Average-strike Options – Single Asset

Overview

An Average-strike option is an option whose payoff is based on the difference between the spot price at expiration and an average strike price determined over the life of the option. These options can assure that the average price paid (or received) for an asset over a certain time period is not greater than the final price. Average-strike options are said to be path dependent options because their final value (payoff) depends not only on the spot price at expiry, but on points in time prior to the expiration. In general (but not always), average-strike 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.

Most average-strike price options have European exercise, use the 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.

Formulas & Technical Details

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 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.

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

The Average-strike payoff is defined as:

where:

is 1 for a call and –1 for a put

is the spot price at expiry

is the average strike price of the underlying for the sampling dates

FINCAD Functions

The FINCAD library contains two functions for single asset Average-strike options:

aaAver_strk_MC (price_u, d_v, d_exp, d_aver, average, vlt, rate_ann, cost_hldg, freq, option_type, num_rnd_stat)

aaAver_strk_fs_MC (price_u, d_v, d_exp, d_aver, average, vlt, rate_ann, sam_seq, freq, option_type, num_rnd_stat)

These functions return, by Monte Carlo simulation, the fair value and delta of a European style arithmetic average-strike option. A statistical measure of the accuracy of the approximations is also provided. For periodic sampling dates (annual, semi-annual, quarterly, etc), use aaAver_strk_MC() and for non-standard sampling dates, use aaAver_strk_fs_MC().

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

aaGeo_Aver_strk_fs (price_u, d_v, d_exp, d_aver, average, vlt, rate_ann, cost_hldg, sam_frequ, option_type, stat)

These functions return the fair value and delta of a European style geometric average-strike option with either periodic or user defined sampling dates. A statistical measure of the accuracy of the approximations is also provided. For periodic sampling points (annual, semi-annual, quarterly, etc), use aaGeo_Aver_strk() and for non-standard sampling dates use aaGeo_Aver_strk_fs().

Example

Calculate the fair value of an Average-strike call option with a spot price of 120, an average price to-date of 80, and a volatility of 25%. Assume further that the value date is December 1, 1999, the expiry is June 1, 2000 and the quarterly averaging started on May 1, 1999. Given 1000 random trials and a risk-free rate and holding cost of 5% each, the option yields a fair value of 19.97503322 and an accuracy of 0.15731503.

We can interpret these results as follows: Given a 95% confidence interval, one can expect the option to be worth $19.97503322 ± $0.15731503.

Note: It is important that the confidence interval for the accuracy output is set to 95%. The second output table, which extends to return the delta of the option, will effectively double the calculation time. As a result, users who do not need to calculate the delta should always opt to have the first table returned.

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

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

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

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

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

Disclaimer

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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.