Skip to content
All library documents

Shared Risk Shocks Require Equal Sharpe Ratios to Avoid Arbitrage

Article Quant Q&A · Author: Amrit Prasad
Read the original

Summary

The document applies Girsanov’s theorem to asset dynamics under physical and risk-neutral probability measures. For a single risky asset, the market price of risk can be chosen as its Sharpe ratio so the asset’s expected return under the transformed measure equals the risk-free rate. It then considers two tradable assets driven by the same Brownian risk source and asks which market price of risk should be used.

The answer is that, under the stated shared-risk setup, absence of arbitrage requires both assets to have the same Sharpe ratio. If they did not, a portfolio long one asset and short a volatility-scaled amount of the other would eliminate the common risk while earning a nonzero excess return, creating a risk-free arbitrage. This argument explains why choosing one asset’s Sharpe ratio cannot leave the other with a different risk-neutral expected return in a consistent no-arbitrage market. The result depends on the assets being tradable and exposed to the same single source of risk.

Key ideas

  • Girsanov’s theorem changes the drift of asset dynamics through a market price of risk.
  • With one source of risk shared by two tradable assets, no arbitrage requires equal Sharpe ratios.
  • A volatility-scaled long-short position can cancel their common risk exposure.
  • Different Sharpe ratios in this setup imply a risk-free arbitrage opportunity.

Tags

Derivatives pricingArbitrageStatisticsRisk management
Full text
# Girsanov's Theorem for Multiple Risky Assets


# Girsanov's Theorem for Multiple Risky Assets












Girsanov's theorem provides the measure transformation from probability measure P to Q such that-

$dW_t^Q=dW_t^P+\lambda dt\implies \xi_tW_t^Q$ is a martingale under the P measure where $\xi_t=e^{-\lambda W_t^P-\frac{\lambda^2}{2}t}$ and $W_t^X$ is the Weiner's Process under probability measure $X$.

For single assets this can be utilised to convert their dynamics from objective probability measure P to risk-neutral measure Q.

$\frac{dS}{S}=\mu dt+\sigma dW^P=(\mu-\lambda\sigma)dt+\sigma dW^Q$.

Setting $\lambda=\frac{\mu-r_f}{\sigma}=$ Sharpe Ratio of asset S, we get-

$\frac{dS}{S}=r_fdt+\sigma dW^Q$.

Suppose we have two tradeable assets in a market with the same source of risk $W$.

$\frac{dS_1}{S_1}=\mu_1 dt+\sigma_1 dW^P$

$\frac{dS_2}{S_2}=\mu_2 dt+\sigma_2 dW^P$

What $\lambda$ should be chosen in this case for making the transformation from P to Q? If we choose the Sharpe Ratio of the 1st asset, then we'll end up getting-

$\frac{dS_1}{S_1}=r_f dt+\sigma_1 dW^Q$

$\frac{dS_2}{S_2}=(\mu_2-\sigma_2\frac{\mu_1-r_f}{\sigma_1}) dt+\sigma_2 dW^Q$

Similarly for choosing the other Sharpe Ratio.

It seems intuitive to choose the greater of the two ratios for making the transformation, but that'll result in the other asset having an expected return lesser than $r_f$ in the risk-neutral measure. This would imply that participants simply choose to never trade that asset. Does this reasoning seem correct?

## Answer by Antoine Conze (score 1)

https://quant.stackexchange.com/a/39742

If two assets have the same source of risk $W^P$ then the no arbitrage opportunity condition implies that their Sharpe ratios are the same, i.e. $\frac{\mu_1-r_f}{\sigma_1} = \frac{\mu_2-r_f}{\sigma_2}$. This is a textbook exercise and is easily proven by building a risk free portfolio long one share of the first asset and short $\frac{S_1 \sigma_1}{S_2 \sigma_2}$ shares of the second asset.

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.