Distribution of Forward returns

[stable christian]

Dear all,

maybe someone can help me here. I'm searching for literature about the distribution of forward returns of stocks. There's a vast literature about fitting distributions directly on stocks to calulate VaR, but I don't find any about forwards.
Maybe it's just because of I'm searching for the wrong keywords? Or is this not a relevant topic because of the strong connection between underlying and Forward in case of stocks?

Thanks for all hints!

christian

 

[DStahl]

In the case of constant interest rates the forward returns are normally distributed with mean 0 and variance the same as the distribution of the stock return, assuming constant variance. This is applying Ito's lemma to the definition of forward return; ie,
( e^{r(T-t)} S_t )
Actually this also applies to forward rates with random interest rates but the proof is slightly more complicated.

 

[stable christian]

Thank you for this fast reponse. This is exactly what I'm searching. Can you please tell me where can I find this proof or some papers with the theory about this topic?

 

[DStahl]

Actually I lied, I was thinking about martingales and risk neutral valuation.

If stock follows a GBM ( dS=\alpha S dt+\sigma S dW ) then stock returns are normally distributed with mean ( \large(r-\frac{\sigma^2}{2} \right)t ) and variance ( \sigma^2 t )

Then the forward price is related to the stock price in the following manner:

( f=e^{r(T-t)} S\,\, \forall \,\,0 \leq t \leq T )

Using Ito's lemma,

( df=f(\alpha-r)dt+\sigma f dW )

The solution to this SDE is ( f=f_0 e^{\large( \alpha-r-\frac{\sigma^2}{2} \right)t+\sigma dW} )

Taking logs, the return of the forward price is normally distributed with mean ( \large(\alpha-r-\frac{\sigma^2}{2} \right)t ) and variance ( \sigma^2 t )

Under the risk neutral measure, ( \alpha ) and ( r ) cancel and the forward price is a martingale.

 

[stable christian]

Thanks again for this detailed answer. Ok, this is the strong academic line. My problem starts at the beginning: stock returns are not normally distributed (fat tails, volatility clusters,...). My question is - keeping this in mind - what are the consequences for forward returns? Is there an analytical transformation possible inspite of this?
Maybe an example: Let's say I'm using some kind of GARCH model with non-normal innovations to fit some stock returns. Now, there is a forward on this stock. Of course, I can start the same fitting procedure again. But is this really necessary or is this connection indeed just "trivial" as in the GBM case you mentioned?

 

[DStahl]

GARCH estimates sigma. Even if sigma is random, as long as it is assumed that the stock follows dS=alpha S dt+sigma S dw, the volatility of the forward price returns should be the same as the stock (ie, conditionally normally distributed with variance sigma*t)

Note that the variance of the forward price itself is proportional to the forward price, and has greater price swings than the stock (if interest rates are positive).

 

[stable christian]

ok, that's interesting. Thanks very much for your help!