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

Let_{}be a portfolio of assets where:

_{} and _{} is the number of units_{
}of _{} in the portfolio.

Let_{}equal 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.

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

This function returns, by ** 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 ** 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 ** 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 ** 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 ** 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 ** 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.

This example illustrates that buying a basket option is
cheaper than buying options on the individual components. Suppose the valuation date is

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

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

[2]

Disclaimer

With respect to this document,
FinancialCAD Corporation (“FINCAD”) makes no warranty either express or
implied, including, but not limited to, any implied warranty of merchantability
or fitness for a particular purpose. In no event shall FINCAD be liable to
anyone for special, collateral, incidental, or consequential damages in
connection with or arising out of the use of this document or the information
contained in it. This document should not be relied on as a substitute for your
own independent research or the advice of your professional financial,
accounting or other advisors.

This information is subject to change
without notice. FINCAD assumes no responsibility for any errors in this
document or their consequences and reserves the right to make changes to this
document without notice.

Copyright

Copyright © FinancialCAD Corporation 2008.
All rights reserved.