# Option pricing when stock price follows binomial tree

Assume that the stock price is currently trading at $$S_0$$. It is known that the stock price follows a binomial tree, such that its price will be either $$S_0e^{\theta_u}$$ or $$S_0e^{−\theta_d}$$ over the next month. The monthly risk-free rate is $$r$$ and is continuously compounded. Let's say that I have the simulated monthly price paths as an input. By the paths I can calculate the $$r$$ and $$p=\frac{e^{r\Delta t}-d}{u-d}$$.

Now my question is how do I calibrate $$\theta_d$$, $$\theta_u$$, and $$p$$. In order to do so, I think I need to utilize the binomial properties and the method of moments (maybe to get the $$\sigma$$?). The ultimate goal is to get an analytical solution for each one.

• One month time-period sounds too long, I'd go for one-day as your time period. Then, each day is a Bernoulli trial when the stock can go either up or down. Take a historical sample of n-days, define $p$ as the probability of the stock going up. Each day, the Bernoulli indicator is $X_i$, where $X_i \epsilon {1,0}$ with probability $p$. The estimator of $p$, that I denote $\tilde{p}$, will be distributed as follows: $$\tilde{p}=\frac{1}{n}\sum_{i=1}^{n}X_i$$ Because $\tilde{p}$ is effectively the sample mean, we know that as $n \to \infty$, $\tilde{p} \to^{d} N(p,\sqrt{n}pq)$. Commented Sep 19, 2021 at 16:06
• PS: I think that because we can only estimate $p$ via a historical sampling, you'll need to make an assumption on $d$ and $u$: you can choose these to be symmetric, so that they give you the estimated $p$. Maybe somebody else will provide a better solution, this one is the first one that came to mind :) Commented Sep 19, 2021 at 16:08