Deriving the Short Rate from Two Correlated Risky Assets

Summary

The document considers two risky assets driven by the same Brownian motion, with different volatilities, alongside a bank account earning the short rate. It constructs a self-financing portfolio from the assets and bank account, then imposes the condition that the portfolio be riskless. Setting the portfolio's Brownian exposure to zero and matching its drift to the bank-account rate yields a formula for the short rate in terms of the assets' expected returns and volatilities.

The derivation relies on the shared source of randomness and on being able to choose holdings that cancel the two assets' diffusion exposure. It is a model-based no-arbitrage argument, not an empirical estimate of an interest rate. The setup does not discuss whether market assumptions such as tradability, admissible portfolio weights, or consistency of the asset parameters hold in practice. Its result should not be carried over directly to assets with independent or more complex risk factors.

Key ideas

• Both risky assets are modeled as driven by the same Brownian motion, with unequal volatilities.
• A riskless portfolio is formed by choosing asset weights that cancel its random component.
• Matching the resulting portfolio drift to the bank account's return gives an implied short rate.
• The derivation depends on the shared risk factor and the model's self-financing assumptions.

Tags

# Answer by Focus (score 1)


# For a market with a bank and risky assets $S_1, S_2$ with different volatility, what should be the short interest rate in this market?












Let there be two assets $S_1$ and $S_2$ s.t.for $\sigma_1 \neq \sigma_2$ $$dS_{1t}=\mu_1 S_{1t}dt+ \sigma_1S_{1t}dB_t \\dS_{2t}=\mu_2 S_{2t}dt+ \sigma_2 S_{2t}dB_t$$ . If there exists a bank, what should be the short interest rate in this market ?

I have tried to make use of the following argument;

If $V_t$ is a riskless self-financing portfolio, then $dV_t=rV_t dt$ must be satisfied where $r$ is the short interest rate.

I want to make a riskless self-financing portfolio out of $S_1,S_2,G_t$ where $G_t=e^{rt}$ so put $V_t=a_t S_{1t}+b_tS_{2t}+c_tG_t$. I tried to calculate the differential but couldn't get any useful result.

Any help is appreciated.

## Answer by Focus (score 1)

https://quant.stackexchange.com/a/39970

Thanks to @Antoine Conze, here's my answer.

Using self-finance condition, straight forward calculation shows, with some sloppy notation, $$dV=(a\mu_1S_1+b\mu_2 S_2+crG)dt+(a\sigma_1 S_1+b\sigma_2S_2)dB =r(aS_1+bS_2+cG)dt\\ \Rightarrow a(\mu_1-r)S_1+b(\mu_2 -r)S_2=0,\ a\sigma_1S_1 +b \sigma_2 S_2=0 \\ \Rightarrow r=\frac{\mu_1\sigma_2-\mu_2\sigma_1}{\sigma_2 - \sigma_1} \text{ comparing the coefficients }$$.

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