Option definitions and classifications often find themselves overlapping one another and certain options can be classified under various different categories. Basket options are an example which fit into this category. They often overlap other options such as Mountain Range options and Rainbow options because of its multi-asset characteristic.

Introduction || Monte Carlo || Copula Methods || Known Names & Variants || References || Pricing Models

Option definitions and classifications often find themselves overlapping one another and certain options can be classified under various different categories. Basket options are an example which fit into this category. They often overlap other options such as Mountain Range options and Rainbow options because of its multi-asset characteristic.

We can generalize the case of basket options because they often refer to options which have an underlying being, for example, a stock index. This makes valuation somewhat simpler than cases where there are several distinct random variables such as a rainbow option on 3 stocks or mountain range options on n-assets, for example.

The main concern when pricing basket options is that with a large number of assets, the correlation matrix becomes very large:

In order to effectively price multi-asset options, one must first decompose the correlation matrix.

Perhaps the most commonly used method of pricing basket options. Although not computationally efficient (particularly as the number of underlying assets is large), Monte Carlo has the ability of handling numerous assets which otherwise would be difficult to evaluate under analytical, tree or finite difference models. By simulating each underlying asset and then discounting the associated payoff of the basket option, we know that a solution can be reached.

As with all other multi-asset problems pricing basket options requires one to decompose the correlation matrix:

1) Determine a suitable correlation matrix for the underlying assets.

2) Set up a simulation for the paths.

3) Generate random normal variables with mean of 0 and variance of 1.

4) Standardise the variables.

5) Apply Cholesky decomposition – but noting that some correlation values cannot be used as they are not positive.

6) For each path, loop the simulations from 0 to n and 0 to i, where n is the number of underlying assets and i represent the number of time periods.

Pellizzarri (1998) shows that the use of control variates as a form of variance reduction can help to reduce the dimensionality of the multi-asset problem with and increase computational efficiency.

A vast amount of recent literature has been focused on the use of Gaussian copulas to price multivariate options such as basket options. [Work in Progress]

American Basket Options

Asian Basket Options

Index Options

Mountain Range Options

Multifactor Options

Portfolio Options

Rainbow Options

Spread Options

**Additional/Useful List of resources**

Papers:

**Black, F. & Scholes, M.*** "The Pricing of Options & Corporate Liabilities", *The Journal of Political Economy (May '73)

Cox, J., Ross, S., & Rubinstein M.,*“Option Pricing: A Simplified Approach."* Journal of Financial Economics, 7. (Sept '79).

**Dahl, L.O., & Benth, F.E.,** "*Valuation of Asian Basket Options With Quasi-Monte Carlo Techniques and Singular Value Decomposition", *Pure Mathematics*, 5* (2001)

**Hull, J.***, "Options, Futures & Other Derivatives"*, 5th Edition 2002 - Chapter 12

**Laamanen, T.,** "*Options on the M Best of N Risky Assets"*, Working Paper (2000)

**Merton, R.,***"Theory of Rational Option Pricing"*, Bell Journal of Economics & Management (June '73)

**Pellizzari, P.,** "*Efficient Monte Carlo Pricing of Basket Options",* Working Paper (1998)

**Salmon, M., & Schleicher, C., **"*Pricing Multivariate Currency Options with Copulas*", Working Paper (2006)