Pricing Equity Basket Options Beyond a One-Dimensional Binomial Tree
Summary
The document asks how to price a European option on a portfolio of equities when the constituent volatilities and correlations are known. Its answer explains why a standard one-dimensional binomial tree is generally unsuitable under Black–Scholes: individual securities may be lognormal, but their sum is not generally lognormal. An exact tree or grid representation would need to track the portfolio’s multiple underlying assets, making the state space grow with the number of constituents.
It outlines alternatives rather than deriving a formula. These include estimating portfolio volatility from historical data, simulating each constituent with Monte Carlo, or approximating the basket distribution with a moment-matched lognormal or shifted lognormal and then applying standard option pricing formulas. The answer favors moment matching as a practical route and mentions its application to American basket options, but supplies no derivation, numerical comparison, or implementation details. Historical estimates also face limitations such as survivorship effects and missing price history around corporate events, while Monte Carlo and moment matching introduce their own modeling approximation or computational considerations.
Key ideas
- A sum of lognormal equity prices is generally not itself lognormal.
- An exact lattice method for a basket with multiple constituents must track multiple state variables.
- Monte Carlo can model the constituent assets individually to price a basket payoff.
- Moment matching approximates the basket distribution with a lognormal or shifted lognormal for closed-form pricing.
- Historical portfolio volatility can be limited by survivorship effects and incomplete histories.
Tags
Full text
# How to use binomial tree for portfolio of equity products # How to use binomial tree for portfolio of equity products How can I use a binomial tree to price a European option that's based on a portfolio of equity products? I have volatility and correlation matrix of all underlying products? Looking for a formula based solution so that I use in Matlab. Thanks. ## Answer by Brian B (score 4, accepted) https://quant.stackexchange.com/a/280 Under the typical Black-Scholes model, you "cannot" do it, because the assumption is that each of the securities in the portfolio has a lognormal terminal distribution, and the sum of lognormally distributed variables it not itself lognormally distributed. In theory one needs an N-dimensional tree (or grid) to treat an N-element portfolio. I write "cannot" in quotes because this problem is actually quite commonly encountered and solved in one of a few ways, none of which involves a binomial tree: - If you are comfortable using historical estimates, simply look at the volatility of the portfolio hypothetically over history. This has two significant disadvantages: (i) historical volatility is generally smaller than forward volatility due to survivorship bias, and (ii) there may have been IPOs or other corporate events that make the portfolio value unknown before some date - Use Monte Carlo to simulate every element of the portfolio, pricing the option by the usual MC methods. - Use the trick of moment-matching, where a little mathematics tells you the equivalent lognormal (or sometimes shifted lognormal) distribution to your portfolio. You can then use the usual closed form option pricing formulas. The technique has been around since at least the mid-90s. Since not all the papers from back then are easy to see online, here's an URL to a recent rediscovery of the trick in which they go so far as to run a binomial tree for American basket options. The final technique is almost certainly what you want to use.
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.