Calibrating a Two-Branch Binomial Model for Option Pricing

Summary

The document frames a calibration problem for a one-period stock-price binomial model. The two possible next-period prices are represented as multiplicative up and down moves from the current price, with continuously compounded risk-free growth over the period. Given simulated monthly price paths, the question asks how to estimate the up and down move parameters and the probability, potentially using moment matching to infer volatility.

It also states the risk-neutral probability relation linking the risk-free growth factor to the up and down factors. However, the text contains only the setup and question; it provides no calibration solution, moment equations, assumptions about the simulated paths, or evidence that a particular specification is suitable. Readers would need to distinguish physical path probabilities from risk-neutral pricing probabilities before using such estimates for option valuation.

Key ideas

• A binomial model represents the next stock price with an up branch and a down branch.
• The up and down parameters can be expressed as log moves from the current price.
• The risk-neutral probability is linked to the risk-free growth factor and the two branch factors.
• Moment matching and volatility are proposed as possible calibration tools, but no derivation is supplied.
• Simulated path probabilities and risk-neutral pricing probabilities may represent different quantities.

Tags

# Option pricing when stock price follows binomial tree


# 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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.