Why Black–Scholes Does Not Require Trading the Option

Summary

The document asks whether Black–Scholes pricing requires the option itself to be tradable, using management incentive units as an example. It distinguishes that question from the model’s assumption that the underlying stock can be traded. The proposed reasoning is that, under market completeness, a portfolio of stock and a risk-free bond can replicate an option’s payoff, and no-arbitrage pricing then determines its value without trading the option directly.

The text presents this as a question rather than a resolved explanation. It gives no formal derivation or discussion of how restrictions on trading, market incompleteness, or the option’s contractual features might affect valuation. Its central learning point is the distinction between trading an instrument and replicating its payoff with other assets.

Key ideas

• The document distinguishes the tradability of an option from the tradability of its underlying stock.
• It proposes that stock and a risk-free bond can replicate an option payoff under market completeness.
• The question argues that replication and no-arbitrage may price an option even when the option itself cannot be traded.
• The document raises the issue but does not provide a formal answer or address limitations to replication.

Tags

# Tradeability of Option (not underlying) necessary assumption in BSM?


# Tradeability of Option (not underlying) necessary assumption in BSM?












Working with the Black Scholes Model to value european Call and Put Options I encountered a question that came up during the valuation of a (european Call) Option, which itself cannot be traded (e.g. management incentive units).

To the best of my knowledge (and understanding) it is of course a crucial assumption in BSM that the underlying stock is tradeable. To my understanding, however, tradeability of the option itself is not needed.

Reason: the idea behind the pricing in BSM is based on the fact that option-payoffs can be reproduced (under market completeness) by a portfolio consisting of units of the risk-free bond and the underlying stock (+ an no arbitrage argument). For this approach, we only need the possibility to buy and sell units of the underlying stock, NOT the option itself. Therefore, I don't see, why the assumption of tradeability of the option should be needed.

Nevertheless, this question came up more than just once and I would be very interested in your thoughts.

Thank you very much in advance.

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