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
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.
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