Expected Return .... Black Scholes

[sipanivishal]

Hi David,

Can you explain to me the reason why expected value E(x) is different from Mue (u) . The reason mentioned in Hull 13.3 is not very clear.

Thnx Sipani

 

[David Harper CFA FRM]

Please review this question. You can see the XLS shows exactly the difference.

The difference is between arithmetic and geometric return. In regard to geometric returns, volatility erodes returns.

The future stock price has a lognormal distribution: mean does not equal median (distribution is asymmetric). The median < mean. That's the difference between the two "expected returns," one is median (reduced by one half variance) the other is mean.

David

 

[ashm07]

David,

Based on "Price levels are lognormally distributed. If LN(tomorrow's price/today's price) is normally distributed, then [tomorrow's price] is lognormally distributed."
Is there not a difference between lognormal returns versus lognormal prices?
Is there an implicit assumption that lognormal of tomorrow's price divided by today's is still lognormal?

Thanks,

 

[David Harper CFA FRM]

Hi seaTurtle,

Most of this is mathematically a given: LN(price/prior price) is a continuously compounded return.
Then we could make all sorts of assumptions about the distribution of this CC return; e.g., we can follow classic tradition -- in violation of empirical reality -- and assume this log return is normally distributed (i.e., the classic and dubious "returns are normal" assumption)

There is a whole theory of the underlying process (Brownian motion) but we don't really need it: we can start from the DUBIOUS ASSUMPTION that the log return is normally distributed. Given that assumption,

Ln(St/S) ~ N(.); i.e., is approximately normal implies
LN(St) - LN(S) ~ N(.) implies
LN(St) ~ N(.)

So given the initial dubious assumption (used for convenience!) about log returns ~ N(.), it follows as mathematical truism that the future price (St) is lognormal.

Hope that helps

David

 

[ashm07]

Hi David,

Thanks for the great explanation!