September 2016 Question 8 part v

Hi all,

I have just finished this question, and although the solutions make sense i am stumped as to why my approach isn't correct.

we are trying to find the value of an option v_0=exp(-0.02)*[50*P(S_1/S_0 <0.8) *P(R_1/R_0 <0.6))
we are given r=0.02 and sigma_s=0.32 and sigma_r=sqrt(0.15)
my approach would be to use that
S_1/S_0-lognormal(muT, \sigma^2T)
and since under black scholes mu is = r, we know mu=0.02
S_1/S_0-lognormal(0.02, 0.32^2)
R_1/R_0-lognormal(0.02, 0.15)

so P(S_1/S_0 <0.8)=P(Z<ln0.8-0.02/0.32)=0.22386
and P(R_1/R_0<0.6)=P(Z<ln0.6-0.02/sqrt(0.15))=0.08525
so v_0=exp(-0.02*)*[50*0.22386*0.08525)
=0.9346 which is not equal to the 1.61 which is the correct answer.

If anyone is able to let me know why we cant use this method that would be amazing please - is my assumption that under black scholes mu=r wrong?

Thank you

 

Hi John, thank you for this! that makes complete sense, and has actually made me realise that i was getting a little confused with the whole \mu=r but \mu also being the drift thing.
for example E(S_t)=S_0*exp(\mu t+1/2\sigma^2) (call this formula 1)
Im also saying that \mu=r (call this formula 2)
I cant necessarily say that E(S_t)=S_0*exp(rt+1/2\sigma^2) because the mu in formula 1 and mu in formula 2 aren't the same thing (i think?).
If I go with the mindset of mu being the drift (aka the coefficient of t in the log normal model S_t), and if not given mu directly but instead given r then mu(aka drift)=r-0.5, then that should be ok i think?
Thank you!

 

Yes that part makes a lot of sense now, thank you!!