When Risk-Neutral Pricing Requires a Tradable Option

Summary

The document asks whether an option must itself be tradable for its value to equal the discounted risk-neutral expectation of its payoff. It frames the question in the Black–Scholes setting, where the underlying follows a constant-volatility process and the risk-free asset earns a constant rate. A European call serves as the example of a payoff determined by the asset price at maturity.

It then extends the question to path-dependent contracts whose maturity payoff depends on prices observed at multiple earlier times. The post raises a useful distinction between pricing a terminal payoff from a specified risk-neutral model and establishing that price through replication or market trading. However, it provides no answer, derivation, assumptions about market completeness, or evidence. Readers should treat it as a question prompt rather than a resolved pricing method; the validity of the expectation formula depends on the pricing framework and the availability of a suitable risk-neutral measure.

Key ideas

• The post asks whether the option itself must be traded for risk-neutral valuation to apply.
• It presents a European call as a payoff depending on the underlying price at maturity.
• It extends the question to payoffs that depend on the underlying's price path.
• The post supplies no answer or derivation, so it does not establish the required assumptions.

Tags

# Does an option need to be tradable for Black Scholes pricing formula to hold?


# Does an option need to be tradable for Black Scholes pricing formula to hold?












Given the classic Black-Scholes model, e.g.

$dS(t)/S(t)=rdt+\sigma dW^{\mathbb{Q}}(t)$ with $S(0)=S_0$ and $dB(t)=rB(t)dt$ with $B(0)=1$,

whereby $r$ and $\sigma$ are constants and $\mathbb{Q}$ denotes the risk-neutral measure. Given now an option with payout $\Phi_T$ only at $T$, e.g. an European call option $\Phi_T(S_{T})=\max(S_T-K,0)$.

- Do we need to assume that the option itself is tradable in order to determine the price of such an option as $e^{-rT}\mathbb{E}^{\mathbb{Q}}[\Phi_T(S_T)]$? If so, why? If not, why not?

- What if the option still can be only exercised at maturity $T$, but the payout depends on previous prices of the asset, e.g. $S_1, S_2,\ldots,S_T$. That is, do we need that the option itself is tradable in order for $e^{-rT}\mathbb{E}^{\mathbb{Q}}[\Phi_T(S_1,S_2,\ldots,S_T)]$?

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