Pricing a European Claim on the Inverse of a Geometric Brownian Stock

Summary

The document poses a derivatives-pricing question: how to value a maturity payoff equal to the inverse of a non-dividend-paying stock price assumed to follow geometric Brownian motion. The proposed approach starts with the stock’s risk-neutral dynamics, applies Itô’s lemma to the reciprocal, and obtains dynamics for the transformed value. Because the reciprocal is determined by the stock, the question asks whether the stock’s risk-neutral measure also suffices for pricing the claim.

The excerpt contains only the question and its proposed derivation; it provides no answer, valuation formula, or discussion of completeness conditions. It therefore does not establish whether the risk-neutral measure is unique in the relevant market. Readers should distinguish transforming an asset’s dynamics under an assumed measure from proving that the market admits a unique equivalent martingale measure.

Key ideas

• The payoff is the reciprocal of the stock price observed at maturity.
• Itô’s lemma can be used to derive the dynamics of the reciprocal from the stock’s risk-neutral dynamics.
• The reciprocal claim depends on the stock, but the excerpt does not prove that the market’s risk-neutral measure is unique.
• A full valuation requires the appropriate risk-neutral expectation and assumptions about the pricing market.

Tags

# Confirm If Risk-Neutral Measure is Unique in My Following Case


# Confirm If Risk-Neutral Measure is Unique in My Following Case












I'm reading a book that discusses about derivatives pricing and have some doubts about a particular problem and really appreciate your advice:

Question:

Assume a non-dividend paying stock follows a geometric Brownian motion. What is the value of a contract that at maturity T pays the inverse of the stock price observed at the maturity?

Here is the solution:

Under risk-neutral measure, $dS = rSdt+σSdW(t)$. Apply Ito's lemma to $V=1/S$, we get $dV = (-r+σ^2)Vdt-σVdW(t)$. So V follows a geometric Brownian motion as well and we can apply Ito's lemma to $lnV$, we can get: $dln(V)=(-r+0.5*σ^2)dt-σdW(t)$. If I understand correctly, we don't need to find out the risk neutral measure for V because V is dependent on stock price S, and we already have the risk neutral measure for S and have $dS = rSdt+σSdW(t)$, due to fundamental theorem of asset pricing, the risk neutral measure is unique in this case and the unique risk neutral measure is derived from stock price S. I'm wondering if my understanding is correct?

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