Pricing contracts on a tradable asset and a non-tradable mean-reverting benchmark: Closed-form formulae and applications

Abstract

We develop a closed-form pricing framework for contracts whose payoff depends on the difference between a tradable underlying asset and a non-tradable mean-reverting benchmark. The framework is designed for applications in which the relevant exercise benchmark or reference variable evolves over time, as in volatility-linked, inflation-linked, weather-related, and real-option settings. The aim is to obtain analytically tractable formulae for an exchange-style payoff involving a tradable underlying asset and a non-tradable mean-reverting benchmark. As shown, this is relevant in sensitive contexts where risks cannot be fully hedged through trading. The framework is applied to real option valuation, examining investment when project value follows geometric Brownian motion and investment cost follows a mean-reverting process. Sensitivity analysis reveals that cost volatility and mean-reversion speed exert distinct and opposing influences on the value of investment options, while the correlation between project value and cost dynamics plays a central role. An empirical calibration using S&P 500 and VIX data shows that benchmark risk premia and mean-reverting benchmark uncertainty can materially affect contract values, particularly when contracts are away from the money.