Direct analytic solution of the Lindquist–Rachev PDE for European call options

Develop a direct analytic solution method for the Lindquist–Rachev partial differential equation governing the price C(t,S,Z) of a European call option in a market with two risky assets S and Z driven by the same Brownian motion and no riskless asset. In this setting, the stock prices follow dS(t)=S(t)(μ(t)dt+σ(t)dW(t)) and dZ(t)=Z(t)(μ̃(t)dt+σ̃(t)dW(t)), the numéraire is S, and the resulting option-pricing PDE features the shadow riskless rate r(t)=(μ(t)σ̃(t)−μ̃(t)σ(t))/(σ̃(t)−σ(t)). The goal is to obtain a closed-form or otherwise explicit analytic solution for C(t,S,Z) without resorting to numerical methods or implicit auxiliary parameters.

Background

Approach One studies option pricing in a market with only risky assets, specifically two perfectly correlated stocks S and Z whose prices are driven by the same Brownian motion. Using S as the numéraire, the authors derive a stochastic PDE (referred to as the Lindquist–Rachev PDE) for the option price C(t,S,Z). The equation depends on the shadow riskless rate r(t)=(μ(t)σ̃(t)−μ̃(t)σ(t))/(σ̃(t)−σ(t)).

The authors report that they could not find a direct analytic method to solve this PDE for a European call option. They therefore construct a solution using the Feynman–Kac formula and verify that it satisfies the PDE, but the resulting expression involves an implicitly defined parameter and is analytically intractable in general. To provide a practical implementation, they develop a corresponding binomial model.