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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.

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  • $\begingroup$ 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 $n \to \infty$, $\tilde{p} \to^{d} N(p,\sqrt{n}pq)$. $\endgroup$
    – Jan Stuller
    Sep 19, 2021 at 16:06
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    $\begingroup$ 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 :) $\endgroup$
    – Jan Stuller
    Sep 19, 2021 at 16:08

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