Reconstructing the CRR model knowing put and call prices

Question

In an arbitrage-free single-period CRR model, the following options on a share are offered:

[They are all European]

(i) Call option at strike price $100$, price: $C_{0,1}=7.44$

(ii) Call option at strike price $110$, price: $C_{0,1}=3.72$

(iii) Put option at strike price $100$, price: $P_{0,1}=23.59$

(iv) Put option at strike price $110$, price: $P_{0,1}=29.49$

Show that given this information the model is fully specified.

Using the Put-Call parity I got $7.44-23.59=S_0-100/(1+r)$ and $3.72-29.49=S_0-110/(1+r)$ which yields $S_0\approx 80$ and $r\approx 0.04$.

How can I get $u,d$?

Answer 1

I suppose by finding martingale measure. If then $u = 1+b, d = 1+a$, the probability of going to the up state is given by $p = \frac{r-a}{b-a}$. $(r \in (a,b))$. The second equation needed to solve for u and d is $ud=1$. Not 100% sure about this approach so please correct me if I'm wrong