Asian Options

Iris Mack, MBA/PhDOptions Trading Deskx3-6711

Outline of Asian Options Module

Introduction

What's in a Name? - Why the name "Asian Options"?

Classification of Asian Options

Payoffs of Asian Options

Pricing and Hedging of Asian Options

Mini Case Study #1: Asian average price call option

Mini Case Study #2: Asian average price put option

Mini Case Study #3: Asian collar

Asian Strip Example

Cinergy Cal '02 Example

Pricing Models

References

Introduction

"Exotic" options are becoming more popular as a means of managing exposure to energy prices. Some factors contributing to this increased popularity are:

* High price volatility of many energy commodities, which results in uncertainties in future costs or revenues;* Deregulation of the US energy markets, which has affected both producers and consumers of energy;* Deregulation is resulting in intense competition and a sensitivity to price fluctuations;* Some exotic options - such as Asian options - are attractive because they tend to cost less than some vanilla options.

One of the main reasons that exotic options have been accepted by the energy industry is that options are in embedded in many energy contracts . For example, many contracts in the energy industry contained averaging provisions based on the monthly or weekly averages of the commodity prices. Hence the risk exposure of most producers and end-users is to an average price level over periods of time, and this facilitated the acceptance of Asian options.

What's in a name? - Why the name "Asian Options'?

During the 1980's two investment bankers - Mark Standish (a former Bankers Trust's VP in interest rate derivatives) and David Spaughton (of CSFP) - created the first commercially used option-pricing methodology to be based on the average price of crude oil over time.They realized that there was no compelling reason to have an average price option model until commodity derivatives came along. Whereas in financial markets you have explicit risk on a particular day or at the end of the year, a standard physical contract in crude is typically based on an average price for the month. They were in Tokyo at the time they developed this pricing methodology, so they called it the "Asian option." It was as simple as that!

One of the most common occurrences of Asian options is the component options in caps and floors. However, in general, the mainuse of Asian options is hedging an exposure to the average price over a period of time. For example, large buyers of electricity may be required to hedge their average fuel cost as the prices they charge customers are based on the average purchase prices. Asian options also fit therisk profile of energy producers who need to meet budget targets based on average prices. Such exotic options make it possible for dealersto cope with the historical volatility of the energy commodity markets.

Classification of Asian Options

Note: The Enron Options trading desk will only be offering average price options at this time.

Realizing the value in an option is dependent upon the form of option exercise that has been stipulated. The two most familiar types of vanilla option exercise are:

Asian (or average rate) options settle in cash based upon an average price. They are generally exercised automatically if they arein the money. The exchange-traded energy options on the NYMEX and IPE are of the American type, while most OTC energy options are of the Asian variety, because of the popularity of the averaging mechanism. A few OTC options are American or European style.

There are two basic styles of Asian options:(1) Average price options - also known as average rate or "fixed strike" Asian options(2) Average strike options - also known as "floating strike" Asian options

* American options - which can be exercised at any time up to the maturity date* European options - which can only be exercised on the maturity date.

Payoffs of Average Price Asian Options

Asian options are options whose final payoff is based in some way on the average level of an energy price (spot, forward, or future) during some or all of the life of the option. The following table details the payoff structures for Asian options:

Payoffs of the Average Price Asian Options

Name Payoff

Average Price Call Option max (average(P) - K, 0)

Average Price Put Option max (K - average(P), 0)

Please note that K denotes the strike priceaverage(P) = (P1 + P2 + … + Pm)/m denotes the average value of the forward prices calculated over a predetermined

averaging period.P1, P2, …, Pm denote the forward power prices at m points in time.

The averaging period may correspond to the entire life of the option, or it can be shorter. If an Asian option is traded when it'swithin its averaging period, pricing the option requires the average-to-date price of the underlying. For such options, averaging effectively starts from the buyer's point of view - prior to the purchase of the contract. Averaging is typically calculated using an arithmetic average.

Pricing and Hedging of Asian Options

The volatility of an Asian option is lower than that of the underlying prices used in the calculation of the average. Hence an Asian option at inception is similar to a European option with a lower volatility. Therefore an Asian option will be less expensive than the corresponding European options, since premiums increase with increasing volatility. In addition to the lower premiums, another advantage of Asian options is that their payoffs (as defined in the previous table) is less sensitive to any extreme market conditions that may prevail on the expiration day (due to random shocks or outright manipulation).

An important question asked by a potential client may be how to price such exotic options contracts and also, in the case of EPMI, how to hedge them. Asian options can't be priced using the Black-Scholes formula since an average of prices will not be lognormally distributed even though the individual components prices are. To price any type of exotic option, one should

* First attempt to replicate the exotic option with a package of vanilla options. If this is possible, then each component option should be priced individually, and the sum of all the long and short positions should give the desired exotic option price.

* If the replication approach doesn't work, in some cases one may find an analytical solution that's comparable to the Black-Scholes formula.

* If an analytical solution can't be found, one may be able to find an approximation method that gives "acceptable" pricing accuracy.* If none of the above works, then it's necessary to use some type of numerical method. The numerical methods used for option

valuation fall into 3 categories: Monte Carlo simulation methods; tree (binomial or multinomial) methods; and finite-difference or numerical integration methods.

Asian options are preferred products because they're easier to hedge. Such options with long averaging periods don't have the high gamma risk that ATM European options may have near expiry. After the Asian option enters its averaging period and the average begins to "set", the gamma risk of the option decreases and approaches zero near the end of averaging for options with reasonably long averaging periods. However, if the averaging period is only 2 or 3 days, the gamma may still be sizeable at expiration.

which contain models for the pricing of various types of exotic options - including Asian OptionsThe options trading desk has at its disposal two software libraries (Exotica and Financial Engineering Associates (FEA), Inc.)

Mini Case Study #1: Asian average price call option

A load-serving entity (LSE) pays the day ahead or real time price and would like to limit its maximum cost. An Asian option is purchased as a hedge covering the remainder of the year. At the end of each month, the strike is compared to the average settlement price (or Megawatt Daily index). Enron pays the purchaser any difference the average rate exceeds the strike price for the quantity covered by the call option.

Characteristics of this Asian average price call option:* Limits worst-case scenario and still maintains benefits if prices decline.* Protection is not for an individual spike on any given day, but the average over the selected period.* Do not have to exercise/notify a day ahead. Option is automatically exercised if it's ITM.* Cheaper than a daily option that would make a payment for each day the market is above the strike, rather than the average.* Price can be adjusted lower by increasing averaging period since the volatility of a longer-term average rate is less than the volatility of a shorter-term average of spot.

Mini Case Study #2: Asian average price put option

A generator receives the day ahead or real time prices, but would like to ensure at least a certain minimum revenue. An Asian optionis purchased as a hedge covering the remainder of the year. At the end of each month, the strike is compared to the average settlement prices (or Megawatt Daily index). Enron pays the purchaser any difference the strike price exceeds the average rate for the quantity covered by the put option.

Characteristics of this Asian average price put option:* Guarantees worst case scenario and still maintains upside potential.* Protection isn't for an individual spike down on any given day, but the average over the selected period.* Do not have to exercise/notify a day ahead. Option is automatically exercised if it's ITM.* Cheaper than a daily option that would make a payment for each day the market is below the strike, rather than the average.* Price can be adjusted lower by increasing averaging period since the volatility of a longer-term average rate is less than the volatility of a shorter-term average of spot.

Mini Case Study #3: Asian collar

may guarantee that its revenues will be within the collar band. For example, suppose a generator receives the day ahead or real time price, but would like to ensure a minimum revenue. An Asian collar could be executed to hedge a portion of the exposure. At the end of each selected average period, the strikes of the collar is compared to the average prices. Enron pays the generator any difference the strike exceeds the average rate for the quantity covered by the lower band, and the generator pays Enron any difference the average exceeds the higher strike.

Characteristics of this Asian average price put option:* May be structured as to have no up-front or low up-front premium since the purchased option is financed by the sale of the other option.* Best and worst case hedge prices are known upfront.* Option averaging may be customized, I.e., daily, monthly, or weighted.* Some spot moves downward within band may not be taken advantage of due to the average feature.* Why use an Asian collar? It provides price stability while allowing for some upside potential within a range customized by the generator.

An Asian collar is a regular collar** except that settlement is made against the average of any given period. A generator

** Recall the following: * A regular collar is a combination of a long position in a cap and a short position in a floor. * A cap provides price protection for the buyer above a predetermined level - the cap price - for a predetermined time period.* A floor guarantees the minimum price that will be paid or received at a predetermined level - the floor price.

Asian on Multiple UnderlyingsFunction: ASTRIP, ASTRIP2m

EffDt 2/10/1997INPUTS

Premium Fwd Price Ann.Vol Set Days Strike Ann.IntRt Correlation#ADDIN? 1-Jan-97 31-Jan-97 32.50 75.38% -0.110 -0.027 23 29 40 4.50% 0.25

1-Feb-97 28-Feb-97 32.50 68.17% -0.025 0.049 201-Mar-97 31-Mar-97 24.75 52.75% 0.052 0.134 211-Apr-97 30-Apr-97 21.50 45.21% 0.137 0.216 22

1-May-97 31-May-97 21.50 47.25% 0.219 0.301 221-Jun-97 30-Jun-97 24.40 52.51% 0.304 0.383 211-Jul-97 31-Jul-97 32.00 59.96% 0.386 0.468 23

1-Aug-97 31-Aug-97 32.00 57.65% 0.471 0.553 211-Sep-97 30-Sep-97 23.10 46.40% 0.556 0.635 221-Oct-97 31-Oct-97 21.30 38.48% 0.638 0.720 231-Nov-97 30-Nov-97 22.40 39.97% 0.723 0.802 201-Dec-97 31-Dec-97 26.35 44.99% 0.805 0.887 23

Here we find the premium and risk parameters for an Asian option where averaging is over multiple underlyings. The averaging periods needn't be continuous. For example, averaging could be over the last 3 days of NYMEX for 6 months. However, the averagingperiods must not overlap (I.e., the samples from the first underlying must all be determined before the price samples from the secondunderlying begin, and so on.), and the strip of underlyings must be supplied in chronological order in which they contribute to the averageprice.

The number of price samples taken from each underlying is explicitly specified by the user and, once defined, will never changeover the life of the option.

The input parameter "Set Days" tells the valuation routine the number of price points which have already been determined.The correlation to be specified by the user is taken to be the correlation between all pairs of strip underlyings.

Avg. Start Date

Avg. End Date

Time to Avg. Start

Time to Avg. End

# Settle Prices

Description:

OUTPUTSINPUTS 0 1 2 3 4

Month Price Delta Gamma Vega Rho32.5 0.887671 0 1 #ADDIN? #ADDIN? #ADDIN? #ADDIN? #ADDIN?

2 #ADDIN? #ADDIN? #ADDIN?3 #ADDIN? #ADDIN? #ADDIN?4 #ADDIN? #ADDIN? #ADDIN?5 #ADDIN? #ADDIN? #ADDIN?6 #ADDIN? #ADDIN? #ADDIN?7 #ADDIN? #ADDIN? #ADDIN?8 #ADDIN? #ADDIN? #ADDIN?9 #ADDIN? #ADDIN? #ADDIN?

10 #ADDIN? #ADDIN? #ADDIN?11 #ADDIN? #ADDIN? #ADDIN?12 #ADDIN? #ADDIN? #ADDIN?

xASTRIP2m#ADDIN?

Avg. Price to Date

Time to Expiry

Call/Put (1/0)

Asian on Multiple UnderlyingsCal '02 30 Strike for Cinergy

EffDt 9/24/2001INPUTS

Premium Fwd Price Ann.Vol#ADDIN? 1-Jan-02 31-Jan-02 33.47 75.00% 0.271 0.353 23

1-Feb-02 28-Feb-02 33.12 75.00% 0.356 0.430 201-Mar-02 31-Mar-02 32.02 70.00% 0.433 0.515 211-Apr-02 30-Apr-02 32.97 70.00% 0.517 0.597 22

1-May-02 31-May-02 37.00 70.00% 0.600 0.682 231-Jun-02 30-Jun-02 47.00 75.00% 0.684 0.764 201-Jul-02 31-Jul-02 60.75 90.00% 0.767 0.849 23

1-Aug-02 31-Aug-02 60.75 90.00% 0.851 0.934 221-Sep-02 30-Sep-02 32.55 50.00% 0.936 1.016 211-Oct-02 31-Oct-02 30.36 50.00% 1.018 1.101 231-Nov-02 30-Nov-02 30.56 50.00% 1.103 1.183 211-Dec-02 31-Dec-02 30.76 50.00% 1.185 1.268 22

Avg. Start Date

Avg. End Date

Time to Avg. Start

Time to Avg. End

# Settle Prices

OUTPUTSINPUTS 0

Set Days Strike Ann.IntRt Correlation Month Price0 30 4.50% 0.25 0 1.268493 1 1 #ADDIN?

23456789

101112

xASTRIP2m#ADDIN?

Avg. Price to Date

Time to Expiry

Call/Put (1/0)

OUTPUTS1 2 3 4

Delta Gamma Vega Rho#ADDIN? #ADDIN? #ADDIN? #ADDIN?#ADDIN? #ADDIN? #ADDIN?#ADDIN? #ADDIN? #ADDIN?#ADDIN? #ADDIN? #ADDIN?#ADDIN? #ADDIN? #ADDIN?#ADDIN? #ADDIN? #ADDIN?#ADDIN? #ADDIN? #ADDIN?#ADDIN? #ADDIN? #ADDIN?#ADDIN? #ADDIN? #ADDIN?#ADDIN? #ADDIN? #ADDIN?#ADDIN? #ADDIN? #ADDIN?#ADDIN? #ADDIN? #ADDIN?

Pricing Models

In the previous two worksheets we gave a detailed example of how to price an Asian strip. Other Asian pricing models are as follows:

spread between two average prices.)

Example worksheets:ASTRIP - O:\research\exotica\xll\xll_templates\asianstr.xls and M:\exotica\xll\xll_templates\asianstr.xlsASTRIP2m - O:\research\exotica\xll\xll_templates\astrip2m.xls and M:\exotica\xll\xll_templates\astrip2m.xlsASV, ASN - O:\research\exotica\xll\xll_templates\assian.xls and M:\exotica\xll\xll_templates\asian.xlsAGC - O:\research\exotica\xll\xll_templates\agc.xls and M:\exotica\xll\xll_templates\agc.xlsAsnSprd - O:\research\exotica\xll\xll_templates\asnsprd.xls and M:\exotica\xll\xll_templates\asnsprd.xlsAsnSprd2 - O:\research\exotica\xll\xll_templates\asnsprd2.xls and M:\exotica\xll\xll_templates\asnsprd2.xls

Example worksheets:asian.xls (Basics) - Examples of APO, STRIPAPO, ASO, and STRIPASO.spread.xls (Advanced) - Examples of SPREADAPO, SPREADASO.

EXOTICA:ASV, ASN, AGC - fast volatility approximation

AsnSprd, AsnSprd2 - spread option on Asian spreads (Finds the premium and risk parameters for an option on the

Financial Engineering Associates (FEA), Inc.:APO - Average-price optionsASO - Average-strike optionsSTRIPAPO, STRIPASO - Strip of Asian OptionsSPREADAPO, SPREADASO - spread options on Asians

References

Lacima Publications.

Journal of Banking and Finance 14, pp. 113-129.

(October). Pp. 474-491.

* Clewlow, L. and C. Strickland, 2000, "Energy Derivatives: Pricing and Risk Management,"

* Enron's Houston Research Group, Exotica Options Library

* Enron Power Marketing, Product Descriptions

* Financial Engineering Associates (FEA), Inc., 2001, User Guide

* Hull, J., 2000, "Options, Futures, and Other Derivatives," Fourth Edition, Prentice Hall.

* Kaminski, V., 1999, "Managing Energy Price Risk," Second Edition, Risk Publications.

* Keman, A.G. Z. and A. C. F. Vorst, 1990, "A Pricing Method for Options Based on Average Asset Values," Fianance,

* Levy, E., 1991, "Pricing European Average Rage Currency Options," Journal of International Money and Finance,

* Levy, E. and S. Turnbull, 1992, "Average Intelligence," Risk, (February)

* Turnbull, S. M. and L. M. Wakeman, 1991, "A Quick Algorithm for Pricing European Average Options," Journal of Financial and Quantitative Analysis, 26 (September), pp. 377-389.