Bermudan and American Style Basket Options

Overview

Basket options are options on a basket of assets. The assets can be stocks, commodities, indices and other financial securities. A commonly traded basket option is a vanilla call/put option on a linear combination of assets. To clarify, suppose are the prices of stocks at time and are constants.

Set:

A vanilla basket option on this stock basket is simply a vanilla option on . In more detail, on an exercise date , the payoff of the option is:

where

is an exercise price and

=1 for a call and –1 for a put.

Such a vanilla basket option is also known as a portfolio option. Note that if , and , the basket option is simply a spread option.

There are many other types of basket options. For example, corresponding to each asset, there can be an exercise price and a vanilla call or put option and the payoff of a basket option can be the maximum of the payoffs of all these single asset options. Basket options can also be path dependent. The popular ones are the Asian basket options, average strike basket options and lookback basket options. Basket options can be quantos. For such baskets, the underlying assets can be assets of different currencies. The value of a basket is treated, without any currency translation, as the value in units of a certain currency, which may be different from the currencies in which the assets are priced.

For European-style basket options, FINCAD provides several sets of functions to value various types of basket options. See the Basket Options FINCAD Math Reference document for details. For Bermudan and American style basket options, FINCAD provides functions to value vanilla and Asian basket (portfolio) options. Details are given below.

Formulas & Technical Details

There are many methods for valuing Bermudan and American style options. The most popular ones are the Rubinstein binomial tree method and its extended version, the Hull-White trinomial tree method. Unfortunately, both methods and other lattice based numerical methods have their limitations. Options that involve multiple factors cannot be valued with these methods, due to the exponential growth of the number of nodes or grids that are used to approximate option values. A lot of effort in the research community of financial engineering has been made to find appropriate methods to value options involving multiple factors. Among the methods that have been researched so far, the most practically useful and popular one is the Monte-Carlo method introduced by Professors Longstaff and Schwartz.

Monte Carlo simulation methods have been widely used in financial instrument valuation. However, their use in option valuation has been mostly restricted to the European style. To value a European style option with Monte Carlo simulation, paths are generated first and intrinsic option values are calculated path by path at the option’s maturity date. The option value is then simply the discounted average of the intrinsic values of the paths. Such a simple method is not transferable to a Bermudan or American option. Valuation of such an option requires dynamically determining whether it is optimal to exercise at an exercise date. This cannot be done on a single path. To make an optimal decision, all the paths of the underlyings must be taken into consideration to determine the continuation value of an option and then determine if it is optimal to exercise.

In the Longstaff and Schwartz Monte Carlo method (LSMC), joint paths are first generated at all exercise time points for the assets in the basket. Then one starts from the last exercise date and goes backward step by step to determine the current intrinsic value and the continuation value and thus determining if it is optimal to exercise, given that it is not exercised at the prior time points. For the last exercise date, the optimal exercise value for each path is simply the option’s intrinsic value at the date. At other exercise time points, the intrinsic value is also calculated for each joint path. The key is to find the continuation value at an exercise date. According to Longstaff and Schwartz, this can be calculated with the relevant values, denoted values, of all joint paths by regressing these values against some properly chosen variables, denoted variables. The values are determined as follows: if a joint path has a future optimal exercise value, the value is calculated by discounting the optimal exercise value to the current exercise time point; otherwise, the value is simply the intrinsic value at the current exercise time point. The selection of the variables depends on the model used and also on the option definition as well. For example, for a vanilla call/put option on assets modeled with the Black-Scholes lognormal model, the spot asset prices can be used as the variables. Another freedom in LSMC is the selection of the regression function, also known as a basis function. Longstaff and Schwartz suggest second order polynomials or second order Laguerre polynomials.

After the regression calculation, the continuation value of the option, i.e., the estimated value, is determined for each joint path. By comparing this continuation value with the intrinsic value, one can decide path by path if it is optimal to exercise. If it is, mark this time point as an optimal exercise point and at the same time delete the future optimal mark if there is one. When these steps are finished, calculate the discounted intrinsic value at each optimal exercise time point for each joint path. If there is no optimal exercise time point for a particular path, the option value of the path is 0. The average of the discounted optimal exercise values is then the value of the option. For further details of LSMC, the reader is referred to the paper of Professors Longstaff and Schwartz [1].

FINCAD Functions

aaBasket_am_MC (ast_info, ex, corr_matrix, d_v, d_exp, d_exer_list, r_or_df_crv, intrp, option_type, option_style, time_steps, MC_para, stat):

Calculates the fair value and risk statistics of an American or Bermudan style option on a portfolio of assets.

aaBasket_mstrk_am_MC (ast_info, corr_matrix, d_v, d_exp, tbl_ex_d_p, r_or_df_crv, intrp, option_type, option_style, time_steps, MC_para, stat):

Calculates the fair value and risk statistics of an American or Bermudan style option on a portfolio of assets. The exercise price can be time-dependent.

aaAsian_bskt_am_MC (ast_info, ex, corr_matrix, d_v, d_exp, d_aver, d_exer_list, r_or_df_crv, intrp, option_type, option_style, sam_freq, hl, time_steps, MC_para, stat):

Calculates the fair value and risk statistics of an American or Bermudan style Asian option on a portfolio of assets. The sampling dates are periodic.

aaAsian_bskt_fs_mstrk_am_MC (ast_info, corr_matrix, d_v, d_exp, d_aver, tbl_ex_d_p, r_or_df_crv, intrp, option_type, option_style, sam_seq, time_steps, MC_para, stat):

Calculates the fair value and risk statistics of an American or Bermudan style Asian option on a portfolio of assets. The sampling dates are user-defined and strike prices can be time-dependent.

aaAsian_bskt_mstrk_am_MC (ast_info, corr_matrix, d_v, d_exp, d_aver, tbl_ex_d_p, r_or_df_crv, intrp, option_type, option_style, sam_freq, hl, time_steps, MC_para, stat):

Calculates the fair value and risk statistics of an American or Bermudan style Asian option on a portfolio of assets. The sampling dates are periodic and the exercise price can be time dependent.

aaAsian_fs_am_MC (price_u, ex, d_v, d_exp, d_aver, average, vlt, d_exer_list, r_or_df_crv, intrp, cost_hldg, option_type, option_style, sam_seq, time_steps, MC_para, stat):

Calculates the fair value and risk statistics of an American or Bermudan style Asian option with free-style sampling dates.

aaAsian_am_MC (price_u, ex, d_v, d_exp, d_aver, average, vlt, d_exer_list, r_or_df_crv, intrp, cost_hldg, option_type, option_style, sam_freq, hl, time_steps, MC_para, stat):

Calculates the fair value and risk statistics of an American or Bermudan style Asian option with periodic sampling dates.

Description of Inputs

Input Argument	Description
price_u	underlying price
ex	exercise price
d_v	value (settlement) date
d_exp	expiry date
d_aver	date when averaging starts
average	average price
vlt	volatility
d_exer_list	exercise dates table. For a Bermudan option this is the exercise date table; for an American option many exercise time points will be inserted into this table. See the parameter time_steps.
r_or_df_crv	rate or discount factor curve. It can be a single value of risk-free rate or a risk-free discount factor curve. See note 298 in the Function References for more details.
intrp	interpolation method
cost_hldg	holding cost - annual compounding
option_type	option type
sam_freq	sampling frequency, a switch. It has 16 values. The commonly used frequency switch values are the first 15. For the last value, the time interval from the averaging start date to the option expiry date is divided equally by the number of time steps that is provided by the parameter time_steps. The end points of these intervals are exercise time points.
hl	Holidays
time_steps	time steps. For an American style option it is the total number of exercise time periods to be used to approximate an option value. For an Asian option with equally spaced exercise time points it is the total number of exercise time periods. For a Bermudan style option it is not used.
MC_para	Monte Carlo simulation parameters. It can be a one- or two-entry array. The first is the number of random trials. If it is an integer, the function uses a random seed in simulation, otherwise a fixed seed is used. The second is the selection of basis functions where 1 = polynomial, 2 = Laguerre polynomial. The default value of the second entry is 1.
stat	statistics. Any subset of {1,2,…,12}. See the section Description of Outputs for details.
ast_info	underlying basket data table. A five column table (price, number of units or weight, holding cost, volatility, average price up to and including d_v).
corr_matrix	correlation matrix of the underlying assets. Note that a correlation matrix must be nonnegative definite.
sam_seq	sampling date sequence for a function with a free-style sampling sequence.
tbl_ex_d_p	strike price table for a function with a time-dependent strike price. It is a two-column table of dates and strike prices. The date must be increasing. For an American option (option_style = 1) or an option exercised on sampling dates only (option_style = 3), the exercise price at an exercise date is the price in the table at the closest future date or the closest prior date if no future date exists. For a Bermudan option (option_style = 2), the table defines the exercise dates and prices. In this case, the expiry date (d_exp) must be an exercise date.

Description of Outputs

Output Statistic	Description
1	fair value
2	delta
3	gamma
4	theta
5	vega
6	rho of rate
7	rho of holding cost
8	accuracy. This is the accuracy value corresponding to a 95% confidence interval. For example, if an option value is 3.5 and the accuracy is 0.01 then the 95% confidence interval is (3.5 - 0.1, 3.5 + 0.1) = (3.4, 3.6).
9	value of European exercise
10	value of early exercise. This is simply the difference of the American or Bermudan exercise value and the European exercise value
11	probability of early exercise
12	probability of exercise

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

Example

Consider an American Asian basket call option on three stocks , and . The option expires on Dec 1, 2006. It can be exercised on semi-annual sampling dates only. The sampling starting date is Dec. 1, 2004. The spot prices of the stocks are 90, 70 and 80, with volatilities 0.2, 0.3 and 0.28, respectively. The stocks don’t pay dividends. The weights of the stocks in the basket are 0.35 and 0.4 and 0.25, respectively. Today’s date is Dec 1, 2005 and the historical average prices of the stocks in the past three sampling dates are 91, 68 and 82, respectively. Assemble the stock data into a table as follows:

price	weight	dividend yield	volatility	average price
90	0.35	0	0.2	91
70	0.4	0	0.3	68
80	0.25	0	0.28	82

Suppose that the instant correlation matrix of the stocks is:

1	0.2	0.65
0.2	1	0.75
0.65	0.75	1

and the risk-free rate is 0.05. Call FINCAD function aaAsian_bskt_am_MC with a number of random trials 10000.5 to get the following results:

aaAsian_bskt_am_MC

Argument	Description	Example Data	Switch
ast_info	underlying basket data table	table given above
ex	exercise price	75
corr_matrix	correlation matrix of the underlying assets.	correlation matrix given above
d_v	value (settlement) date	1-Dec-2005
d_exp	expiry date	1-Dec-2006
d_aver	date when averaging starts	1-Dec-2004
d_exer_list	exercise dates table	0
r_or_df_crv	rate or discount factor curve	0.05
intrp	interpolation method	1	linear
option_type	option type	1	call
option_style	option style	3	exercise on sampling points
sam_freq	sampling frequency	2	semi-annual
hl	holidays	0
time_step	time steps	0
MC_para	Monte Carlo simulation parameters	10000.5
stat	stat list	1, 2…12

Results

Statistic	Description	Value
1	fair value	5.892968
2	delta	0.114046
3	gamma	-0.00417
4	theta	-0.00491
5	vega	0.032108
6	rho of rate	0.119575
7	rho of holding cost	-0.07146
8	accuracy	0.166405
9	value of European exercise	5.720506
10	value of early exercise	0.172461
11	probability of early exercise	0.3136
12	probability of exercise	0.9652

Related Functions

Many functions are also available in FINCAD products that can be used to value European-style basket options of different types. See the Basket Options FINCAD Math Reference document for details.

References

[1] Longstaff, F. and Schwartz, E. (2001), ‘Valuing American options by simulation: a simple least-squares approach’, The Review of Financial Studies, 14 No.1, 113 – 147.

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