# returns of Bonds and exchange rates

which are the best distributions in order to model the bonds and exchange rate returns distributions. I am searching for a distribution such as the log-normal one of the stocks ( N(m-0.5*v),Sqrt[v])

## 2 Answers

Do you want to model the returns in a risk-neutral framework (for derivatives) or in the real world measure (for risk analysis/portfolio construction)?

For the first approach (say modelling under $Q$) you should go to the literature on bond and FX-derivatives. I would go more into detail if this is your aim. The formulation $N(\mu-\sigma^2/2,\sigma)$ suggests this a bit.

For the second (say modelling under $P$) I have 2 things to say:

1. don't confuse it with risk neutral pricing. Looking at log-returns the expected return and the variance can be estimated from the sample directly, say as $\mu$ and $\sigma$. You don't have to plug-in $\mu-\sigma^2/2$ for the expected value
2. A very flexible family of distributions is the Generalized Hyperbolic distrbution. There is also an R package for this ghyp.

EDIT after comment of OP:

If you look at the log-return of a stock price. ie. $X_i = \log(S_i)-\log(S_{i-1})$ then you can assume that has unlimited support (no left or right end point). If you look at simple returns $S_i/S_{i-1}-1$ then you have a left endpoint of $-1$ (if $S_i=0$). Thus if you want to use something like a normal-distribution then you should use log-returns. If you have the expected value $\mu$ and the variance $\sigma^2$ then you can model the log-return $X$ by a normal distribtion $N(\mu,\sigma^2)$.

The $N(\mu-\sigma^2/2,\sigma^2)$ comes from the SDE approach (world of $Q$) where $$S_t = S_0 \exp( (\mu-\sigma^2/2) t + \sigma B_t )$$ solves the SDE $$dS_t/S_t = \mu dt + \sigma dB_t$$ as due to Ito's lemma you get a quadratic variation term $\sigma^2 t$.

EDIT 2: If you look at the log-returns and the mean is $\mu$ then it is $\mu$ and not $\mu - \sigma^2/2$. The latter is only used to identify $\mu$ as the drift of the SDE. If $\mu$ is the drift, then $\mu - \sigma^2/2$ is the expected value of $$\log(S_{t+1}/S_t) = \mu-\sigma^2/2) t + \sigma B_t$$ as you can derive from the equation above.

if $\hat{\mu}$ is the mean log-return then your SDE should have drift $\hat{\mu} - \sigma^2/2$ in order to be consistent with this statistic.

You can use SDEs for risk mgmt. But if you just look at returns - why do you need a continuous time framework? VIX tells you something about the options market. The implied vol is often different from realized vol. These are 2 different but connected things.

• I would like to measure the returns and risk of any instrument for a portfolio construction. Thus, i will use your second approach. But, why it is not different to apply the formulation μ−σ2/2 ? for the mean of of a log-normal distribution, in order to model the returns of a stock? – dimos Jun 1 '15 at 12:07
• For instance, assume that i have a data set for the daily stock log- returns. I perform a test such as the jarque bera to see if the returns are normally distributed. After that, i calculate the sample mean and variance. FInally, i plug in the sample variance into the formula μ−σ2/2 to get the real mean. So, all this methodology is wrong? – dimos Jun 1 '15 at 12:13
• I edited the answer. If you model the log-returns as normally distributed then you don't have to change the mean in any way. – Richard Jun 1 '15 at 12:25
• So,the first one method comes from a statistical framework with a MLE estimation and the second one from the SDEs. What i cannot understand is that, is wrong to use the SDEs framework for risk management? i believe intuitively that this approach is more correct that the other one, due to the fact that you can use variance expectations such as the VIX index – dimos Jun 1 '15 at 12:34
• one more edit. .. – Richard Jun 1 '15 at 13:31

To a first approximation bond and FX returns are also assumed log-normal, but of course with different mu and sigma.

• So, the mu and sigma are the sample ones? – dimos Jun 1 '15 at 12:08
• Yes, but be careful in the case of mu. In a short time sample mu can be too extreme (too high or too low); you need to "shrink it" towards a reasonable long term value, empirical or theoretical. – noob2 Jun 1 '15 at 17:41