South-Western College Publishing ©2000 All rights reserved

South-Western College Publishing ©2000 All rights reserved

1

STOCK INDEXES AND COMMODITIES

Jarrow & Turnbull

Chapter 12


Chapter 12, Page 2Jarrow-Turnbull

IntroductionDerivatives on stock indexes and commodities are widely used. Money managers of mutual funds, pension plans, and hedge funds regularly use them to insure fund performance and/or enhance fund returns.

Buy puts on stock indexes for “portfolio insurance.” Pick up extra returns in flat markets by writing call options on stock indexes.



Introduction (cont’d)Commodity derivatives can be used to hedge future commodity price uncertainties due to production or delivery lags.

Farmers sell corn futures to fix a selling price.Oil companies regularly use oil futures and options to hedge price risks associated with long-dated, fixed-price contracts to deliver oil.

Fortunately, little new theory is needed because earlier results can be applied with minor changes.



Stock Market IndexesA stock market index is an average price of a portfolio of stocks. Indexes are aplenty these days

Dow Jones Industrial Average (the “Dow”)is a price average of 30 major, representative U.S. stocks. S&P 500 index is a weighted average of 500 major U.S. common stocks. New York Stock Exchange composite index includes all common stocks that trade on the NYSE. The Value Line Composite Average has 1,700+ stocks. Japanese Index is composed of 225 companies traded on the Tokyo Stock Exchange.



Example: Stock Market Index Quotes

Assume today is March 26. Consider S&P 500 Stock Index.

HIGH for the day 1100.05, LOW 1089.77CLOSE for the day 1098.06NET CHANGE from previous day 2.15Change FROM DEC 31 is 127.15 % CHANGE from Dec 31 is 13.3%

Business newspapers similarly report other indexes.



Derivatives on Stock Market Indexes

Derivatives trading on stock indexes are: futures on the index options on the index options on stock index futures.

When these 3 mature together on the same Friday, the market often becomes very volatile in the last hour of trading (triple witching hour). As pricing and hedging of “derivatives on foreign currencies” and “derivatives on stock indexes and commodities” are very similar, last chapter results readily apply.



Example: Stock Market Index Futures Quotes

Futures on Dow trade in the CBTContract size $10 times average. Contracts trade for June, Sept, Dec, Mar.

Assume today is May 26. Consider June contract.OPEN is day’s 1st price, say 8850.HIGH price for the day 9051, LOW 8800.SETTLEMENT is usually the last price 9015.CHANGE of price from last trading day 303.LIFETIME HIGH 9313 and LOW 7053 for contract.OPEN INTEREST 13,050 contracts. EST. VOLUME 14,000. In all, 15,050 contracts open.



Stock Market Index Futures Price Consider a futures contract that matures at date TF. Let F(0,TF) be the stock market index futures price at date 0. Assume interest rates are deterministic. Then forward and futures prices equal. We can relate the futures price to the value of the spot index via the cost-of-carry relationship of Chapter 2.



Stock Market Index Futures Price (cont’d)

Futures price can be found fromB(0,TF)F(0,TF)

where I(0) is the current value of the index, B(0,TF) is the date-0 value of a Treasury bill that pays $1 for sure at date TF.

contract futures theof life theoverindex

in the stocks by the paid dividends all of)0( 0PVI



Stock Market Index Futures (cont’d)

This cost-of-carry has 3 important implications. Futures price depends on the index level.. Everything else constant, if index rises, futures price also rises. If stocks in the index increase the level of dividend payments over futures’ life, then futures price will fall, everything else constant. This happens because long futures does not receive dividends.If the level of interest rates increases, the futures price will increase, everything else constant. This happens because discount factors decline, and as the value of the index remains unchanged, futures price must rise.



Index ArbitrageThis cost-of-carry relation is the basis of index arbitrage, a form of program trading, often engaged in by various investment banks. Cost-of-carry violation gives arbitrage opportunity

For example, if futures price is too large relative to index; sell futures and buy spot index to lock in profits.

Stock index arbitrage involves big trades and can cause market swings.

It has been blamed for October 1987 crash. But it increases market efficiency by keeping spot and futures linked.



Example: Spread TradingSame indexes

If we anticipate interest rates will rise, we can sell short-maturity stock index futures and buy the more rate sensitive long-maturity futures.

Different indexes If we believe that given the current economic conditions, blue-chip firms will do poorly relative to smaller firms; we can sell June S&P MIDCAP 400 index futures and short June DJ Industrial Average index futures.

In each case, right bet will yield trading profits.



Index Option PayoffConsider a call written on the S&P 500 index [I(t)] that matures at date T and has strike K. Then call value at maturity is

As contract size is $100 for S&P 500 index option, if the index exceeds the strike, its value is 100 [I(T) – K]. Otherwise, it is worthless. Put payoffs are similarly defined.

.)( 0

)( )(sizecontract )(

KTIifKTIifKTI

Tc



Example: Index Option QuoteExpiration: Mar, Jun, Sep, and Dec, plus 2 near-term months. Active contracts have more months.Example: Assume today is May 26. Options on S&P 500 index are European. Trade in Chicago.

June 1050 call has VOL 1,509; LAST price 54 3/8NET CHANGE is 1 5/8, OPEN INTEREST 16,257June 1050 put has VOL 7,163; LAST price 5 1/2 NET CHANGE is 2, OPEN INTEREST 30,057Also trade Jun 400, Jun 700, Jul 1050, Aug 1050, Sep 1050, Jun 1055, Jun 1060, ..VOL and OPEN INT for the market are also given.



Example: Portfolio Insurance Using Index Put Options

Stock index puts are in high demand for protecting a portfolio against decline (portfolio insurance).

Example: A portfolio manager wants to protect her $5 million portfolio from dropping below $4,480,000 over the next 3 months.

Assume that the portfolio perfectly tracks the index. We can write V(t) = a + bI(t),

• where V(t) is the value of the portfolio at date t • a and b are constants that must be determined.

Suppose, empirical analysis gives a = 0 and b = 4,458.



Example: Portfolio Insurance Using Index Put Options (cont’d)

If I(T) is index value in 3 months, V(t) = bI(t). Now, bI(T) = 4,480,000,

or I(T) = 4,480,000/4,458 = 1,005. Manager insures against this index level. She buys 3 months puts on S&P 500 with strike 1,005. How much do the puts cost? Each put costs $1,500 because by contract specification, put price 15 is multiplied by 100. How many options should she buy? As the portfolio is worth bI(T) = 4,458 I(T) and each option pays off 100 I(T), she needs n = b/100 = 44.58 options.



Example: Portfolio Insurance Using Index Put Options (cont’d)

Does this work? Yes, the insured portfolio’s payoff when the index is below 1,005 is always $4,480,000: bI(T) + n[1,005 I(T)] 100 = b 1,005 = 4,480,000. (because n = b/100)

If portfolio imperfectly tracks the index, then index puts provide partial insurance. We need to use portfolio’s Beta (Beta measures covariance of a portfolio with the market portfolio; if market portfolio changes 1%, portfolio changes Beta%)If portfolio beta is 0.8, she would need 0.8 44.58 = 35.66 index puts.



Pricing Index OptionsPricing stock market index options is similar to pricing foreign currency options. Suppose that index stocks pay dividends that can be approximated by a constant dividend yield, dy. Cost-of-carry relation from Chapter 2 holds: F(0,T)B(0,T) = I(0)exp(-dy)

where F(0,T) is toady’s forward price B(0,T) is a zero coupon bond that matures at time TI(0) is today’s index level.

A1-A7 holds. Add A8 (constant dividend yield over option life), and we can value index options.



Pricing Index Options: Closed Form Solution

Merton’s Model for pricing options with constant dividend yields can be used to value European call and put options on stock indexes. Call value is

where I(0) is today’s index level

21 ),0()0(exp)0( dNTKBdNITdc y

)/(})2/()],0(/)0({ln[ 21 TTdTKBSd y

Tdd 12



Example: Options on Stock Index Futures Quotes

Options on Dow Jones Industrial Average futures trade in the Chicago Board of Trade.

Contract size $100 times premium Assume today is May 26. June, July, August maturity options trade.

STRIKE prices are 88, 89, …,93.June 88 call’s SETTELMENT price is 27.89.June 88 put’s SETTLEMENT price is 9.55.



Pricing Index Futures Options Consider a call that matures at date T. It is written on an index futures contract that matures at date TF which is later than date T. By Black’s Model, value of the European call is c(0) = B(0,T)[F(0,TF)N(d1) - KN(d2)]

where K = call’s strike price d1 {ln[F(0,TF)/K] + 2T/2}/( T) d2 = d1 - T and N(.) is the cumulative normal distribution function.

And put, p(0) = B(0,T)[KN(-d2) - F(0,TF)N(-d1)]



Commodity DerivativesCommodity futures and option on futures trade for many assets:

Grains and oilseeds (corn, oats, soybeans, …)Livestock (cattle, pork bellies, …)Food and fiber (cocoa, sugar, cotton, …)Metals and Petroleum (copper, gold, oil, …)

Fortunately, pricing and hedging of commodity derivatives is very similar to that for derivatives on foreign currency and stock index options.



Commodity Futures PriceConsider a commodity futures contract that matures at date TF. Let F(0,TF) be the futures price at date 0. Assume interest rates are deterministic. Then forward and futures prices equal. We can relate the futures price to the value of the spot index via the cost-of-carry relationship of Chapter 2.



Commodity Futures Price (cont’d)

Futures price can be found fromF(0,TF) = S(0)exp[(r - yN)TF]

where S(0) = spot commodity price,r = continuously compounded rate of interest, yN = net convenience yield.

When other variables are given, net convenience yield can be solved from this expression.Comparing with earlier results, the net convenience yield is analogous to the foreign interest rate or the dividend yield.



Futures OptionsPricing of commodity futures options is similar to pricing of index futures options. We only need to modify assumption A8 as follows.A8. The net convenience yield is known and constant over the option’s life.This is analogous to a constant dividend yield on a stock index.

For European options, formula for pricing index futures options can be used.For American options, models like a binomial lattice approximation to the lognormal model can be used.



Pricing Commodity Futures Options

Consider a call that matures at date T. It is written on a commodity futures contract that matures at date TF which is later than date T. By Black’s Model, value of the European call is c(0) = B(0,T)[F(0,TF)N(d1) - KN(d2)]

where K = call’s strike price d1 {ln[F(0,TF)/K] + 2T/2}/( T) d2 = d1 - T and N(.) is the cumulative normal distribution function.

And put, p(0) = B(0,T)[KN(-d2) - F(0,TF)N(-d1)]Net convenience yield is accounted for in futures price.



SummaryGiven assumptions A1 through A6, pricing of derivatives on stock indexes and commodities is analogous to pricing of foreign currency derivatives.This helps us to construct the binomial lattice of spot prices for different assets and futures prices. Merton’s Constant Dividend Yield model and Black’s Model give closed form solutions. Many of these assets have an implicit dividend, so early exercise of American options may be optimal.

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net convenience

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constant dividend

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futures price