Reconstructing the CRR model knowing put and call prices

External question — Quantitative Finance Stack Exchange Author: Analysis Original post: https://quant.stackexchange.com/questions/76463 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. 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 $u,d$ ?

External answer — Quantitative Finance Stack Exchange Author: Ledog Original post: https://quant.stackexchange.com/a/76475 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. 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