Take the 2-minute tour ×
Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. It's 100% free, no registration required.

$S_t$ is the random variable representing the risky asset price at time $t$.

M_t is the riskless asset. They are governed by the equations

$\frac{dS_t}{dt}=\mu dt + \sigma dZ_t$ and

$dM_t = rM_t dt$

where $Z_t$ is Brownian motion. If we define the discounted risky asset by $S_t^{*}=S_t/M_t$. How does the process $S_t^{*}$ become governed by

$\frac{dS_t^{*}}{dt}=(\mu-r) dt + \sigma dZ_t$ ?

I cannot see why you subtract $rdt$.

share|improve this question
    
Do you mean S(t)udt in the GBM equation? –  Student T Aug 18 at 4:30

1 Answer 1

up vote 5 down vote accepted

$$ \textbf{Preface} $$ I am assuming log normal asset but this is not clear from the question? Or rather I have misinterpreted the question!

Well as I see it from a a purely mathematical exercise $$ d\left(\dfrac{S_t}{M_t}\right) =\frac{1}{M_t}dS_t - \frac{S_t}{M_t^2}dM_t +O(dt^2) $$

using Ito's lemma.

Then we can sub in the original processes yields

\begin{align} d\left(\dfrac{S_t}{M_t}\right)&=&\frac{1}{M_t}S_t\left(\mu dt + \sigma dZ_t\right) - \frac{S_t}{M_t}\frac{1}{M_t}\left(M_t r dt\right)\\ &=& S^{*}_t\left(\mu dt +\sigma dZ_t\right) - S^{*}_trdt \\ &=& S^*_t\left[(\mu-r)dt+\sigma dZ_t\right] \end{align}

or finally $$ \frac{dS^*_t}{S^*_t} = (\mu-r)dt+\sigma dZ_t $$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.