European options correctly, and [Formula: see text] is the price of the American option under the model [Formula: see text]. Our result generalizes the example given in Hobson & Neuberger (2016) that the highest model-based price can be considered as a randomization over models.

Download Full-text