[Paper Review] Option Pricing in a Regime Switching Jump Diffusion Model

Anindya Goswami, Omkar Manjarekar+1 more|arXiv (Cornell University)|Nov 28, 2018

Stochastic processes and financial applicationsEconomics, Econometrics and Finance23 references3 citations

TL;DR

This paper develops a regime-switching jump diffusion model for European option pricing, where market regimes follow a semi-Markov process with duration-dependent transitions and asset prices experience jumps. Using Föllmer-Schweizer decomposition, it establishes the existence and uniqueness of a classical solution to a degenerate, non-local, linear parabolic integro-partial differential equation, enabling optimal hedging in incomplete markets with both regime shifts and jumps.

ABSTRACT

This paper presents the solution to a European option pricing problem by considering a regime-switching jump diffusion model of the underlying financial asset price dynamics. The regimes are assumed to be the results of an observed pure jump process, driving the values of interest rate and volatility coefficient. The pure jump process is assumed to be a semi-Markov process on finite state space. This consideration helps to incorporate a specific type of memory influence in the asset price. Under this model assumption, the locally risk minimizing price of the European type path-independent options is found. The Föllmer-Schweizer decomposition is adopted to show that the option price satisfies an evolution problem, as a function of time, stock price, market regime, and the stagnancy period. To be more precise, the evolution problem involves a linear, parabolic, degenerate and non-local system of integro-partial differential equations. We have established existence and uniqueness of classical solution to the evolution problem in an appropriate class.

Motivation & Objective

• To address the gap in existing models that do not simultaneously capture jump discontinuities and duration-dependent regime switching in asset price dynamics.
• To establish the absence of arbitrage in a regime-switching jump diffusion model driven by a semi-Markov process.
• To derive the locally risk-minimizing price of European path-independent options under this model.
• To prove existence and uniqueness of a classical solution to the resulting evolution problem involving a non-local, degenerate, parabolic system of integro-partial differential equations.
• To enable optimal hedging strategies by characterizing the solution in terms of time, stock price, market regime, and stagnancy period.

Proposed method

• Modeling asset price dynamics using a regime-switching jump diffusion process where the regime is governed by a finite-state semi-Markov process with general holding time distributions.
• Assuming the jump size distribution is bounded and satisfies η(z) > -1 to ensure positivity and integrability of the price process.
• Applying the Föllmer-Schweizer decomposition to derive the locally risk-minimizing price, which leads to a system of integro-partial differential equations (IPDEs) in time, stock price, regime, and stagnancy period.
• Proving existence and uniqueness of classical solutions to the IPDE system in an appropriate function space using boundedness and continuity assumptions on the coefficients.
• Establishing square integrability of the asset price process by analyzing the moment-generating function of the jump component and using the finiteness of the Lévy measure.
• Using the continuity of the coefficient functions in time and uniform boundedness to show continuity of the associated linear operator in the space of bounded linear maps.

Experimental results

Research questions

• RQ1 Does a regime-switching jump diffusion model driven by a semi-Markov process allow for arbitrage-free pricing under appropriate conditions on the jump distribution?
• RQ2 How does the inclusion of both regime switching with duration dependence and jump discontinuities affect the option pricing PDE in incomplete markets?
• RQ3 Can the Föllmer-Schweizer decomposition be applied to derive a locally risk-minimizing price in this hybrid model setting?
• RQ4 What are the regularity and existence properties of the solution to the resulting non-local, degenerate, parabolic IPDE system?
• RQ5 What conditions ensure the square integrability of the asset price process under this model?

Key findings

• The model ensures no-arbitrage under the condition that the jump size distribution satisfies η(z) > -1 and has finite second moments, which is verified via the positivity and integrability of the solution to the SDE.
• The locally risk-minimizing price of a European option satisfies a degenerate, non-local, linear parabolic system of integro-partial differential equations involving time, stock price, regime, and stagnancy period.
• Existence and uniqueness of a classical solution to the evolution problem are established in a suitable function space, ensuring mathematical well-posedness of the pricing problem.
• The solution is shown to be square-integrable over finite time horizons, which is essential for the validity of the Föllmer-Schweizer decomposition and optimal hedging.
• The optimal hedging strategy is derived from the Föllmer-Schweizer decomposition, minimizing the quadratic residual risk under continuous trading.
• The continuity of the coefficient functions in time and their boundedness ensure the continuity of the associated linear operator in the space of bounded linear maps, supporting the regularity of the solution.