PRICING BASKETS
Pricing a basket of stocks from one country proceeds essentially along the lines of pricing a basket of groceries. If the basket contains stocks priced locally in different currencies, in order to compute the price of the basket in a common currency, one must first multiply each share price in its local currency price by the price of that currency in the common currency.
BASKET SWAPS, FORWARDS, AND FUTURES
An at-the-money equity basket swap, forward, or futures contract has zero value at creation. After that, the mark-to-market value of the swap or forward contract is the discounted value of the forward value of the net payoff. If interest rates are nonstochastic, then we can price the futures contracts the same way.
With stochastic interest rates we would want to adjust this computation for the correlation of movements in the underlying risk factor and interest rates. Sometimes the correlation is significant enough that professionals make this adjustment, and sometimes they ignore it.
BASKET OPTIONS
Two popular ways of pricing a basket option are moment matching and multivariate integration. Moment matching assumes that we can model the basket price as we ordinarily model a share price, allowing us to use a standard, univariate pricing model--for example, Black (1976). Multivariate integration allows us to recognize that the basket price depends on multiple, lognormal underlying stock prices. Theoretically, the two approaches are inconsistent, because the price of a stock basket is a linear combination of lognormal stock prices, and a linear combination of lognormal random variables is not lognormal, as a general rule. However, in practice, the two approaches often give approximately the same result.
MOMENT MATCHING
If we assume that the basket price is S = Q1S1+ ... + QnSn, where Qi is the number of shares in the basket of stock i, Si is the lognormal price of the ith stock, and S behaves like a share price, we need to know its forward price and its volatility. The forward price for each share is the expected valueusing the equivalent martingale measureof the spot price at the delivery date (TD):
assuming that interest rates are not stochastic. The forward price, F(t), of the basket is the obvious weighted sum of the forward prices of the shares in the portfolio. For example, if a basket consists of a single share in each of two corporations, then its spot (forward) price is the sum of the two constituent spot (forward) share prices.
Computing the basket's volatility is straightforward, but requires more steps. A fundamental mathematical relationship among the moments of the prices of the portfolio and its component shares is:
Assuming that all the prices are lognormal,
where T is the option expiration date, * is the annualized volatility for the basket, and *ij is the covariance rate for two rates of return. The problem is to solve for the basket volatility, then use it and the basket's forward price in Black's (1976) model. This gives us a quick--and sometimes dirty--method for pricing a basket option.
MULTIVARIATE INTEGRATION
If we decide to go the more complicated, multivariate route, we would typically assume that the vector of spot prices follows a multivariate lognormal diffusion with a mean vector and a covariance matrix. By the standard arbitrage argument we can derive the relevant multivariate analog of the Black-Scholes-Merton (1973) or Margrabe (1978) partial differential equation (PDE). This PDE and the basket option's payoff function define an initial value problem. We can solve this problem a variety of ways.
For a European basket option, spot value is the discounted, expected value--a multivariate integral--of its random payoff. Numerical integration using something like the trapezoidal rule works quickly for a basket of two stocks, but extremely slowly for a basket of, for example, 10 stocks. If it took 20 points to adequately evaluate each one-dimensional integral, then pricing an option on a basket of five stocks would take about 160,000 times as long as pricing an option on one.
Monte Carlo integration handles one or two risk factors slower than other methods of numerical integration, but handles many risk factors much quicker, and is the only way to go for about five or more risk factors. If we needed 10,000 paths to obtain sufficient accuracy, then pricing an option on a basket of five stocks would take about five times as long as pricing an option on one.
For an American basket option, a multinomial model can work nicely. It is an extension of the familiar binomial model, which has one risk factor and in which each node branches to two "children". With two risk factors we would need at least three branches from each node, etc. Multinomial models were able to handle up to six risk factors back when a 286 PC was top of the line.
A finite difference scheme pops out fairly naturally from the PDE after we approximate each partial derivative with the appropriate ratio of finite differences in the option value, time, and the price variables. Theoretically, the multinomial model is (approximately) a special case of this approach, and other special cases--if you can find them--promise to outperform the multinomial model. In practice the computational demands of this approach at higher dimensions can be overwhelming.
The figure in the lower left-hand corner below--based on the assumptions in the table on the previous page--illustrates two points about pricing basket options:
(1) A trapezoidal rule, Monte Carlo integration, and moment matching often produce similar values. In this figure the models produce values that differ by less than one percent for each correlation from 0.30 to +0.95. Moment matching performed less well for extremely negative correlations. The trapezoidal rule with 20 points broke down for the extreme correlation of 0.999.
(2) The value of this call option on a basket of two stocks is an increasing function of the correlation between the two rates of return.
The figure above, for multiple assets with the same positive correlation between each pair of asset returns, shows how basket option value is (1) an increasing function of the correlation and (2) a decreasing function of the number of assets. A basket with a large number of uncorrelated assets has little risk, and an at-the-money-forward option on that basket has little value.
With any number of stocks having the same volatility, as the typical correlation between an arbitrary pair of stocks goes to unity, the basket value behaves increasingly like a single, lognormal stock price with that volatility, and the value of the European basket option approaches the value of an option on that stock using the Black (1976) model.
This week's Learning Curve is by Bill Margrabe, president of the Margrabe Group(www.margrabe.com), a risk management and financial engineering consulting firm.
