September 2006, q8

Looking at the ASET solution for this (p34), slightly confused as to how they jump from the definition of log(St) = log(So) + etc to the conclusion that log(St) must have a normal distribution with those parameters.

Can anyone put me out of my misery?

 

Hi.

Here we're working under the set of risk-neutral probabilities, or equivalent martginale measure (EMM) Q. By definition, this is the set of probabilities that ensure that the expected rate of (continuously compounded) return/drift of the underlying risky share is equal to the risk-free force of interest, r.

So, under Q, the stochastic differential equation for S(t) must be:

dS(t) = S(t).[r.dt + sigma.dZ(t)~]

where Z(t)~ is a standard Brownian motion under the Q probabilities.

(This is what you have already stated in part (i) of the question.)

So, if we apply Ito's Lemma to the function:

G(S(t) = ln S(t)

as per the example on p22 of Chapter 7 of the Course Notes, then we can see that the corresponding formula for S(t) itself is:

S(t) = S(0). exp[(r - sigma/2)t + sigma.Z(t)~]

{Note that whenever you have a formula of this type, if you ignore the -sigma/2 term, then whatever else is in the coefficient on t (ie r here), tells you what the drift of S(t) is. The -sigma/2 term appears because of the 2nd-order term in Ito's Lemma.}

Also, if we log the equation for S(t), then we can show that lnS(t) has a Normal distribution, as shown on p34 of the ASET, and hence that S(t) itself has a logNormal distribution.

Hope this helps.

Graham

 

Thanks Graham - I get the bulk of the question, but just confused as to the last part, i.e. "we can show that lnS(t) has a Normal distribution", exactly how do we know that lnS(t) has a Normal distribution?

 

The key is the equation that Graham presented:
S(t) = S(0). exp[(r - sigma/2)t + sigma.Z(t)~]

Rewrite as
ln(S(t)/S(0)) = [(r - sigma/2)t + sigma.Z(t)~]

Now we now Z(t) is N[0,t] since it is standard BM
so ln(S(t)/S(0)) or (lnS(t) if S(0)=1)
must be N[(r - sigma/2)t,sigma^2t]

 

Got it, being slow, thanks for the replies

 

what about part (v) of that question?! I'm lost!

i don't have ASET, only the Revision notes, so maybe that's my problem but I can't understand beyond the basic fee of 0.1%!