Abstract
This paper extends the Black-Scholes methodology to payoffs that are functions of a stochastically varying variable that can be observed but not traded. The stochastic price process proposed in this paper satisfies a partial differential equation that is an extension of the Black-Scholes equation. The resulting price process is based on projection onto the marketed space, and it is universal in the sense that all risk-averse investors will find that, when priced according to the process, the asset cannot improve portfolio performance relative to other assets in the market. The development of the equation and its properties is facilitated by the introduction of an operational calculus for pricing. The results can be put in risk-neutral form. Perfect replication is not generally possible for these derivatives, but the approximation of minimum expected squared error is determined by another partial differential equation.
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Luenberger, D.G. Pricing a Nontradeable Asset and Its Derivatives. Journal of Optimization Theory and Applications 121, 465–487 (2004). https://doi.org/10.1023/B:JOTA.0000037600.85025.db
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DOI: https://doi.org/10.1023/B:JOTA.0000037600.85025.db