Asian Options

Overview

The payoff for an average price (Asian) option is the difference between the strike price and the average price of the underlying instrument over a certain time period. In essence, these options allow the buyer to purchase (or sell) the underlying asset at the average price instead of the spot price. These options have become prevalent in commodity and foreign exchange markets where a party may have regular and ongoing transactions in a particular instrument and a desire to hedge itself against price fluctuations. Also, average price options are used in situations where the purchaser wants to cover many spot transactions using only one hedging instrument or in situations where it is prudent to reduce the dependence of an option on the spot price of the underlying on only one date. In general (but not always), average price options are less expensive than their European counterparts. Intuitively this makes sense, as the volatility of the average price is less than the volatility of the spot price.

As an example, consider a regular consumer of crude oil whose price of supply is not fixed, but is set weekly from a particular benchmark. He/she is concerned that there may be a spike in oil prices over the next few months and wants to hedge himself/herself using options. He/she requires that the payoff of the hedge reflects the weekly purchases made over a specified time period. An average price option can be tailored to meet these requirements by including the use of weekly price settings over the applicable period. This option captures changes in the commodity over the averaging period and is significantly less expensive than the alternative of purchasing a basket of European options each maturing on a given fixing date.

Most average price options use an arithmetic average and sample at discrete regular time intervals such as daily, weekly or monthly closing prices. However, it is possible that transactions are made with irregular sampling periods. FINCAD provides functions for valuing European, Bermudan and American style Asian options in all of these situations.

Formulas & Technical Details

The Asian option payoff function is:

where:

is the average at the expiry date

is the strike

is 1 for a call, -1 for a put

Suppose an underlying satisfies the same assumptions as those in the Black-Scholes model. In particular, the underlying is assumed to follow a geometric diffusion process with a constant volatility. Unfortunately, the value of an arithmetic average Asian option cannot be written as a compact, easily solvable, analytical formula like the value of a call or put option in the Black -Scholes model. The value of the option must be solved for using iterative techniques (like Monte Carlo methods), or analytically using approximating expansions (see [3], [5]) or by specialized tree based methods.

The function aaAsian uses an analytical approximation for the value of the option based on an Edgeworth expansion (see [3]) of the probability distribution of the undetermined component of the average. This is a fast and reliable technique and can be considered accurate for volatilities up to 30%. As the volatility increases, the approximation becomes less accurate and users may prefer to use other valuation techniques like Monte Carlo (e.g. aaAsian_MC).

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

FINCAD Functions

aaAsian (price_u, ex, average, freq, d_exp, d_v, d_aver, vlt, rate_ann, cost_hldg, option_type, stat)

This function returns fair value and a full slate of risk statistics (delta, gamma, vega and rho) for a European style Asian option. It uses an approximating expansion (see [3]) to value the option and hence is very computationally efficient and in most situations (European style with regular sampling periods), it provides very good results. However, as the volatility increases, the quality of the approximation deteriorates. As well, the approximation is only valid for regular sampling periods.

aaAsian_MC (price_u, ex, d_v, d_exp, d_aver, average, vlt, rate_ann, cost_hldg, option_type, freq, num_rnd, stat)

aaAsian_fs_MC (price_u, ex, d_v, d_exp, d_aver, average, vlt, rate_ann, cost_hldg, option_type, sam_seq, num_rnd, stat)

These functions return, by Monte Carlo simulation, the fair value and delta for a European style average price option with either periodic (aaAsian_MC) or user-defined sampling points (aaAsian_fs_MC). A statistical measure of the accuracy of the approximations is also provided.

aaGeo_Asian (price_u, ex, d_v, d_exp, d_aver, average, vlt, rate_ann, cost_hldg, sam_freq, option_type, stat)

aaGeo_Asian_fs (price_u, ex, d_v, d_exp, d_aver, average, vlt, rate_ann, cost_hldg, sam_freq, option_type, stat)

These functions return the fair value and delta for a European style geometric average price option with periodic (aaGeo_Asian) or user-defined sampling points (aaGeo_Asian_fs). A statistical measure of the accuracy of the approximations is also provided.

Geometric vs. arithmetic price averaging - Consider a stock whose price has been sampled at several dates. The Arithmetic average is the sum of the stock values divided by the number of sampling points. The Geometric average is the nth root of the product of the n sample points. Geometric Asian options, though not used in practice, are useful because of they offer nice analytic properties and they often form the basis of a good initial guess for the price of an arithmetic Asian option. This property is used, in FINCAD functions, to improve the Monte Carlo convergence rate of arithmetic Asian options.

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

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)

These functions return, by Monte Carlo simulation, the fair value and a slate of risk statistics for a Bermudan or an American style average price option with either periodic (aaAsian_am_MC) or user-defined sampling points (aaAsian_fs_am_MC). A statistical measure of the accuracy of the approximations is also provided. The user has control over the interplay between the exercise and sampling dates and can, for instance, vary the accuracy with which American options are valued by varying the number of time steps (at which exercise is possible).

There are also functions that value European style options on the average spread, options on the average of a basket and options on the best of a basket of Asian options on multiple assets.

There are several other types of Asian option functions available.

For European style options:

Spread Optionsm_Spread: aaAsian_spread_MC(), aaAsian_spread_fs_MC()

Basket Optionsm_Basket: aaAsian_basket_MC(), aaAsian_basket_fs_MC()

Quantom_Quanto: aaQuanto_Asian(), aaQuanto_Geo_Asian(), aaQuanto_Geo_Asian_fs()

Quanto Basketm_Basket: aaQuanto_asian_basket_MC(), aaQuanto_asian_basket_fs_MC()

Multi-Assetm_MultiAss: aaMulti_Asian_MC(), aaMulti_Asian_fs_MC()

FX Specificm_fxopt: aaFX_Asian()

For Bermudan and American style options:

aaAsian_bskt_am_MC()

aaAsian_bskt_fs_mstrk_am_MC()

aaAsian_bskt_mstrk_am_MC()

These functions can value basket Asian options and can also handle time-dependent strike prices.

Examples

Calculate the fair value of an average price call option on an underlying with spot price of $18.00 and strike price of $18.27. Today’s date is February 1, 1996 and the option expires on
April 1, 1996. The payoff at expiration is based on the average daily closing prices starting January 2, 1996. Currently, the average is $18.27. The riskless rate is 5.0 %, while the underlying provides no yield. Annual volatility is 20%.

aaAsian

Argument	Description	Example Data	Switch
price_u	spot price of underlying instrument	18
ex	exercise (strike) price	17.5
average	If averaging has already begun, enter the current value of the average. If averaging starts in the future, enter zero.	18.27
freq	sampling frequency of the average	6	daily
d_exp	option expiry date (and date when averaging ends)	1-Apr-1996
d_v	value date	1-Feb-1996
d_aver	date when averaging (sampling) begins.	1-Jan-1996
vlt	annualized volatility of the spot price of the underlying instrument	0.2
rate_ann	risk free discount rate for the period from the value date to the expiry date	0.05
cost_hldg	cost of holding the option rather than the underlying instrument	0
option_type	call or put	1	call
stat	stat list	1, 2

Results

Statistic	Description	Value
1	fair value of the option.	0.66831479
2	delta: sensitivity of the option price to small changes in the spot price.	0.57532008

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.

[3] Kemna and Vorst, A. (March 1990) ‘A Pricing Method for Options Based on Average Asset Values’, Journal of Banking and Finance, 14.

[4] Levy, E. and Turnbull, S. (1992) ‘Average Intelligence, From Black Scholes to Black Holes’, London England, Risk Magazine Ltd.

[5] Levy, E., (1992) ‘Pricing European Average Rate Currency Options’, Journal of International Money and Finance.

[6] Rubinstein, M. (1991) ‘Asian Options’, University of California at Berkeley.

[7] Turnbull S. and Wakeman, L.M., (September 1991) ‘A Quick Algorithm for Pricing European Average Options’, Journal of Financial and Quantitative Analysis, 26.

[8] Wilmott P., (1998) Derivatives, NY, John Wiley & Sons, Inc.

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.