Double Average Options

Overview

Double Average Options are options that combine the features of an average strike option and an Asian option. Used predominately in the currency markets by multinationals, these options are designed to match currency risks more closely as the option creates an average strike and average price based on predefined sampling periods. This is in contrast to traditional plain vanilla options that have a strike based solely on the day the option was executed and a price based solely on the current spot of the underlying.

A typical example of a double average option is a ‘Daro’, short for Double Average Rate Option. Because of the averaging feature, the strike price of a Daro is not set at inception; it becomes the average price of the predefined sampling points. Combine this with an average rate feature, the value of a Daro can be described as the difference between the expected average strike price and the expected average rate. Daro’s tend to be attractive for several reasons: First, the Daro tends to be cheaper (although there may be exceptions) because the averaging features render the option value sensitive to the long-term volatility curve which is typically more stable than the spot curve. Secondly, from a hedging perspective, the Daro can assist to reduce income volatility: Assume a US multinational is motivated to ensure that currency movements will not make Q1-2000 results look poor when compared to Q1-1999, they could purchase a Daro with a strike price based on the average monthly rate for the three months of Q1-1999 which will settle based on the same three month rates in Q1-2000. The desired effect of this hedge is to render reported earnings on an “equal currency basis” regardless of currency fluctuations. Lastly, until the Q1-2000 rate is set, the Daro has no gamma risk (it has no sensitivity to spot rates).

Formulas & Technical Details

The payoff for a double average rate option is:

where:

is 1 for a call and –1 for a put

is the sum of the spot prices

is the number of sampling points for the spot prices

is the sum of the strike prices

is the number of sampling points for the strike prices

The payoff for a double average rate spread option is:

where:

is 1 for a call and –1 for a put

is the sum of the spot prices for asset 2

is the sum of the spot prices for asset 1

is the number of sampling points for the spot prices of asset 2 and asset 1

is the sum of the strike prices for asset 2

is the sum of the strike prices for asset 1

is the number of sampling points for the strike prices of asset 2 and asset 1

The payoff for a double average rate basket option is:

where:

is 1 for a call and –1 for a put

is the sum of the spot prices for the n-asset basket

is the number of sampling points for the spot prices of n assets

is the sum of the strike prices for n assets

is the number of sampling points for the strike prices of n assets

The payoff for a double average multi-asset option is:

where:

is 1 for a call and –1 for a put

is 1 for a call and –1 for a put of asset S1

is 1 for a call and –1 for a put of asset S2

is 1 for a call and –1 for a put of asset SR

is the sum of the spot prices for asset S1

is the sum of the spot prices for asset S2

is the sum of the spot prices for asset SR

is the sum of the strike prices for asset S1

is the sum of the strike prices for asset S2

is the sum of the strike prices for asset SR

is the number of sampling points for the spot prices of asset S1

is the number of sampling points for the spot prices of asset S2

is the number of sampling points for the spot prices of asset SR

FINCAD Functions

The FINCAD library contains eight functions based on Monte Carlo simulation for valuing Double Average Options:

aaDbl_Aver_MC (price_u, d_v, d_exp, d_s_aver_strk, d_e_aver_strk, d_aver, sam_freq, sam_freq, scale_strk, average_strk, average, vlt, rate_ann, cost_hldg, option_type, num_rnd, table_type)

aaDbl_Aver_fs_MC (price_u, d_v, average, strk, average, scale_strk, vlt, rate_ann, cost_hldg, sam_aver_strk_seq, sam_seq, option_type, num_rnd, table_type)

The double average rate functions return fair value, accuracy of fair value, and the delta of the underlying asset. For periodic sampling points (annual, semi-annual, quarterly, etc.) use aaDbl_Aver_MC and for free-style sampling use aaDbl_Aver_fs_MC.

aaDbl_Aver_basket_MC (ast_info, corr_matrix, d_v, d_exp, d_s_aver_strk, s_e_aver_strk, d_aver, sam_freq, sam_freq, scale_strk, rate_ann, Option_type, num_rnd, table_type)

aaDbl_Aver_basket_fs_MC (ast_info, corr_matrix, d_v, sam_aver_strk_seq, sam_seq, scale_strk, rate_ann, option_type, num_rnd, table_type)

The double average rate basket functions return fair value, accuracy of fair value, and the delta for an underlying portfolio of assets. For periodic sampling points (annual, semi-annual, quarterly, etc.) use aaDbl_Aver_basket_MC and for free-style sampling use aaDbl_Aver_basket_fs_MC.

aaDbl_Aver_spread_MC (price_u1, price_u2, average_strk, average, scale_strk, d_v, d_exp, d_s_aver_strk, d_e_aver_strk, d_aver, vlt1, vlt2, rate_ann, cost_hldg1, cost_hldg2, corr, sam_aver_strk_seq, sam_seq, option_type, num_rnd, table_type)

aaDbl_Aver_spread_fs_MC (price_u1, price_u2, d_v, average_strk, average, scale_strk, vlt1, vlt2, rate_ann, cost_hldg1, cost_hldg2, corr, sam_aver_strk_seq, sam_seq, option_type, num_rnd, table_type)

The double average rate spread functions return fair value, accuracy of fair value, and the delta for the relative performance of two assets. For periodic sampling points (annual, semi-annual, quarterly, etc) use aaDbl_Aver_spread_MC and for free-style sampling use aaDbl_Aver_spread_fs_MC.

aaMulti_dbl_Aver_MC (ast_info, corr_matrix, d_v, d_exp, d_s_aver_strk, d_e_aver_strk, d_aver, sam_freq, sam_freq, rate_ann, num_rnd, table_type)

aaMulti_dbl_Aver_fs_MC (ast_info, corr_matrix, d_v, sam_aver_strk_seq, sam_seq, rate_ann, num_rnd, table_type)

The double average rate multi-asset functions return fair value, accuracy of fair value, and the delta for the best of n assets. For periodic sampling points (annual, semi-annual, quarterly, etc) use aaMulti_dbl_Aver_MC and for free-style sampling use aaMulti_dbl_Aver_fs_MC.

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

Example

Calculate the fair value of a double average rate call option as of December 1, 1999 with an underlying price of 50, and an average strike to-date of 45. Both the strike and the spot are averaged monthly and the strike averaging starts November 1, 1999 and ends June 1, 2000. The average price starts averaging on September 1, 2000 and ends at expiry on December 1, 2000. The volatility of the option is 20%, and the holding cost and risk-free rate are both 5%. Using the function aaDbl_Aver_MC with 20000 random trials we have a fair value of 2.970994 with an accuracy of 0.249254.

aaDbl_Aver_MC

Argument	Description	Example Data	Switch
price_u	underlying price	50
d_v	value (settlement) date	1-Dec-1999
d_exp	expiry date	1-Dec-2000
d_s_aver_strk	date when strike averaging starts	1-Nov-1999
d_e_aver_strk	date when strike averaging ends	1-Jun-2000
d_aver	date when averaging starts	1-Sep-2000
sam-freq—strk	sampling frequency – strike	4	monthly
sam_freq	sampling frequency	4	monthly
scale_strk	scale factor of the strike	1
average_strk	strike price average of underlying up to value date	45
average	average price	0
vlt	volatility	20%
rate_ann	rate – annual compounding	5%
cost_hldg	holding cost – annual compounding	5%
option_type	option type	1	call
num_rnd	number of random trials	20000
table_type	output table type	1	fair value, accuracy

Results

This means that given 95% confidence, one can expect the value to be within $2.970994 ± $0.249254.

.Note that the confidence interval for the accuracy output is set to 95%.

References

[1] Falloon, William, (October 1999), ‘Forex managers go for Daros option’, Risk, Volume 12/No. 10.

Disclaimer

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