Lognormal model vs Geometric BM

[Benjamin]

Question based on PEQ April 2011, question 6, part (iii) (using ASET too).

Confused about the ASET explanation in the solution to part (iii) - I understand the notion that if the fair price is based on a risk-neutral probability, that will be lower than real-world probability and so the proposed price is too high. But, not clear specifically on the calculation of the 0.0128 in the solution - why is the risk neutral probability using the Geometric BM drift and why is the mu in the formula in the asset replaced with r?

Thanks in advance!

 

[Steve Hales]

The difference comes as a result of whether you're working in the real word or the risk-neutral world.

In part (ii) of the question we know we're in the real world because we're told the drift of the lognormal model, \(\mu\). The proposed solution to part (iii) involves finding the risk-neutral probability of \(S_1>2.20\) - and so we know that now we're operating in the risk-neutral world. In the risk-neutral setting, risky assets are expected to grow at the risk-free rate. In order to allow for this the drift term in the lognormal model is replaced by \(r-\frac{1}{2}\sigma^2= 1.28\%\).

 

[Benjamin]

Ok cool. Thank you!