[Paper Review] Symmetry reductions of a nonlinear option pricing model

Ljudmila A. Bordag|arXiv (Cornell University)|Apr 9, 2006

Stochastic processes and financial applications7 references3 citations

TL;DR

This paper applies Lie group symmetry analysis to a nonlinear Black-Scholes equation modeling illiquidity effects in option pricing, where large traders' hedging strategies impact underlying asset prices. By reducing the PDE to ODEs under specific parameter conditions—particularly λ(S) = ωS—it derives exact invariant solutions, including novel families not present in the linear Black-Scholes model, which diverge as ρ → 0 due to nonlinearity.

ABSTRACT

The studied model was suggested to design a perfect hedging strategy for a large trader. In this case the implementation of a hedging strategy affects the price of the underlying security. The feedback-effect leads to a nonlinear version of the Black-Scholes partial differential equation. Using the Lie group theory we reduce the partial differential equation in special cases to ordinary differential equations. The found Lie group of the model equation gives rise to invariant solutions. Families of exact invariant solutions for special values of parameters are described.

Motivation & Objective

• To analyze a nonlinear partial differential equation (PDE) arising from illiquidity effects in option pricing, where large traders' hedging strategies affect the underlying asset price.
• To apply Lie group theory to reduce the nonlinear PDE to ordinary differential equations (ODEs) for special parameter cases.
• To classify and construct exact invariant solutions of the model equation, particularly for λ(S) = ωS.
• To investigate the behavior of these solutions, including their singularities and asymptotic properties.
• To demonstrate that the derived solutions are unique up to symmetry transformations and do not exist in the linear Black-Scholes case.

Proposed method

• The study uses Lie point symmetry analysis to determine the symmetry algebra of the nonlinear PDE (1.1), identifying infinitesimal generators ξ(S,t,u), τ(S,t,u), and φ(S,t,u).
• The second-order prolongation of the vector field is computed to extend the symmetry action to jet space M^(2), enabling the analysis of the PDE's differential invariants.
• For λ(S) = ωS (i.e., k=1), two functionally independent invariants are derived, allowing reduction of the PDE to a system of ODEs.
• The reduced ODEs are analyzed for singular points and solution behavior, leading to explicit parametric solutions via trigonometric and hyperbolic functions.
• Solutions are constructed using the first integral method and verified by substitution into the reduced ODEs.
• The complete set of invariant solutions is derived for the case q=4 (equivalently a=σ²/8), with explicit formulas provided for u(S,t) in terms of log S, t, and inverse trigonometric/hyperbolic functions.

Experimental results

Research questions

• RQ1 What Lie group symmetries exist for the nonlinear option pricing PDE (1.1) under general λ(S)?
• RQ2 For which specific forms of λ(S) can the PDE be reduced to ODEs via symmetry reduction?
• RQ3 What are the exact invariant solutions of the reduced ODEs, and how do they behave near singularities?
• RQ4 How do these solutions differ from those of the linear Black-Scholes equation, particularly in the limit ρ→0?
• RQ5 Are the derived solutions unique up to symmetry transformations, and what is their domain of definition?

Key findings

• For λ(S) = ωS, the PDE admits a two-dimensional symmetry algebra, enabling reduction to ODEs via two functionally independent invariants.
• The reduced ODEs yield a complete set of exact invariant solutions, including families involving inverse cosine and hyperbolic functions.
• Three distinct solution families are identified: u₁(S,t), u₂(S,t), and piecewise-defined u₃,₁(S,t) and u₃,₂(S,t), all defined for S > 0 and t ≥ 0.
• Solutions u₁, u₂, and u₃ exhibit a common intersection curve of the form S = const × exp(−σ²t/8), indicating shared asymptotic behavior.
• The solutions are singular along S = (2/|c|)^(4/3) × exp(−σ²t/8), where they cannot be analytically continued.
• In the limit ρ → 0, the solutions diverge due to the 1/ρ factor in their expressions, confirming their nonlinearity and absence in the linear case.