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Setting Up a Risk-Neutral Price for a Minimum of Two Log Returns

Article Quant Q&A · Author: Skyly83
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Summary

The document poses an option-pricing problem with a payoff at maturity equal to the smaller of two assets’ log returns, measured relative to their initial values. It states that the two Brownian motions driving the assets are independent and asks how to use a martingale approach with the risk-free asset as numeraire and a Radon–Nikodym derivative.

No solution, derivation, or hint is included, so the excerpt does not establish a pricing formula or describe how the change of measure should be carried out. It is useful as a problem statement about pricing a payoff linked to the minimum of two stochastic returns. Any answer would need additional model details, including the assets’ dynamics and parameters, and would need to account for the distribution of the minimum under the selected pricing measure. The text offers no numerical evidence or discussion of assumptions beyond independence of the Brownian motions.

Key ideas

  • The payoff is the minimum of two assets’ log returns over a specified horizon.
  • The problem states that the Brownian drivers of the two assets are independent.
  • It proposes a martingale pricing approach using the risk-free asset as numeraire.
  • The excerpt gives no solution, dynamics, parameters, or numerical pricing evidence.

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OptionsDerivatives pricingStatistics
Full text
# Martingale approach - Option pricing with Radom-Nikodym


# Martingale approach - Option pricing with Radom-Nikodym












I would like to get the price of an option which pays at time T the minimum between the logarithm of (S(1,T) / S(1,0)) and the logarithm of (S(2,T) / S(2,0), with the following processes:

(The two brownians motions are not correlated).

I decided to use the martingale approach for this problem. By choosing the risk-less asset as the numeraire, I know I have to use the Radom-Nikodym derivative but I am a little bit stuck. Does anybody have some hints to give regarding this problem?

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