No-arbitrage and the sharpe ratio?

Question

I'm reading a paper and it says that in a no-arbitrage market the sharpe ratio is the same for all bonds. I'm guessing that a difference in two bonds sharpe ratios would open the possibility of arbitrage, but why is that?

Regards

Answer 1

This was proved by Vasiceck in his 1977 paper. If you suppose that the price of a pure discount bond depends only on a markovian short rate $r(t)$ with SDE \begin{equation} dr(t)=\mu(t,r(t))dt + \sigma(t,r(t))dW(t) \end{equation}

then you can assume that $P(t,T)=F(t,r(t);T)$. Now, with similar arguments used in the derivation of the Black-Scholes formula, he made a self-financing portfolio consisting of a $T$-bond and a $S$-bond. Say your portfolio value has SDE: \begin{equation} dV(t)=\theta_T(t)dF(t,r(t);T) + \theta_SdF(t,r(t);S) \end{equation} where $(\theta_T,\theta_S)$ is your self financing strategy. Now for simplicity write $F(t,r(t);T)=F^T(t,r(t))$ and since $P(t,T)>0$ for all $t\le T$ we can use Ito lemma to write his differential in this way: \begin{align} dF^T(t,r(t))&=\alpha^TF^Tdt + \beta^TF^TdW(t) \\ dF^S(t,r(t))&=\alpha^SF^Sdt + \beta^SF^SdW(t) \end{align} By substituting in the self financing portfolio SDE you now search the strategy $(\theta_T,\theta_S)$ that makes this portfolio risk-neutral. If the market doesn't allow for arbitrage, then a risk-less asset must earn the same rate of return of the bank account: \begin{equation} dV(t)=r(t)V(t)dt \end{equation} After substituting you will find that this equals the condition \begin{equation} \frac{\alpha^S(t) - r(t)}{\beta^S(t)}=\frac{\alpha^T(t) - r(t)}{\beta^T(t)} \end{equation} This means that bonds with different maturities have the same Sharpe Ratio. You will find a clearer derivation in the book by Bjork, however this just works for short rate models. Actually I don't know if there are more general derivation of this result.

Answer 2

I would guess you mean that all the expected Sharpe ratios are equal. Here is why.

Consider a market with $d$ assets $(S^1, \dots, S^d)$ which is free ob arbitrage. Let $B$ denote the numeraire. According to the fundamental theorem of asset pricing there exists a martingale measure $\Bbb Q$. In particular: $$ \Bbb E_{\Bbb Q} \Bigg[\frac{S_{1}^j - S_0^j} {S_0^j} \Biggr] = \frac{1}{S_0^j}\Bbb E_{\Bbb Q} \bigl[S_{1}^j\bigr] - 1 = \frac{B_1}{S_0^j}\frac{S_0^j}{B_0} - 1 = \frac{B_1 - B_0}{B_0}, \quad \ \text{for all} \ j \in \{1,\dots, d\}. $$

This is the well known result that under the absence of arbitrage the expected return of each asset is given by the expected return of the bank account.

In particular all Sharpe ratios have to be equal.