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Pricing a Two-Asset Arithmetic Rainbow Option under Bachelier Dynamics

Article Quant Q&A · Author: Vim
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Summary

The document sets up a call option whose payoff depends on the lower of two underlying asset values under a Bachelier model. Both assets are modeled as arithmetic Brownian motions with no drift, constant volatilities, and constant correlation. It writes a two-variable partial differential equation for the option value, including a mixed derivative term that captures correlation, and specifies a payoff based on the minimum asset value relative to a strike.

The author observes that the equation resembles a heat equation and asks whether an explicit pricing formula or prior literature exists, following the notation and approach of Stulz. The material provides a model setup and boundary condition, but no derivation, closed-form solution, numerical results, or discussion of assumptions beyond the stated dynamics. Readers would need further analysis to resolve the pricing question and check details such as the exact maturity notation and risk-free-rate convention.

Key ideas

  • The option payoff depends on the smaller of two underlying asset values at maturity.
  • The proposed model uses arithmetic Brownian motion with constant volatility and correlation.
  • The pricing equation includes a cross-partial term for dependence between the assets.
  • The document poses the search for an explicit formula but does not provide one.

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Full text
# Rainbow option pricing formula under *Bachelier* model


# 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.)

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.