Positive solutions of a Dirichlet problem for a stationary nonlinear Black–Scholes equation

In this paper, we study the existence of stationary solutions of a nonlinear Black–Scholes type equation that concerns an option pricing model with stochastic volatility. More precisely we consider in Ω⊂(R+)2Ω⊂(R+)2 the nonlinear elliptic PDE 12σ2S2∂2f∂S2+12σ2V2∂2f∂σ2+ρσ2VS∂2f∂S∂σ−12ρσ2V∂f∂σ+rS∂f∂S=rγ(f) with the boundary condition f(S,σ)=h(S,σ)on ∂Ω, where γγ is Hölder continuous and the variables SS and σσ are respectively the asset value and the market volatility [P. Amster, C.G. Averbuj, M.C. Mariani, Solutions to a stationary nonlinear Black–Scholes type equation, J. Math. Anal. Appl. 276 (2002) 231–238; M. Avellaneda, Y. Zhu, Risk neutral stochastic volatility model, Internat. J. Theor. Appl. Finance 1 (1998) 289–310]. We prove the existence of a positive solution ff for this problem assuming certain conditions on the primitive ΓΓ of γγ. The method of the proof, which is based on the construction of upper and lower solutions obtained as solutions of an auxiliary initial value problem, also yields information on the localization of ff.