Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. It only takes a minute to sign up.
Sign up to join this community
You are given a $5\%$ call option worth $\$2.66$. The strike price $k$ is $\$41.00$. $S(0)=40$, $Sd=35$ (i.e the lower price of the stock at $t=1$) find $Su$ (i.e the high price of the stock at $t=1$).
How would this be done? I know $Cu=Su-\$41$, $C(0)=\$2.66$ and I have something written in my notes that $c(0)= B/(1+r) +ΔS(0)$, but I'm not sure what $B$ is here. I assume to find $Su$ I need to first find $Cu$, but I'm not sure how to do this from the given information. Any help is appreciated.
By no arbitrage, $S(0)=(Su q+Sd (1-q))/(1+r)$ and $C(0)=((Su-k)^+ q+(Sd-k)^+(1-q))/(1+r)$. Simplifying and rearranging (and assuming $Su>k$), $$\left[\begin{array}{c} S(0) \\ C(0) \end{array} \right]=\frac{1}{1+r}\left[\begin{array}{cc} Su & Sd \\ (Su-k) & 0 \end{array} \right]\left[\begin{array}{c} q\\ 1-q \end{array} \right] $$ Clearly, $q=C(0)(1+r)/(Su-k) $. Substituting this back into the first equation, $$S(0)=\left(\frac{Su C(0)(1+r)}{Su-k}+Sd (1-\frac{C(0)(1+r)}{Su-k})\right)/(1+r)$$ After simplifying and solving for Su, $$Su=\frac{kS(0)(1+r)-kSd-C(0)Sd(1+r)}{S(0)(1+r)-C(0)(1+r)-Sd}$$
