Convergence of reward functionals in a reselling model for a European option. (English. Russian original) Zbl 1252.60064

Pupashenko, M. S.

Theory Probab. Math. Stat. 83, 135-148 (2011); translation from Teor. Jmovirn. Mat. Stat. 83, 113-124 (2010).

In the context of the classical Black-Scholes model, a Cox-Ingersoll-Ross process is used to describe the implied volatility \(\sigma(t)\), i.e., \(\sigma(t)=\sqrt{\widetilde\sigma^2(t)+\delta_0^2}\), where \(\widetilde \sigma^2(t)\) is the CIR-process correlated with the Black-Scholes asset price \(S(t)\). If the Black-Scholes call-price is given by \(C(t, S(t), \sigma)\), then the optimal reselling problem is defined like the optimal exercise problem for the American option: \[ \Phi(\mathcal{M}_T)=\sup_{\tau\in\mathcal{M}_T} \operatorname{E}\mathrm{e}^{-r\tau}C(\tau, S(\tau), \sigma(\tau)), \] where \(\mathcal{M}_T\) is the class of Markov stopping times. A two-dimensional approximation, binomial for the log-price and trinomial for the implied volatility, is constructed. The weak convergence in the Skorokhod space of the reward functional is proved. The proposed discrete model is arbitrage-free.

Reviewer: Nikita E. Ratanov (Bogotá)

MSC:

60H30	Applications of stochastic analysis (to PDEs, etc.)
60J05	Discrete-time Markov processes on general state spaces
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
91G20	Derivative securities (option pricing, hedging, etc.)
91B70	Stochastic models in economics

Keywords:

European option; American option; reselling problem; reward; convergence; optimal stopping time; discrete approximation; Cox-Ingersoll-Ross process

References:

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