Pricing Bermudan Option with Variable Transaction Costs under the Information-Based Model

Abstract

The Bermudan option pricing problem with variable transaction costs is considered for a risky asset whose price process is derived under the information-based model. The price is formulated as the value function of an optimal stopping problem, which is the value function of a stochastic control problem given by a non-linear second order partial differential equation. The theory of viscosity solutions is applied to solve the stochastic control problem such that the value function is also the solution of the corresponding Bellman equation. Under some regularity assumptions, the existence and uniqueness of the solution of the pricing equation are derived by the application of the Perron method and Banach Fixed Point theorem.

Unraveling Market Inefficiencies: Weak Arbitrage and the Information-Based Model for Option Pricing

“…The PDE can then be obtained through the application of Itô formula together with the assumption that the portfolio is risk-less and must earn the risk-free rate of interest. For a comprehensive derivation of this PDE along with the existence and uniqueness proofs, please refer to [39]. The resultant no-arbitrage pricing equation, accounting for variable transaction costs, is presented as follows:…”

“…Since its inception, IBM has undergone extensions and modifications to accommodate various market dynamics in option pricing and to price different types of options. In [39], the first attempt was made to price early exercise Bermudan-style options under the information-based framework while considering variable transaction costs of trading.…”

Numerical Approximation of Information-Based Model Equation for Bermudan Option with Variable Transaction Costs

“…However, [7] suggested that the IBM could be used to price early exercise options with American or Bermudan-type exercise rights. In [8], we extended the IBM to include variable transaction costs and derived a partial differential equation (PDE) for valuing a Bermudan call option on an asset driven by the Lévy Random Bridge information process. This is the first attempt to incorporate transaction costs under the IBM and allow for the possibility of pricing an early exercise option.…”

“…The subscript p in p t is dropped henceforth, for ease of notation. The partial differential operator is derived in [8] for a call option with variable transaction costs and given as…”