Two scenarios are foreseen for a certain stock after one period: one in which the stock value is $110E$ and another in which the value is $90E$. Its current value is $S_{0}=100E$. Furthermore:
Each operation of selling stock to the market carries a fee of $2\%$ (there is no fee to buy from the market).
Borrowing money costs $12\%$.
Now I want to know what the risk neutral probability is of a call option.
I know the formula: $p=\dfrac{R_{0}S_{0}-S_{1}(t)}{S_{1}(H)-S_{1}(T)}$ with
$S_{1}(H)=110$
$S_{1}(T)=90-2\%90=88.2$
and $R_{0}=1+r$ where $r$ is the interest rate which I assume is zero .
Where does the $12\%$ borrow rate come into play? I tried using the $12\%$ as interest rate but then I get that $p>1$ which is impossible because its a probability.
I gave you the below solution for one of your questions earlier. In the same solution, substitute the value of 12% for r and you get the answer. If you want the derivation, let me know I shall do it.
The solution for this would be
Risk Neutral Probability $= \frac{(1-d-(1+r)k)}{u-d-(1+r)k}$
Fair Price of the Option $ = \frac{1}{1+r}\left(p\psi{(u)}+(1-p)\psi{(d)}\right)$
where $\psi{(u)} = Max((110-100),0) = 10$
$\psi{(d)} = Max((90-100),0)= 0$
Solution for the said problem is
$p = \frac{(1-.9-(1+0.12)0.02)}{1.1-0.9-(1+0.12)0.02} = \frac{0.0776}{0.1776} = 0.43694$
fair price of the option $= (10*0.43694+0*(1-0.4444)) = 4.3694E$