Basket Options

A basket option is an exotic option that combines several assets into one underlying asset. In its valuation and risk management, the correlation assumed between the assets drives the price and the hedge. The classification of exotic options as a whole is covered in Overview of Exotic Options.

How the basket value and the payoff are set

The underlying asset of a basket option is the sum of the prices of several assets at weights set in the contract, called the basket value. Let $S_i$ be the price of asset $i$ and $a_i$ its weight. The basket value $B$ is then given by the following formula¹.

$$B = \sum_{i=1}^{n} a_i S_i$$

The payoff of a call at expiry is the amount by which the basket value exceeds the strike price $K$, that is, $\max(B - K,\ 0)$. The payoff of a put is $\max(K - B,\ 0)$. A paper by the Institute for Monetary and Economic Studies of the Bank of Japan uses a European call on the sum of the prices of 32 stocks as its example².

Options on the highest or lowest of several prices use the maximum (max option) or the minimum (min option), not the sum, as the underlying asset. These are options for which over-the-counter quotes were available early on. There are also spread options on the difference between two prices³.

Consider a hypothetical example in which Company C buys a European call on a basket that sums the prices of stocks X and Y at weights of 0.5 each. X and Y are both 100 yen now, so the basket value is 100 yen. The strike price is 100 yen, expiry is in 1 year, and the figures are per unit. The payoff is $\max(0.5X + 0.5Y - 100,\ 0)$.

Figure description

The payoff at expiry of the basket call (strike price 100 yen) bought by Company C. The horizontal axis is the basket value at expiry and the vertical axis is the payoff per unit, excluding the premium. The payoff is 0 yen when the basket value is 100 yen or below. Above 100 yen, the payoff rises by 1 yen for each 1-yen rise in the basket value, reaching 20 yen at 120 yen and 40 yen at 140 yen.

Company C pays the premium at the start. In cases 1 and 3 there is a payoff, and the payoff is a gain if it exceeds the premium and a loss if it is smaller. In cases 2 and 4 the payoff is 0 yen, and the loss is the premium paid. In case 2, X rises by 30 yen but Y falls by 30 yen and the basket value stays at 100 yen, so there is no payoff.

Why it is cheaper than the sum of the individual options

The payoff of a basket call never exceeds that of a combination of calls on each asset at the same weights. Compare it with the case where Company C buys calls on 0.5 unit each of X and Y separately (both with a strike price of 100 yen).

The combination receives only the rise on each asset, and the fall cannot be used to offset it. The basket adds the rises and falls together before setting the payoff. When the weights are positive and the strike price of the basket equals the weighted sum of the strike prices of the individual assets, the following relation holds in every case.

$$\max\Bigl(\sum_i a_i S_i - \sum_i a_i K_i,\ 0\Bigr) \le \sum_i a_i \max(S_i - K_i,\ 0)$$

Equality holds when all the assets are above their strike prices or all are below, which applies to cases 1 and 4. Because the payoff is always the same or smaller, the premium of the basket call does not exceed that of the combination. However, the basket cannot pick the movement of each asset, so, as in cases 2 and 3, it may fail to receive an amount that the combination would have received.

How correlation affects the premium

The lower the correlation between the assets, the more easily a rise in one is offset by a fall in the other, and the smaller the variation in the basket value (its volatility). The price of a derivative with several underlying assets is affected by the correlation between the prices of those assets⁴.

Consider a hypothetical example in which the current price of X and Y is 100 yen, the weight is 0.5, the annual volatility of each is 20%, the strike price is 100 yen, expiry is in 1 year, and interest rates and dividends are 0%. The table compares the volatility of the basket and the price of the call for each correlation coefficient. The prices are values from the moment-matching approximation described below.

When the correlation coefficient is 1, the volatility of the basket is 20%, the same as each asset, and the price of 8.0 yen is the same as that of the combination of calls on X and Y. The lower the correlation coefficient, the further the price falls below that of the combination⁵. In this example, the price of the call falls as the correlation falls.

Moment matching and the Monte Carlo method

The pricing formula for a European option on several assets usually contains a multidimensional integral and needs numerical computation⁶. Two methods are often used for basket options.

The first is moment matching, which replaces the distribution of the basket value with a lognormal distribution. Assuming that the price of each asset follows a correlated geometric Brownian motion, the distribution of the basket value is approximated by a lognormal distribution, which is easy to handle. The volatility $\bar\sigma$ of the basket is chosen so that the mean and the mean of the square of the basket value at expiry agree with those of the lognormal distribution used in the approximation. With no dividends and constant interest rates and volatilities, $\bar\sigma$ is given by the following formula⁷.

$$\bar\sigma^2 = \frac{1}{T}\ln\left(\frac{\sum_{i,j} a_i a_j S_i S_j\, e^{\rho_{ij}\sigma_i\sigma_j T}}{\bigl(\sum_i a_i S_i\bigr)^2}\right)$$

Here $\sigma_i$ is the volatility of asset $i$, $\rho_{ij}$ is the correlation coefficient between assets $i$ and $j$, and $T$ is the number of years to expiry. When the weights are positive, the larger the correlation coefficients, the larger the numerator and so the larger $\bar\sigma$. With this $\bar\sigma$ as the volatility of the underlying asset, the price of the call is found with a formula of the same form as the Black-Scholes model⁸. The table in the previous section was calculated with this method.

Deep diveDerivation of the formula from the mean of the square

Consider the case with no dividends and a constant interest rate $r$. The expected value of the product of asset prices is $E[S_i S_j] = S_i S_j\, e^{(2r + \rho_{ij}\sigma_i\sigma_j)T}$, where $S_i$ and $S_j$ on the right-hand side are the current prices. Using this, the expected value of the square of the basket value is as follows⁷.

$$E[B^2] = \sum_{i,j} a_i a_j S_i S_j\, e^{(2r + \rho_{ij}\sigma_i\sigma_j)T}$$

The lognormal distribution used in the approximation has a mean of $B_0 e^{rT}$ and a mean of the square of $B_0^2 e^{(2r + \bar\sigma^2)T}$ (with $B_0 = \sum_i a_i S_i$). Setting the mean of the square equal to $E[B^2]$ and solving for $\bar\sigma$ gives the formula in the main text.

This approximation matches only the first two moments of the distribution of the basket value. In the numerical experiments of Brigo et al., the approximation agreed well with the Monte Carlo price when the volatilities of the assets were close and the correlation was high. When the volatilities differ widely or the correlation is negative, the quality of the approximation falls sharply⁹.

The second is the Monte Carlo method. Uniform random numbers are converted into standard normal random numbers, and then into multidimensional normal random numbers that follow the variance-covariance matrix $\Sigma$, to generate many prices of the underlying assets. The payoffs of the runs are averaged to give the price¹⁰. The variance-covariance matrix contains the volatility of each asset and the correlation coefficients between the assets.

There is also a technique that fits the mean and the variance-covariance matrix of the random numbers to their theoretical values (first- and second-order sampling). In the 32-stock call example in the same paper, using this technique together reduced the standard deviation of the price over 10,000 trials to about 1/5.4 of that without it¹¹.

With either method, the price changes with the correlation coefficient assumed. How to set it is covered in the section on correlation risk.

Correlation risk and the difficulty of hedging

The value of a basket option also moves when the correlation changes. This risk, which single-asset derivatives do not have and which is specific to derivatives with several underlying assets, is called correlation risk¹². In the example in the table in the previous section, the price of the basket call falls as the correlation falls.

For options on a single underlying asset, a common hedge is to bring the total vega on the same underlying asset to zero. Correlation derivatives have low liquidity, and it can be hard to find a correlation derivative on the same underlying assets in the market. Firms therefore try to hold the vega and the correlation risk as they are and to control the total risk, for example by combining correlation derivatives on similar underlying assets¹³.

In practice around 2003, the risk of individual stocks or baskets of stocks was often hedged with stock index futures or equity index options. This hedge relies on the correlation between the index and the stocks. If the correlation is not as assumed, the hedge may not work and market risk may remain¹⁶. The general discussion of hedging is covered in Hedging Exotic Products.

Notes

1. Damiano Brigo et al., "Approximated moment-matching dynamics for basket-options simulation" (Banca IMI, October 2002 version), Section 2 "Assumptions and notation" (page 3). The paper treats the weights $a_i$ as constants set by the contract. ↩

2. 石川達也、内田善彦「モンテカルロ法によるプライシングとリスク量の算出について」(Pricing and calculating risk amounts by the Monte Carlo method; our translation), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (June 2002), Section 5 (1) "バスケット型オプションのプライシング" (Pricing of basket-type options), page 66 (in Japanese). The example is a European call on the sum of the prices of 32 stocks, with the current price of each stock at 100 yen and expiry in 1 year. ↩

3. Mark Broadie and Jérôme Detemple, "Option Pricing: Valuation Models and Applications," Appendix A.3 "Multiasset Options" (page 1172). The paper explains that quotes for options on the maximum or minimum have long been available over the counter, and that spread options on the difference between two prices can also be traded on exchanges. ↩

4. 中村一岳「相関デリバティブのリスク特性やリスク管理手法に関する考察」(A study of the risk characteristics and risk management methods of correlation derivatives; our translation), Institute for Monetary and Economic Studies, Bank of Japan, Discussion Paper No. 2003-J-7 (May 2003), Section 1 "はじめに" (Introduction), page 1 (in Japanese). ↩

5. Damiano Brigo et al., cited above, Section 6 "An empirical analysis of the two- and three-moment matching approximations" (page 16). When the correlation coefficient is 1 and the volatilities are equal, the basket actually follows a geometric Brownian motion, so the approximation is exact. The figures in the table were calculated under the conditions in the main text and rounded to one decimal place. ↩

6. Mark Broadie and Jérôme Detemple, cited above, Section 5.1.3 "Multiasset Options" (page 1166). ↩

7. Damiano Brigo et al., cited above, Section 3.1 "Case with continuous dividends, constant rates and volatilities" (pages 4–5). The formula in the paper includes the continuous dividend yield, and the formula in the main text is the case where the dividend yield is 0. ↩ ↩²

8. Damiano Brigo et al., cited above, Section 4.1 "The call option value in the three-moment matching procedure" (page 13). The paper explains that the price of a call on the approximated geometric Brownian motion takes the Black-Scholes form (the paper writes this within the procedure that matches three moments). ↩

9. Damiano Brigo et al., cited above, Section 6 (page 16) and Section 9 "Conclusions" (page 25). The experiments cover baskets of two assets. The approximation agrees well with the Monte Carlo price when the volatilities are similar and the correlation is high, and its quality falls sharply when the correlation is negative. ↩

10. 石川達也、内田善彦, cited above, the procedure for generating normal random numbers (Section 3; steps 1 to 5; page 7 of the PDF). ↩

11. 石川達也、内田善彦, cited above, Figure 14 in Section 5 (1) (page 67). For the at-the-money case, where the strike price equals the current price index, the standard deviation of the price over 10,000 trials is 4.141 without adjustment and 0.769 with first- and second-order sampling together (a ratio of about 5.4). ↩

12. 中村一岳, cited above, Section 4 (1) (page 8). The paper calls the risk in its eighth item "correlation risk." The paper's subject is composite options, but the statement is about derivatives with several underlying assets in general. ↩

13. 中村一岳, cited above, Section 4 (1) (page 18). ↩

14. 中村一岳, cited above, Section 5 (3) "The effect of the estimate of the correlation coefficient on pricing" (page 29). ↩

15. 中村一岳, cited above, Section 6 (2) "Handling of the correlation coefficient" (page 31). ↩

16. Board of Governors of the Federal Reserve System, "Trading and Capital-Markets Activities Manual" (April 2003), Section 4345.1 "Equity Derivatives," "Hedging" (page 3). ↩