Risk-Neutral Valuation

In Principles of Corporate Finance the risk neutral valuation is calculated by assuming that investors are indifferent to risk and expect return on stock to be equal to the risk-free rate of interest. Furthermore it assumed that stock have a certain downside potential $d$ and and and upside potential $u$. There is a certain chance $p$ that the stock will go up with percentage $u$ and a chance of $1-p$ that the stock will go down with percentage $d$. This gives:

${\text{Expected Return}}$ = $p\cdot u+(1-p)\cdot d=r$

This gives the general probability of rise of the stock:

$p$ = ${\frac {(1+r)-d}{u-d}}$

Valuing a call this way gives ($k$ is the exercise price):

${\text{Expected value of a call}}$ = ${\frac {p\cdot max(Su-k,0)+(1-p)\cdot max(Sd-k,0)}{1+r}}$

Using the same example as in Replicating Portfolio we get:

$p$ = ${\frac {(0.025)-(-0.25)}{0.33-(-0.25)}}$ = $.471$

${\text{Expected value of a call}}$ = ${\frac {0.471\cdot max(106.67-80,0)+(1-0.471)\cdot max(60-80,0)}{1+r}}$ = $12.26$