Options on Stock Indices, Currencies and Futures

Transcription

1 Options on Stock Indices, Currencies and utures It turns out that options on stock indices, currencies and utures all have something in common. In each o these cases the holder o the option does not get the same thing that the holder o the underlying asset gets. his aects the value o the option. In this note we will cover the valuation implications o this or options on stock indices, currencies and utures. Stock Indices One o the main issues or valuing options on stock indices is dividends. While it might be the case that an individual stock might not pay dividends, or stock indices it would be very unusual i at least one o the stocks included in the index wasn t paying a dividend. As o December 23 there were options available on over 3 stock indices. Some o the more popular include Dow Jones Industrial Average (DJX, NASDAQ (NDX, Russell 2 (RU, and the S&P 5 (SPX. See Exhibit or some estimates o number o securities included in the each index, basis on which the index is calculated 2, the recent returns, volatilities, the percentage o dividend paying stocks and the average dividend yield. Note that the similarities among the large cap indices, Dow Jones and the S&P 5, and among the small cap indices, Russell and the Nasdaq. In each case the returns are similar as are the dividend yields. igure shows the relative 5-year perormance or each index. As we saw earlier 3 on options on individual stocks, we could account or the dividends by deining the Underlying Asset Value ( as the stock price less the present value o the dividends. his is relatively easy or options on individual stocks. However, when he CBOE oers options on a wide variety o dierent indices. See or a complete lisitng. 2 In their most general orm, indexes are simply an average value relative to some base. or example, we could construct an index with a base value o today and then rom that point orward have a value index relative to. All o the indexes listed above take this approach. All o them except or the Dow Jones Averages, which is a price weighted index, are value weighted indices. his means that the Dow Jones Averages represent a simple average o prices and the others are market value weighted averages. he major dierence between them is that the Dow Jones gives more weight to the high price stocks and the others give more weight to the stocks with the highest market values. 3 See Hull, J., Options, utures and Other Derivatives, 5 th edition, 22, Prentice Hall, Upper Saddle River, NJ, pages his note was prepared by Proessor Robert M. Conroy. Copyright 23 by the University o Virginia Darden School oundation, Charlottesville, VA. All rights reserved. o order copies, send an to dardencases@virginia.edu. No part o this publication may be reproduced, stored in a retrieval system, used in a spreadsheet, or transmitted in any orm or by any means electronic, mechanical, photocopying, recording, or otherwise without the permission o the Darden School oundation.

2 -2- the number o stocks in an index is large, this not very practical. or options on stock indices where the number o stocks in the index varies rom 3 to 2,, we almost always use a dividend yield approach to valuing options. Dividend Yield Approach to valuing options on Stock Indices We can use the Black-Scholes model to value European call options on stock indices. A airly typical call option on an index is the Chicago Board Option Exchange 4, CBOE, S&P5 index option. See Exhibit 2 or details on this option. Its payo is Payo [ Index $ exercise,] Max. It is a cash-settled option which means that there is no buying or selling o the underlying asset but rather the writer pays in cash to the holder upon exercise o the option at maturity. or example, suppose that the S&P5 index value 5 was,94 and we have a European call option with a maturity o 3 months, and an exercise value o,. he value o this European call option will depend on the exercise price (X, the time to maturity (3 months, 3-month risk-ree rate 6 (r.9%, volatility (σ.2 and the underlying asset value. As noted above, it is necessary to account or dividends in the underlying asset value. We do this by incorporating a dividend yield. In order to do this, we can use an approach similar to the one we used or the known dividend, i.e., deduct the present value o the dividend rom the stock price or index value 7. Here let S stand o the current value o the index, R is the expected return on the index, is the time to maturity on the option and d y is the dividend yield. We can estimate the total dividend step by step. irst the index value should grow at a rate R-d y. R is the total return and d y is the dividend yield. he Index value increases but at the same time it is growing, it is also paying out a dividend. As such, the net capital gains (index value growth would be R-d y and the index value at time would be S S ( R d y his represents the capital gain in the index. he total return would be the capital gains plus dividend yield or ( R d y d y R return S S he dividend would be equal to the total return less capital gains. 4 or inormation on the exchange please see, 5 See Exhibit or the values or the S&P 5 index as o December 23, month continuously compounded spot rate or US Government securities. 7 he original work to price European Call options on stocks with a dividend yield was irst derived by R. Merton, heory o Rational Option Pricing, Bell Journal o Economics and Management Science, 4 (Spring 973, 4-83.

3 -3- Dividend ( R d R d ( ( R y y S S S e. inally the underlying asset value would be the current value o the index less the present value o the dividend, which is the dividend discounted at the required rate o return 8, R, or S S S S S S S S S d y R d y e ( e d y ( e d y ( e + S d y R R R Hence or the dividend yield model, setting the equal to taking out the present value o the dividends. d y S is equivalent to Let s return to the example above; S&P5 index value,94, 3 months (.25 years, X,, index volatility.2, and a dividend yield.8%. he inputs or a Black- Scholes model would be Index X, r.9%.25years σ.2 d y 94 d y Based on these inputs the Black-Scholes value or this call option would be 4.68 and based on a payo o $ times the index, the option should sell or $4,68.. Model o Call Options on Indices Sometimes you ind that authors 9 will present an option pricing model speciically to deal with the dividend yield model. he orm o the model is d y r Call Option Value S N( d X N( d 2 8 Note that we are taking the present value o the dividends using the expected return on the index. he key here is that the expected total return on the index is the same as the discount rate. his is perectly consistent with the basic concept o required rate o return. 9 Hull presents such a model on page 268.

4 -4- Where N ( Cumulative Standard Normal unction, d d S ln d σ d y 2, + r X σ + 2 S ln r d X + σ + σ σ y, Underlying Asset Value, ime to maturity, σ volatility X Exercise (Strike Price r Risk-ree rate Dividend yield d y It is useul to note that this model is just the Black-Scholes model with d y S. Currency Options Currency options are widely available. In the United States the Philadelphia Exchange (PHLX is one o the largest markets or standardized option contracts. he PHLX trades contracts in Australian Dollars (A$, British Pounds (, Canadian Dollars (C$, Euros (, Japanese Yen ( and the Swiss ranc (CH. Besides these standardized options, a wide variety o customized options are available on the PHLX and in the over-thecounter markets. or valuation purposes, Currency options are actually very similar to Index options. In each case the holder o the option does not get the same thing as the holder o the underlying asset. In the case o an index it is the dividend. or currency it is the interest that could be earned by holding the underlying currency. Here it is useul to be very speciic. We need to speciy a domestic currency, D and a oreign currency,. Do not think o the US dollar as the domestic currency and everything else as the oreign currency. We need to be more general and as such, the deinition o the Domestic currency is the currency in which the option price will be quoted. Consequently, all o the currency related inputs to the model,, X and r must be expressed in terms o the domestic currency. or example, suppose we wanted to have a call option with a maturity o 6-months which gives the right to buy Swedish ona at a rate o.429 Euros per Krona. he call option is valuable i the Euro depreciates against the Krona, i.e., it takes more Euros to Contract sizes: A$5,, 3,25, C$5,, 62,5, 6,25,, and CH62,5. his is an exchange rate o 7. Krona per Euro.

5 -5- buy one Krona. I the exchange rate at the maturity o the option were.5 / (6.67 / then the call option would be exercised and the payo would be ( We can price this option using the Black-Scholes model. he most important actor is that the option price will be quoted in Euros. his means that the, the exercise price and the risk-ree rate must all be in Euros. Assume that the current exchange rate is.98 / (9.5 /, the exercise price is. / (. /, the historic volatility 2 or the exchange rate, σ ( /.467, the Swedish 6-month risk-ree rate 3 is 2.62% and the Euro 6-month risk-ree rate is 2.9%. Hence the inputs or valuing this call option would be: euro? euro X. r euro 2.9%.25years σ.467 or the we use the same approach that we did with the dividend yield. I we have Swedish Kronas then we would earn the Swedish rate o interest. I we hold an option on Kronas we do not earn the Swedish risk-ree interest rate. Hence the should be the current value less the present value o the interest we do not get. In this example we start with. I we invest at the Swedish risk-ree rate, r or time the result is S r he total interest, I, we would have earned would be I r e he present value o the interest is r r r ( e e e PV ( I and the underlying asset value in terms o Swedish ona would be would be 2 Please see exhibit 3 or an estimate o the historic volatility or the / exchange rate. Note that or the exchange rate volatility it does not matter i it is expressed in / or /. Since volatility is expressed in terms o percentage change, the volatility estimate would be the same. Exchange rate volatilities are much lower than stock index volatilities. or a comparison see igure 3 which shows the relative volatility o the Nasdaq index and the / exchange rate. 3 hese are the risk-ree rates or December 23. In addition, assume that these rates are continuously compounded rates.

6 -6- PV ( I ( r r Since all the Black-Scholes model inputs must be in the same currency as the price o the option, we convert the UVA rom ona to euros at the spot rate and the ull set o inputs or the Black-Scholes model would be: X r euro euro euro σ.467 spot. 2.9%.25years ( Euro r Given the values above, the call value would be.96 per Krona covered by the contract and an option contract to purchase, would have a value o 96.. An example o a PHLX traded would be a European Call option on the British Pound. his would give the right to buy 3,25 at an exercise (exchange rate price o US$.7/ and a maturity o 2 months. Assume a spot rate o US$.77, a 2-month US$ risk-ree rate o.9%, a risk-ree rate o 3.9% and an exchange rate (US$/ volatility 4 o.78. he would be r BP Spot( US$ / BP US$.77 US$.7585 Black-Scholes value would be Maturity Call $.6477 X US $ r σ US$ months US$.7.9%.78 he key to valuing currency options is to irst deine the currency in which you wish to express the option price. Call this the domestic currency. All o the inputs must be expressed in this currency. he other currency becomes the oreign currency and the interest rate on this currency represents the income not received by the option holder. As such, the current spot rate (domestic per oreign is adjusted or this oreign interest rate. 4 See Exhibit 3 or a calculation o historic volatility or US$/ exchange rates.

7 -7- Options on utures As noted in prior discussions 5 the basic relationship between the spot price at time t, S t and a uture/orward price with maturity at time, is Asset with no income Assets with income (dividend yield oreign currency Commodities D S S S S D r ( t ( r d y ( r r ( t D ( t ( r + µ + ψ ( t he risk-ree rate, r is the spot rate at time t or maturity, d y is the dividend yield, and r are respectively, the domestic and oreign risk-ree rates, µ is the rate or the commodity storage costs, and ψ is the commodity spoilage rate. In each case, as the time to maturity gets closer, t and -t gets smaller. As such the utures price declines and gets closer to the spot price over time. Pricing options on utures requires that we take into account the decline o the utures price over time. Let s look at the simplest case irst. his is the utures price or the asset with no income. or simplicity 6, consider a utures contract to buy a share o XYZ stock. he current stock price o XYZ is $45 and it pays no dividend. Assume that today s date is September, and there are 3-month utures contracts which mature on December 3 st or this stock. he current level o risk-ree rates relects a low level o interest rates. he 4- month spot rate is i 4 2.% based on continuous compounding and the -month spot rate is i.% also based on continuous compounding. In order or there not to be an arbitrage opportunity on the utures contract, the uture price would have to be D r r ( t S $ 45 $45.3. Now suppose that there was a call option that matured on October which gave the holder the right to take the buy-side o a December 3 st utures contract on XYZ stock with at a price o $ his option contract will have value to the holder, i on October, the utures price or delivery on December 3 st is higher than $ he table shown below demonstrates the timing o the option. he underlying utures 5 See the technical note on orward and utures. 6 Note there are no utures contracts on individual stocks.

8 -8- contract matures on December 3 st. However, the option contract has a maturity o October. As such, there is still three months let on the utures contract. Date Sept. Oct. Dec. 3 ime t t utures price S r ( t S r ( t S Option on utures Contract C Max( X, or this example, suppose the utures price or December 3 st delivery on October st is $47. Here one would exercise the option and take the buy-side o the December utures contract at a price o $ his utures contract would be marked to market and the holder would realize a payo 7 o the dierence between the $47 and the $45.25 or payo equal to $.75. We can value this using the Black-Scholes model 8. It is a European call option with an exercise price o $45.25 and a maturity o one month. he underlying asset is the utures price on October or delivery on December 3. However, we know that regardless o what happens to the spot price, the utures price will decline at a rate r over the t t time period. his is exactly the same as a dividend yield. As such, the or this option on a utures contract is r ( t t. In this case the risk ree rate is the -month spot rate and the would be s (. ( 2 2 $ 45.3 e $ he remaining Black-Scholes model inputs would be; the time to maturity ( 4 months, the risk-ree rate or the maturity o the option (s.%, the exercise price (X$45.25, and the volatility o the utures price (here assume σ Note that i the underlying asset had been a orward contract, the payo would be dierent. In the case o a orward contract the holder would realize the proit by taking the sell-side o a orward contract that matures on December 3 st. his would lock in the payo o $.75 but it would not be received until 4 months later when the orward contracts mature. We will deal with options with delayed payos later in the course. 8 See. Black, he Pricing o Commodity Contracts, Journal o inancial Economics, 3 (March 976,

9 -9- $ Maturity month Call $.72 X $45.25 r.% σ.22 Using this basic result, we can value call options on all o the dierent types o utures contracts that we listed above. Assuming t is the time to maturity o the option and i is the appropriate spot rate or that maturity, the in each case would be: Asset with no income ( t i t Assets with income (dividend yield ( i d y ( t t oreign currency Commodities D ( D i i ( t t ( i + µ + ψ ( t t All o the other inputs, (X, σ,, and r would be identical to any call option that we might value using a Black-Scholes model. One important caveat worth noting here is the volatility measure or options on utures. Generally, the volatility o utures contracts on inancial assets, such as stock indices, is essentially the same as the volatility o the price o the assets and a constant volatility is usually not a big problem. or commodities, interest rates, and oreign currencies this turns out not to be the case. Exhibit 4 shows the volatility estimates or crude oil orward prices taken rom the MG Reining & Marketing, Inc. case 9. he volatilities shown on this exhibit demonstrate a airly typical result. Volatilities or near term orward/utures prices is greater than the volatilities or orward prices urther out in time. Hence it is not the case that volatility is constant over the lie o a call option on a commodity utures price. In act volatility increases as the time to maturity o the underlying utures contract decreases. An unsatisactory but practical solution is to use an average volatility over the lie o the option. or example, assume today is September, 993 and we had a call option on the September, 994 crude oil utures price and the option maturity is in May, 994. During the lie o the option the underlying utures contract will start as a 2 month orward price, then become an month orward price and so on until at the maturity o the option itsel the underlying utures price will be a 4 month orward price. he volatility or all o these is dierent and one solution is just to take the average. rom Exhibit 4, the average o the 2-month, 9-month, 6-month, and 4-month orward price 9 MG Reining & Marketing, Inc. (A, (UVA--227

10 -- volatilities is.276. Again this is not entirely satisactory but it does relect the act that none o the single estimates is correct. Black s model or valuing utures Options Sometimes you will see people reerring to Black s model or valuing options on utures. Call e where, d d 2 r ( [ N( d X N( ] 2 ln + σ X 2 σ d σ d 2 his is not really a dierent model but rather totally equivalent to the Black-Scholes r model with the e. Hence we can always use the Black-Scholes model with the appropriately deined. Summary In valuing options on stock indices, currencies and utures, we used the same approach. We adjusted the to account or the income stream that we do not get by holding the option instead o the underlying asset.

11 -- Index Standard & Poor s 5 Exhibit Stock Indices Data Dow Jones Industrial Average Russell 2 Nasdaq Index Value*,94.4, ,443.9 Number o stocks Weighting Market value Price Market value Market value racking Large Cap Large Cap Small Cap Small Cap year return* 22.5% 2.98% 44.8% 4.3% 7-yr. Volatility** yr. Volatility*** Average Dividend.5% 2.2%.%.8% yield* % paying Dividend 73.4%.% N.A. 26.% *As o December 24, 23 ** Based on weekly data 2/23/996-2/23/23 ***Based on weekly data 2/26/2-2/23/23

12 -2- igure. Relative Perormance o Major Indices 2/23/996-2/23/ Index (2/23/ S&PCOMP DJINDUS NASN RUSSL2 Dec-96 Mar-97 Jun-97 Sep-97 Dec-97 Mar-98 Jun-98 Sep-98 Dec-98 Mar-99 Jun-99 Sep-99 Dec-99 Mar- Jun- Sep- Dec- Mar- Jun- Sep- Dec- Mar-2 Jun-2 Sep-2 Dec-2 Mar-3 Jun-3 Sep-3 Dec-3 date igure 2. Relative Perormance o Major Indices 2/26/2-2/23/ Dec- eb- Apr- Jun- Aug- Oct- Dec- eb-2 Apr-2 Jun-2 Aug-2 Oct-2 Dec-2 eb-3 Apr-3 Jun-3 Aug-3 Oct-3 Dec-3 Index (2/26/2 S&PCOMP DJINDUS NASN RUSSL2 Date

13 -3- Exhibit 2 S&P 5 M Index Options Symbol:SPX Underlying: he Standard & Poor's 5 Index is a capitalization-weighted index o 5 stocks rom a broad range o industries. he component stocks are weighted according to the total market value o their outstanding shares. he impact o a component's price change is proportional to the issue's total market value, which is the share price times the number o shares outstanding. hese are summed or all 5 stocks and divided by a predetermined base value. he base value or the S&P 5 Index is adjusted to relect changes in capitalization resulting rom mergers, acquisitions, stock rights, substitutions, etc. Multiplier: $. Premium Quote: Stated in decimals. One point equals $. Minimum tick or options trading below 3. is.5 ($5. and or all other series,. ($.. Strike Prices: In-,at- and out-o-the-money strike prices are initially listed. New series are generally added when the underlying trades through the highest or lowest strike price available. Strike Price Intervals: ive points. 25-point intervals or ar months. Expiration Months: hree near-term months ollowed by three additional months rom the March quarterly cycle (March, June, September and December. Expiration Date: Saturday ollowing the third riday o the expiration month. Exercise Style: European - SPX options generally may be exercised only on the last business day beore expiration. Last rading Day: rading in SPX options will ordinarily cease on the business day (usually a hursday preceding the day on which the exercise-settlement value is calculated. Settlement o Option Exercise: he exercise-settlement value, SE, is calculated using the opening (irst reported sales price in the primary market o each component stock on the last business day (usually a riday beore the expiration date. I a stock in the index does not open on the day on which the exercise & settlement value is determined, the last reported sales price in the primary market will be used in calculating the exercise-settlement value. he exercise-settlement amount is equal to the dierence between the exercise- settlement value, SE, and the exercise price o the option, multiplied by $. Exercise will result in delivery o cash on the business day ollowing expiration. Margin: Purchases o puts or calls with 9 months or less until expiration must be paid or in ull. Writers o uncovered puts or calls must deposit / maintain % o the option proceeds* plus 5% o the aggregate contract value (current index level x $ minus the amount by which the option is out-o-the-money, i any, subject to a minimum or calls o option proceeds* plus % o the aggregate contract value and a minimum or puts o option proceeds* plus % o the aggregate exercise price amount. (*or calculating maintenance margin, use option current market value instead o option proceeds. Additional margin may be required pursuant to Exchange Rule 2.. Cusip Number: rading Hours: 8:3 a.m. - 3:5 p.m. Central ime (Chicago time.

14 -4- Exhibit 3. Exchange Rates and Volatilities Exchange Rate Exchange Rate Month end / ln(price relative US$/ ln(price relative Dec Jan-.28 (.5.46 (.25 eb-.5 ( (.34 Mar-.93 (.2.42 (.85 Apr May-.96 ( (.3 Jun-.82 ( (. Jul-.76 ( Aug-.54 ( Sep-.28 ( Oct (.64 Nov (.22 Dec Jan (.35 eb Mar Apr-2.8 ( May (.22 Jul Jul-2.73 ( Aug (.92 Sep Oct (.6 Nov (.48 Dec-2.94 ( Jan-3.82 ( eb (.456 Mar-3.8 ( Apr May Jun-3.88 ( Jul-3.82 (.56.6 (.282 Aug (.243 Sep Oct-3.3 ( Nov Dec-3.98 ( σ(monthly.35 σ(monthly.225 σ(annual.467 σ(annual.78

15 -5- irgure 3 Comparison o variability Nasdaq index and Euro/Krona exchange rate Index (/ Jan-99 Apr-99 Jul-99 Oct-99 Jan- Apr- Jul- Oct- Jan- Apr- Jul- Oct- Jan-2 Apr-2 Jul-2 Oct-2 Jan-3 Apr-3 Jul-3 Oct-3 NASDAQ / exchange rate dates

16 -6- Exhibit 4. Volatility o Crude Oil orward Prices Crude Oil orward Prices Months orward Spot Date Prices // // // // // // // // #N/A 5// // // // // // #N/A // #N/A 2// // // // // #N/A 5// // // // #N/A 9// // // // // #N/A 2// // // // #N/A 6// // // // Volatility* *Volatility is the standard deviation o the natural logarithm o the monthly price relatives. It is annualized by multiplying by the square root o 2.