# Rainbow option pricing formula under *Bachelier* model

Let's consider a call on min option on two underlying arithmetic Browniation motions $$V_t$$ and $$H_t$$ (no drift). Let $$P_t$$ denotes the price process of the option, $$r$$ the riskfree rate, $$\tau$$ the time to maturity, then following the notation and the procedure in [Stulz, 1982] (eq (3) - (7) in particular), we obtain a similar PDE

$$-P_\tau = r_f(P-P_VV-P_HH)-\frac12(P_{VV}\sigma_V^2+P_{HH}\sigma_H^2+2P_{VH}\rho_{VH}\sigma_V\sigma_H)$$

and the boundry conditions are given by $$P_T:=(\min(V,H)-F,0)_+$$

It looks like the PDE is a heat PDE. Is there any available literature that has already dealt with this PDE, or even better, already given the explicit pricing formula for the arithmetic rainbow option? (Assume constant vol, constant corr etc.)