Chapter 131 CHAPTER 13 Options on Futures In this chapter, we discuss option on futures contracts....

Chapter 13 1

CHAPTER 13Options on Futures

In this chapter, we discuss option on futures contracts. This chapter is organized into:

1. Characteristics of Options on Physicals and Options on Futures.

2. The Market for Options on Futures

3. Pricing of Options on Futures

4. Price Relationship Between Options on Physicals and Options on Futures

5. Put-Call Parity for Options on Futures

6. Options on Futures and Synthetic Futures

7. Risk Management with Options on Futures

Chapter 13 2

Characteristics of Options on Physicals and Options Futures

Recall from Chapter 12 that options are written for a pre-specified amount of a pre-specified asset at a pre-specified price that can be bought or sold at a pre-specified time period.

Call Options

The buyer of a call option has the right but not the obligation to purchase.

The seller of a call option has the obligation to sell.

Put Options.

The buyer of a put option has the right but not the obligation to sell.

The seller of a put option has the obligation to purchase.

Chapter 13 3


Prices of options on futures are closely related to prices of options on the underlying good.

Call Option on Futures

Upon exercising a option on futures, the call owner:

– Receives a long position in the underlying futures at the settlement price prevailing at the time of exercise.

– Receives a payment that equals the settlement price minus the exercise price of the option on futures.

The call owner would not exercise if the futures settlement price did not exceed the exercise price.

Upon exercise, the call seller:

– Receives a short position in the underlying futures at the settlement price prevailing at the time of exercise.

– Pays the long trader the futures settlement price minus the exercise price.

Chapter 13 4


On February 1, a trader buys a call option on a MAR euro futures contract with an exercise price of $0.44 per euro. On February 15, the call owner decides to exercise the call option. The futures settlement price is $.48. After gathering all the information, the owner has:

Future settlement price = $.48The exercise price = $.44/euro The euro futures maturing = MarchEuro contract amount = 125,000 euros

Upon exercise, the call owner:

– Receives a long position in the MAR euro futures contract.

– Receives a payment = F0 – E

$.48 - .44 (125,000) = $5000

Upon exercise, the call seller:

– Receives a short position in the euro futures.

– Pay $5,000.

The traders can offset or hold their futures positions.

Chapter 13 5


Put Option on Futures

Upon exercising a option on futures, the put owner:

– Receives a short position in the underlying futures contract at the settlement price prevailing at the time of exercise.

– Receives a payment that equals the exercise price minus the futures settlement price.

The put owner would not exercise unless the exercise price exceeded the futures settlement price.

Upon exercise, the put seller:

– Receives a long position in the underlying futures contract.

– Pays the exercise price minus the settlement price.

Chapter 13 6


On April 1, a trader buys a put option on a MAY wheat futures contract. The exercise price is $2.40/bushel and wheat contract is for 5,000 bushels. On April 4, the owner of the call option decides to exercise. The futures settlement price is $2.32/bushel.

Exercise price = $2.40/bushelWheat contract = 5,000 bushelsFutures settlement price = $2.32/bushel.The wheat futures matures = May

Upon exercise, the put owner:

– Receives a short position MAY Wheat futures contract.

– Receives a payment = F0 – E

$2.40-$2.32 (5,000) = $400

Upon exercise, the put seller:

– Receives a long position MAY Wheat futures contract.

– Pays $400.

The traders can offset or hold their futures positions.

Chapter 13 7


The following table summarizes the option examples discussed previously.

The overall profitability of the transactions depends upon the original premium and the prices that become available before expiration of the option.

Results of Futures Option Exercises

Option Futures Results Cash Flows

Call Owner holds long futures position. Seller holds short futures position.

Owner receives F0 - E. Seller pays F0 - E.

Put Owner holds short futures position. Seller holds long futures position.

Owner receives E - F0. Seller pays E - F0.

where: F0 = futures settlement price at time of exercise E = exercise price of the futures option

Chapter 13 8

The Market of Options on Futures

Figure 13.1 presents some illustrative quotations for options on futures.

Insert figure 13.1 here

Chapter 13 9


Table 13.1 shows the trading volume for options on futures by type of commodity in the fiscal year ending September 30, 1995.

Table 13.1

Trading Volume for Futures Options (Year Ending September 30, 2003)

Commodity Group

Number of Contracts Traded (millions)

Grain Oilseeds Livestock Other Agricultural Energy/Wood Metals Financial Instruments Currencies Total

6.8 5.3 0.9 5.3

20.7 4.3

173.9 2.1

219.2

Source: Commodity Futures Trading Commission, Annual Report, 2003.

Chapter 13 10


Product Profile: The NYMEX=s Crude Oil Futures Options

Contract Size: One NYMEX light, sweet, crude oil futures contract Strike Prices: Twenty strike prices in increments of 50 cents per barrel above and below the at-the-money strike price. The next 10 strike prices are in increments of $2.50 above the highest and below the lowest strike prices for a total of 61 strike prices (including the at-the-money strike price). Tick Size: One cent per barrel ($10 per contract) Price Quote: U.S. dollars and cents per barrel. Contract Months: Thirty consecutive months plus long-dated futures initially listed 36, 48, 60, 72, and 84 months prior to delivery. Expiration and final Settlement: Last trading day is three business days prior to the last trading day for the underlying futures contract. Trading Hours: Open outcry trading is conducted from 10:00 AM until 2:30 PM. Daily Price Limit: None.

Chapter 13 11


Product Profile: The CME=s S&P 500 Futures Options

Contract Size: One S&P 500 stock index futures contract Strike Prices: Generally12 strikes, including the at-the-money strike. Increments between strike price generally are 25 index points. Number of strike prices increases as expiration approaches and increments between strike prices is reduced to a minimum of 5 index points. Tick Size: .1 index points or $25.00. Price Quote: Price is quoted in terms of Standard & Poor=s 500 Index. Contract Months: Four months in the March, June, September, December cycle plus the first two serial months not in the cycle for a total of 6 contract months. Expiration and final Settlement: Options that expire in the March, June, September, December cycle expire at the same time as the underlying futures contract. The two non-March cycle options expire on the third Friday for the contract month. Trading Hours: Floor: 8:30 a.m. to 3:15 p.m.; Globex: Monday through Thursday 3:30 p.m. to 8:15 a.m. with a shutdown period from 4:30 p.m. to 5:00 p.m. nightly. Sunday and holidays 5:30 p.m. to 8:15 a.m. Daily Price Limit: Trading halted when futures trading is halted

Chapter 13 12


Product Profile: The CME=s Eurodollar Futures Options

Contract Size: One Eurodollar futures contract Strike Prices: Generally12 strikes, including the at-the-money strike. Increments between strike price generally are 25 index points. Number of strike prices increases as expiration approaches and increments between strike prices is reduced to a minimum of 5 index points. Tick Size: .01 index points or $25.00. Price Quote: Price is quoted in terms of the IMM 3-month Eurodollar index, 100 minus the yield on an annual basis for a 360-day year. Contract Months: Eight months in the March, June, September, December cycle plus the first two serial months not in the cycle for a total of 10 contract months. Expiration and final Settlement: Options on the March, June, September, December cycle cease trading at 5:00 a.m. Chicago Time (11:00 a.m. London Time) on the second London bank business day immediately preceding the third Wednesday of the contract month. The two non-March cycle options expire on the Friday immediately preceding the third Wednesday for the contract month. Trading Hours:Floor: 7:20 a.m.-2:00 p.m; Globex: Mon/Thurs 2:10 p.m.-7:05 p.m.; Shutdown period from 4:00 p.m. to 5:00 p.m. nightly; Sunday & holidays 5:30 p.m.-7:05 p.m. Daily Price Limit: No limit

Chapter 13 13

Pricing Options on Futures

Recall from Chapter 12:

European Options

European options can be exercised only on the maturity date.

American Options

American options can be exercised any time prior to maturity.

The Black-Scholes model focus best on European options which avoids problems with early exercise and dividends.

When there is a dividend and the dividend rate varies, the Black-Scholes model is not suitable for valuing options on futures.

The Black-Scholes model can be modified for forward option pricing.

Chapter 13 14

Graphical Approach to American Options on Futures

Figure 13.2 illustrates how European options prices are good approximations for American futures option prices

Insert figure 13.2 here

Chapter 13 15

Black-Scholes Model for Options on Forward Contracts

The Black-Scholes equation for option on forward contracts is:

)]dN( E - )dN( F[ e = C *2

*1t0,

rt-

Where

r = risk-free rate of interestt = time until expiration for the forward and the optionF0,t = forward price for a contract expiring at time tα = standard deviation of the forward contract’s price

t

tEFd

2

1* 5.)/ln(

tdd 1*

2*

If there were no uncertainty, N(d1*) and N(d2*) will equal 1 and the equation would simplify to:

Cf = e-rt[F0,t - E]

Chapter 13 16

European Versus American Option on Futures

European Options

Early exercise of an option on a non-dividend paying stock is not recommended:

– Recall that upon exercising, the call owner receives the intrinsic value (S – E).

– Exercising a call discards the excess value of the option over and above S – E.

American Options

Early exercise of a dividend paying futures option has benefits and costs

– Benefit: exercise provides an immediate payment of F – E which can earn interest until expiration ert [F - E].

– Cost: sacrifice of option value over and above intrinsic value F – E.

Chapter 13 17

Approximating European and American Futures Option Values

Table 13.2

Comparison of European and Approximate American Futures Option Call Values

r = .08 σ = .20 t = .5 years E = 100

Futures Price

European

Approximate

American

80

0.30

0.30 90

1.70

1.72

100

5.42

5.48 110

11.73

11.90

120

19.91

20.34 Source: G. BaroneBAdesi and R. Whaley, AEfficient Analytic Approximation of American Option Values,@ Journal of Finance, 42:2, June 1987, pp. 301B320.

Table 13.2 compares the theoretical values for European and American options on futures. The table assumes that the option on futures expires in half a year and has an exercise price of $100. The risk-free rate of interest is 8% and the standard deviation of the percentage change in the futures price is 0.2.

Chapter 13 18

Efficiency of The Option on Futures Market

Most tests of efficiency examine whether market prices match the prices of a theoretical model.

A test of market prices against a theoretical model is a joint test of the market's efficiency and the model's ability to correctly represent the price.

The results of Whaley’s test for efficiency are presented in Table 13.3.

Table 13.3 Pricing Discrepancies for S&P 500 Futures Options

Observed Market Price C Theoretical Price Summary of average pricing errors of American futures option pricing models by the option's moneyness (F/E) and by the option's time to expiration in weeks (t) for S&P 500 futures option transactions during the period January 28, 1983, through December 30, 1983.

Calls

Puts

t < 6

6 t < 12

t 12

All t

t < 6

6 t < 12

t 12

All t

F/E < 0.98

B0.0630

B0.1372

B0.0872

B0.1028

B0.1064

B0.0914

B0.1056

B0.1014

0.98 F/E <1.02 B0.1228

B0.0775

0.0073

B0.0924

B0.0816

B0.0196

0.1336

B0.0406

F/E 1.02

0.0577

0.1175

0.0702

0.0806

0.1286

0.1906

30.3060

0.1929

All F/E

B0.0757

B0.0599

B0.0120

B0.0606

B0.0191

0.0808

0.2287

0.0537

Source: R. Whaley, AValuation of American Futures Options: Theory and Empirical Tests,@ Journal of Finance, March 1986, p. 138.

The differences between the theoretical and market price are significant here.

Chapter 13 19

Efficiency of The Option on Futures Market

Some of the studies summarized in Table 13.4 compare actual prices with Black model prices.

Table 13.4

Tests of Efficiency for Futures Options Study

Key Results

Whaley (1986)

For S&P 500 futures options, market and theoretical prices are systematically different.

Jordan, McCabe, and Kenyon (1987)

For soybeans, compared the difference between actual market prices and the Black model price, with average differences being 4/100 of a cent per bushel.

Ogden and Tucker (1987)

For currencies, futures options appear to be efficiently priced.

Bailey (1987)

For gold, futures options appear to be efficiently priced.

Blomeyer and Boyd (1988)

In early trading of TBbond futures options, inefficiencies may have existed. However, inefficient prices were rare and difficult to exploit.

Wilson and Fung (1988)

For grain futures options, prices closely conformed to the Black model. In periods of high volatility, actual prices did not rise as much as Black model prices.

Chapter 13 20

Price Relationship Between Options on Physicals and Options on Futures

In this section, the pricing relationship between options on physicals and options on futures is considered, specifically for call options. The analysis is organized as follows:

1. European options

2. American options on underlying assets with no cash flow

3. American options on underlying assets with cash flow

Chapter 13 21

Price Relationship Between Options on Physicals and Options on Futures

The following assumption will be held for this analysis:

1. The options have the same expiration and exercise price.

2. The options are on the same underlying commodity.

– One option is on the commodity itself.

– One option is on the futures on the commodity.

Chapter 13 22

European Options on Physicals and Futures

Recall from Chapter 12 that at expiration a call option on the physical will be worth:

S - E

For European options on futures, exercise can occur only at expiration, so it must be that:

Ft,t - E = St - E

For European options the exercise value for options on physicals and options on futures is the same.

Chapter 13 23

American Options on Physicals and Futures with No Underlying Cash Flows

For American options, any difference in value between options on physicals and options on futures results from the early exercise privilege. Table 13.5 shows the exercise values that the option on the futures can have given the option on physicals in percentage terms. The risk-free rate is assumed to be 15% and the percentage change in the underlying assets is .25.

Table 13.5 Percentage Difference in Value

for Call Options on Futures and Options on Physicals Assumptions:

Underlying asset has no cash flows. r = .15 σ = .25

Ratio of Physical Price to

Days Until Expiration

Exercise Price

30

60

90

180

270

0.8

0.00

0.00

0.00

1.20

2.02 0.9

0.00

0.00

0.47

1.58

3.15

1.0

0.29

0.56

1.02

2.48

4.51 1.1

0.61

1.15

1.72

3.79

6.34

1.2

1.22

2.13

2.89

5.52

8.70 Source: M. Brenner, G. Courtadon, and M. Subrahmanyam, AOptions on the Spot and Options on Futures,@ Journal of Finance, 40:5, 1985, pp. 1303-1317.

Chapter 13 24

American Options on Physicals and Futures with Underlying Cash Flows

This analysis is particularly relevant to options on stock indexes and options on stock index futures.

Cash flows from the underlying good reduce its value.

– When stock pays a dividend, the stock price drops by approximately the amount of the dividend.

These cash flows affect both the option on the physical and the option on the futures.

The analysis focuses on underlying physical asset paying a continuous dividend (cash flow) equal to the risk-free rate of interest.

Under conditions of certainty, a futures call option is worth the present value of:

F0,t – E, t = 0

Based on the perfect markets Cost-of-Carry Model the futures price will be:

F0,t = S0(1 + C)

Chapter 13 25

American Options on Physicals and Futures with Underlying Cash Flows

For financial futures, the cost of carry is the risk-free interest rate. Assume a continuous dividend equal to the risk-free rate of interest. In this case, the cost of carry is zero, so the futures call option price equals:

F0,t = S0erte-rt = S0

Substituting the value of F0,t into the Black-Scholes OPM gives an adjusted Black-Scholes OPM of:

)]dEN( )dN( S[ e = C *2

*10

rt f

where:Cf = the price of a call option on the futures

After adjusting the Black-Scholes model for continuous paying dividend:

The values for the call option on the futures and physical are the same. That is, d1* = d1, and d2* = d2.

)dEN(e - )dN(Se = C *2

rt- *10

-rtf

)dEN( - )dN( Se =C 210-rt

f

Chapter 13 26

Relative Prices of Options on Physicals and Futures

Relative Prices of Options on Physicals and Futures Option Characteristics Call Put European Options Cf = Cp Pf = Pp American Option–No Dividend Cf > Cp Pf < Pp American Option–Continuous Dividend

Dividend Rate < Interest Rate Cf > Cp Pf < Pp Dividend Rate = Interest Rate Cf = Cp Pf = Pp Dividend Rate > Interest Rate Cf < Cp Pf > Pp

where: Cf, Cp = call on the futures and call on the physical Pf, Pp = put on the futures and put on the physical

The implications of this analysis for various dividend rates are:

Chapter 13 27

Put-Call Parity for Options on Futures

Recall that Put-Call Parity specifies a relationship between the price of call and put options.

For non-dividend paying assets put-call parity equals:

C - P = S0 - Ee-rt

where:

C = value of a call with exercise price EP = value of a put with exercise price EE = exercise price of both the call and putS0 = stock pricer = risk-free rate of interestt = time until the options expire

Chapter 13 28


Before expiration, for options on futures, the relationship can be expressed as:

Cf - Pf = (F0,t - E)e-rt

where:

Cf = futures call option with exercise price E

Pf = futures put option with exercise price E

F0,t = current futures price

E = common exercise price for Cf and Pf

r = risk-free rate

t = time until expiration for the futures and options

Comparing both equations shows the similar structure of put-call parity for options on physicals and on futures.

Chapter 13 29


Using continuous compounding, the Cost-of-Carry Model for a perfect market is:

F0,t = S0ert

Substituting this expression for the futures price into the above equation gives:

Cf - Pf = (S0ert - E)e-rt = S0 - Ee-rt

Chapter 13 30

Options on Futures and Synthetic Futures

Synthetic Futures

A position that duplicates the profits and losses from a futures, but consists of positions in other instruments.

Creating synthetic futures equals:

Futures Call - Futures Put = Synthetic Futures

Table 13.6 summarizes the rules for constructing synthetic positions.

Table 13.6

Rules for Creating Synthetic Instruments

Synthetic Futures = Call B Put Synthetic Call = Put + Futures Synthetic Put = Call B Futures Synthetic Short Futures = Put B Call Synthetic Short Call = B Put B Futures Synthetic Short Put = B Call + Futures

Note: A synthetic instrument has the same profit and loss characteristics as the actual instrument. However, the synthetic instrument does not necessarily have the same value as the actual instrument.

Chapter 13 31

Risk Management with Options on Futures

This section explores examples related to risk management including:

– Portfolio Insurance

– Synthetic Portfolio Insurance and Put-Call Parity

– Risk and Return in Insured Portfolios

Chapter 13 32

Risk Management with Options on Futures Example

Assume: a stock index is currently at $100. Stocks in the index pay no dividends, and the expected return on the index is 10% with a standard deviation of 20%. A put option on the index with an exercise price of $100 is available and costs $4. Consider three investment strategies:

Portfolio A : Buy the index; total investment $100.(uninsured)

Portfolio B: Buy the index and one-half of a put; total(half insured) investment $102.

Portfolio C: Buy the index and one put; total(fully insured) investment $104.

At expiration, the three portfolios will have profits and losses computed using the following equations:

Portfolio A: Index Value - $100

Portfolio B: Index Value + .5 MAX{0, Index Value – $100} - $102

Portfolio C: Index Value + MAX{0, Index Value – $100} - $104

Chapter 13 33

Risk Management with Options on Futures

Figure 13.4 graphs the profits and losses of these 3 portfolios.

Insert Figure 13.4 here

Chapter 13 34

Portfolio Insurance

Recall that in portfolio insurance, a trader transacts to insure that the value of a portfolio does not fall below a given amount.

Based on figure 13.4, portfolio C is an insured portfolio:

The value of portfolio C cannot fall below $100. To create portfolio C, a trader bought the index at $100 and bought an index put with an exercise price of $100.

The worst possible loss on portfolio C is $4. Portfolio C must always be worth at least $100 because the value can not fall below $100, so it an insured portfolio.

Chapter 13 35

Synthetic Portfolio Insurance and Put-Call Parity

Recall that a synthetic call could be created from a long position in the underlying good plus a long put. Thus a synthetic call is:

Synthetic Call = Put + Index

From Figure 13.4, the Put + Index portfolio has the same profits and losses as a call option with an exercise price of $100.

Applying the put-call parity equation to the index example:

Call = Put + Index - Ee-rt

where:

E = exercise price on the index option

An instrument with the same value and profits and losses as a call can be created by holding a long put, long index, and borrowing the present value of the exercise price.

Chapter 13 36

Synthetic Portfolio Insurance and Put-Call Parity

From the put-call parity, there is another way to create a portfolio that exactly mimics the insured portfolio’s value at expiration.

Call + E-rt = Put + Index

We can hold a long call plus investing the present value of the exercise price in the risk-free asset.

Synthetic Calls and Put-Call Parity Synthetic Call = Put + Index Put-Call Parity: Call = Put + Index - E-rt

A synthetic call replicates the profits and losses from the call, but it does not have the same value as the call. The long put/long index/short bond portfolio duplicates the value and profits and losses of the call option.

Chapter 13 37

Risk’s Return on Insured Portfolios

Each of the portfolios A-C has different risk characteristics. To explore the risk properties of the portfolios assume that the return on the index follows a normal distribution with a mean of 10% and a standard deviation of 20%.

Terminal Values for Portfolios A-C.

The portfolio values at expiration depend on the price of the index at expiration. For each, the terminal value is:

Portfolio A = IndexPortfolio B = Index + MAX{0, .5(100.00 - Index)}Portfolio C = Index + MAX{0, 100.00 - Index}

What is the probability that each of the portfolios will have a terminal value equal to or less than $100?

Chapter 13 38

Risk Return in Insured Portfolios

Table 13.7 shows some portfolio values and the probabilities that each portfolio will be equal to or less than the given terminal value at the expiration date.

Table 13.7

Probability That the Terminal Portfolio Value Will Be Equal to or Less than a Specified Value

Probabilities

Terminal Portfolio Value

Uninsured Portfolio A

HalfBInsured Portfolio B

Fully Insured

Portfolio C 50.00 60.00 70.00 80.00 90.00

100.00 110.00 120.00 130.00 140.00 150.00 160.00 170.00

0.0014 0.0062 0.0228 0.0668 0.1587 0.3085 0.5000 0.6915 0.8413 0.9332 0.9773 0.9938 0.9987

0.0000 0.0000 0.0002 0.0062 0.0668 0.3085 0.5000 0.6915 0.8413 0.9332 0.9773 0.9938 0.9987

0.0000 0.0000 0.0000 0.0000 0.0000 0.3085 0.5000 0.6915 0.8413 0.9332 0.9773 0.9938 0.9987

Chapter 13 39


Figure 13.5 graphs the terminal portfolio values from $50 to $170 and shows the probability for each portfolio that the terminal portfolio value will be below or equal to the given amount.


Chapter 13 40


Returns on Portfolios A-C

Table 13.8 shows the probability that each portfolio will achieve a return greater than a specified return.

Table 13.8

Probability of Achieving a Return Equal to or Greater than a Specified Return

Probabilities

Portfolio Return

Uninsured Portfolio A

HalfBInsured Portfolio B

Fully Insured

Portfolio C

-0.5000 -0.4000 -0.3000 -0.2000 -0.1000 0.0000 0.1000 0.2000 0.3000 0.4000 0.5000

0.9987 0.9938 0.9773 0.9332 0.8413 0.6915 0.5000 0.3085 0.1587 0.0668 0.0228

1.0000 1.0000 0.9996 0.9904 0.9066 0.6554 0.4562 0.2676 0.1292 0.0505 0.0158

1.0000 1.0000 1.0000 1.0000 1.0000 0.6179 0.4129 0.2297 0.1038 0.0375 0.0107

This is a tradeoff between return and risk on the portfolios. The portfolios having a higher return also have a higher risk.

Chapter 13 41


Figure 13.6 graphs the probabilities for each portfolio for a range of returns from -50% to 50%.


Chapter 13 42

Why Options on Futures

Some reasons for the popularity of options on futures are:

1. A futures position exposes a trader to a theoretically unlimited risk of gain or loss, but this is not true for the buyer of a futures option.

2. Options on futures dominate options on physicals in some markets because the futures market for some goods is much more liquid than the market for the physical good itself.

3. Options on futures generally require less investment than options on the physical good itself.

Chapter 131 CHAPTER 13 Options on Futures In this chapter, we discuss option on futures contracts....

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