Two-Asset Binomial Trees for Pricing Options

Summary

The document describes two approaches to extending binomial or multinomial option pricing to two risky assets over multiple time steps. One approach changes to a stock measure and expresses the pair of asset prices through their ratio, using a common state variable. The answer associates this route with the Garman-Kohlhagen framework for options on two underlyings.

A second approach constructs a multi-outcome one-step tree using asset-specific up factors and solves for probabilities that match each asset’s expected value and variance, along with the covariance implied by their correlation. The document points to course material for further study but does not give the full derivation, probability constraints, or a worked multi-step example. It therefore outlines modeling routes rather than furnishing a complete pricing procedure.

Key ideas

• A two-asset option tree can reduce the state description to the ratio of the two asset prices under a stock measure.
• A multi-outcome tree can be calibrated to match each asset’s expected value and variance.
• The joint tree must also reflect covariance between the risky assets.
• The document sketches two approaches but does not provide a full derivation or worked multi-step solution.

Tags

# I’m trying to construct a binomial model that uses 2 risky model - number of steps varied


# I’m trying to construct a binomial model that uses 2 risky model - number of steps varied












So with this question I am unsure how to even do a binomial model with 2 risky assets never mind having n-steps. All the examples I’ve found are either not containing any risky assets or only have one. Every time I try to google it I get more confused and I really need some help.

## Answer by Kermittfrog (score 2)

https://quant.stackexchange.com/a/59059

I see two possible paths for solving for the price of an option written on two underlying assets in a bi/multinomial tree.

1: Change to a stock measure and express the state space $S^{(1)}_t,S^{(2)}_t$ in terms of a common state variable $Z_t=\frac{S^{(1)}_t}{S^{(2)}_t}$. This is the Garman-Kohlhagen approach to pricing an option on two underlyings.

2: Set $U_i=e^{\sigma_i\sqrt{\Delta t}}$ as in the CRR model and find some probabilities $p_1,p_2,p_3,p_4,p_5$ such that $E(S^{(i)}_{t+\Delta t})=R$ as well as $V(S^{(i)}_{t+\Delta t})=R^2e^{\sigma_i^2\Delta_t}$ and $Cov(S^{(i)}_{t+\Delta t},S^{(j)}_{t+\Delta t})=R^2e^{\sigma_i\sigma_j\rho\Delta_t}$.

I found some recommendable course material on this on the pages of prof. Kwok, starting on p 311 here..

HTH?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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