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Arbitrage and Completeness with Two Assets Driven by One Brownian Motion

Article Quant Q&A · Author: Tera
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Summary

The document asks when a Black–Scholes market with two risky assets driven by the same Brownian motion and one risk-free asset is arbitrage-free and complete. The question highlights a modeling issue: the setup has one source of randomness, which does not fit the respondent’s recollection of a textbook treatment of market completeness.

The short response asserts that the risk-free rate and both assets’ expected returns must be equal for absence of arbitrage. It then claims completeness follows from a comparison between the number of risky assets and random sources. The post supplies no derivation or assumptions beyond the model description, and the response’s count-based explanation is unclear as written. Readers should treat it as a prompt for checking the market price of risk and spanning conditions, rather than as a fully supported result.

Key ideas

  • The model contains two risky assets driven by one shared Brownian motion and a risk-free asset.
  • The question concerns the conditions for absence of arbitrage and market completeness.
  • The response states that both risky assets’ expected returns must equal the risk-free rate.
  • The response’s completeness claim is brief and unsupported by a derivation in the document.

Tags

StatisticsDerivatives pricingArbitrage
Full text
# Black Scholes: two assets, same $W$-process


# Black Scholes: two assets, same $W$-process












Consider a Black Scholes model with two risky assets that are driven by the same $W$-process, and then 1 risk-free asset.

When is this model arbitrage-free and complete?

We have only 1 driving Wiener process, so it does not fit into the theory I learned from Bjork, chapter 13.

## Answer by Chamin (score -3)

https://quant.stackexchange.com/a/32024

You must have $ r= \mu_1 = \mu_2$ for arbitrage-ness. The completeness follows because $n < k$ (k is the number of random sources, n is the number of risky assets).

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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