Basket Options

Overview

A Basket option is an option whose payoff depends on the value of a portfolio (or basket) of assets.  Basket options are growing in popularity as a means of hedging the risk of a portfolio.  Basket options are attractive because of their cost; an option on a basket is cheaper than buying options on the individual components and because of their payoff profile, as a basket option more closely replicates the changes in a portfolio’s value than any combination of options on the underlying assets.

Each underlying asset in the portfolio is assumed to satisfy the Black-Scholes hypotheses.  In particular, each underlying is assumed to follow a geometric (lognormal) diffusion process with a constant volatility.  It is also assumed that the correlation (or covariance) of each asset to all other assets is constant.  Hence required information to value a basket option is the volatility of each asset as well as the correlation between each asset (a correlation matrix).  The difficulty with basket options stems from the fact that the sum of lognormal random variables is not lognormal.  Thus, in general, there are no simple analytical formulas for basket options.  In general, basket options are solved by Monte Carlo simulation.

Formulas & Technical Details

Letbe a portfolio of assets where:

 and  is the number of units of  in the portfolio.

 

Letequal 1 for a call and –1 for a put, let K be a constant (strike price) and let  be the value of the portfolio on the expiry date of the option.  The payoff profiles for the various options are:

 

Asian option:                             where  is the arithmetic average of the sampling points.

 

Average-strike option:               where is the arithmetic average of the sampling points.

 

Lookback call option:               where is the minimum price of the underlying over the sampling points.

 

Lookback put option:                where is the maximum price of the underlying over the sampling points.

 

Double average rate option:     where  is the arithmetic average of the underlying spot prices of the sample points in the first sampling period and  is the arithmetic average of the underlying in the second sampling period. 

 

For more information on individual options, refer to the corresponding FINCAD Math Reference document.

FINCAD Functions

aaBasket_MC (ast_info, ex, corr_matrix, d_v, d_exp, rate_ann, option_type, num_rnd, table_type)

This function returns, by Monte Carlo simulation, fair value and deltas (one for each underlying) and a statistical measure of the accuracy of the approximations for a European style call or put option on any linear combination of any number of assets.

 

aaAsian_basket_MC (ast_info, ex, corr_matrix, d_v, d_exp, d_aver, rate_ann, sam_freq, option_type, num_rnd, table_type)

aaAsian_basket_fs_MC (ast_info, ex, corr_matrix, d_v, d_exp, d_aver, rate_ann, sam_freq, option_type, num_rnd, table_type)

These functions return, by Monte Carlo simulation, fair value and deltas (one for each underlying) and a statistical measure of the accuracy of the approximations for an Asian option on any linear combination of any number of assets.  The function aaAsian_basket_MC() assumes the sampling points are periodic (annual, semi-annual, quarterly, etc.) while the function aaAsian_basket_fs_MC() allows the user to define the sampling dates.

 

aaAver_strk_basket_MC (ast_info, corr_matrix, d_v, d_exp, d_aver, rate_ann, sam_freq, option_type, num_rnd, table_type)

aaAver_strk_basket_fs_MC (ast_info, corr_matrix, d_v, d_exp, d_aver, rate_ann, sam_freq, option_type, num_rnd, table_type)

These functions return, by Monte Carlo simulation, fair value and deltas (one for each underlying) and a statistical measure of the accuracy of the approximations for an Average-Strike option on any linear combination of any number of assets.  The function aaAver_strk_basket_MC() assumes the sampling points are periodic (annual, semi-annual, quarterly, etc.) while the function aaAver_strk_basket_fs_MC() allows the user to define the sampling dates.

 

aaDbl_aver_basket_MC (ast_info, corr_matrix, d_v, d_exp, d_s_aver_strk, d_e_aver_strk, d_aver, sam_freq_strk, sam_freq, scale_strk, rate_ann, option_type, num_rnd, table_type)

aaDbl_aver_basket_fs_MC (ast_info, corr_matrix, d_v, d_exp, d_s_aver_strk, d_e_aver_strk, d_aver, sam_freq_strk, sam_freq, scale_strk, rate_ann, option_type, num_rnd, table_type)

These functions return, by Monte Carlo simulation, fair value and deltas (one for each underlying) and a statistical measure of the accuracy of the approximations for a Double Average Rate option on any linear combination of any number of assets.  The function aaDbl_aver_basket_MC() assumes the sampling points are periodic (annual, semi-annual, quarterly, etc.) while the function aaDbl_aver_basket_fs_MC() allows the user to define the sampling dates.

 

aaLook_basket_MC (ast_info, min_max, corr_matrix, d_v, d_exp, d_sam_start, sam_freq, rate_ann, option_type, num_rnd, table_type)

aaLook_basket_fs_MC (ast_info, min_max, corr_matrix, d_v, d_exp, d_sam_start, sam_freq, rate_ann, option_type, num_rnd, table_type)

These functions return, by Monte Carlo simulation, fair value and deltas (one for each underlying) and a statistical measure of the accuracy of the approximations for a Lookback option on any linear combination of any number of assets.  The function aaLook_basket_MC() assumes the lookback dates are periodic (annual, semi-annual, quarterly, etc.) while the function aaDbl_aver_basket_fs_MC() allows the user to define the lookback dates.

 

aaQuanto_asian_basket_MC(asian_basket_ast_info_b, ex_for, curr_tbl, correlation_matrix, d_v, d_exp, d_aver, sam_freq, option_type, num_rnd, table_type)

aaQuanto_asian_basket_fs_MC(asian_basket_ast_info_b, ex_for, curr_tbl, correlation_matrix, d_v, d_exp, d_aver, sam_seq, option_type ,num_rnd, table_type)

These functions return, by Monte Carlo simulation, fair value and deltas (one for each underlying) and a statistical measure of the accuracy of the approximations for a Quanto version of Asian Basket Option on any linear combination of any number of assets.  The function aaQuanto_asian_basket_MC() assumes the sampling dates are periodic (annual, semi-annual, quarterly, etc.) while the function aaQuanto_asian_basket_fs_MC() allows the user to define the sampling dates. 

 

For details about the calculation of Greeks, see the Greeks of Options on non-Interest Rate Instruments FINCAD Math Reference document.

For more information on quanto options, please refer to the Quanto Forwards and Quanto Options FINCAD Math Reference document.

Example

Basket Option

This example illustrates that buying a basket option is cheaper than buying options on the individual components.  Suppose the valuation date is December 1, 1999 and suppose the expiry date is June 1, 1999.  Consider two equities with the following characteristics: the spot prices are 50 and 60 respectively, the volatilities are 25% and 30% respectively, the dividend yield is 4% for both, while the correlation between the two is 0.48.  Further, assume the risk-free rate, applicable to both, is 4%.  We find that the premium for an at-the-money call option on the basket made up of these two equities is 6.8889 (using the function aaBasket_MC), while the combined premium of two individual at-the-money call options (using aaBSG) is 8.4340. 

 

Quanto Asian Basket Option

For more examples, please refer to FINCAD XL Function Finder, select the function you want, and click Paste Example.

References

[1]          Haug, E.G., (1998), The complete guide to option pricing formulas, McGraw-Hill.

[2]          Hull, J., (1993), Options Futures and Other Derivative Securities, Toronto, Prentice Hall Inc.

 

 

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