# Outline

• Mechanics of Futures Options
• Binomial Trees
• Black's Model
• Put-Call Parity

# Mechanics of Futures Options

• Option on Futures
• Referred to by the maturity month of the underlying futures
• Usually an American option that expires on or a few days before the earliest delivery date of the underlying futures contract
• Call on a futures
• When a holder exercises a call futures, the holder acquires
• A long position in the futures
• A cash amount equal to the excess of the futures price at the most recent settlement over the strike price
• If the futures position is closed out immediately
• Payout from call = F – K
• F is futures price at time of exercise
• K is strike price
• Example - Call futures
• June call option contract on petroluem futures has a strike price of \$105 per barrel
• Holder exercises when futures price is \$115 and most recent settlement is \$113
• One contract is on 1,000 barrels
• Long June futures contract on petroluem
• Cash = ( \$113 – 105 ) x 1,000 = \$8,000
• Put on a futures
• When a holder exercises a put futures option, the holder acquires
• A short position in the futures
• A cash amount equal to the excess of the strike price over the futures price at the most recent settlement
• If the futures position is closed out immediately
• Payout from put = K – F
• F is futures price at time of exercise
• K is strike price
• Example - Put futures
• August put option contract on soybean futures has a strike price of 970 cents per bushel
• It is exercised when the futures price is 950 cents per bushel and the most recent settlement price is 948 cents per bushel
• One contract is for 5,000 bushels
• Short Sept futures contract on soybeans
• Cash = ( \$9.70 –\$9.48 ) x 5,000 = \$1,100
• Potential Advantages of Futures Options over Spot Options
• Futures contract may be easier to trade than underlying asset
• Exercise of the option does not lead to delivery of the underlying asset
• Unlike the spot market, the investor can enter a new contracts and not take delivery of the assets
• Futures options and futures usually trade on the same exchange
• Futures options may entail lower transactions costs
• European Futures Options
• European futures options and spot options are equivalent when futures contract matures at the same time as the option
• It is common to regard European spot options as European futures options when they are valued in the over-the-counter markets

# Binomial Trees

• Example
• A 2-month call option on futures has a strike price of 48
• Since you do not have to put money down for a futures, you only look at gains and losses
• You start at 0 for futures, when F0 = 50
• You do not put money down for the futures (ignoring margin)
• Up branch gains 3 on futures
• Lower branch losses 3 on futures

• Since you do not have to put money down for a futures, you only look at gains and losses
• You start at 0 for futures
• Up branch gains 3 on futures
• Lower branch losses 3 on futures

• Consider the Portfolio:
• Long Δ futures
• Short 1 call option
• Portfolio is riskless when
• 3Δ – 5 = –3Δ – 0
• Δ = 0.8333
• Valuing the Portfolio
• Risk-free interest rate is 4%
• The riskless portfolio is:
• Long 0.8333 futures
• Short 1 call option
• The value of the portfolio in 1 month is
• FV = 3Δ – 5 = 3(0.8333) – 5 = –2.5001
• The value of the portfolio today is
• PV = 0Δ – f
• PV = FV e-rT
• –f = –2.5001e–0.04(2/12)
• f = 2.483
• Generalization of Binomial Tree Example
• Consider the portfolio that is long Δ futures and short 1 derivative
• The difference in futures price F0 u – F0
• The difference in futures price F0 d – F0

• The portfolio is riskless when

• Value of the portfolio at time T is
• FV = F0u Δ –F0 Δ – ƒu
• Value of portfolio today is
• PV = – ƒ
• Value of futures is zero at Time 0
• Hence
• PV = FV e-rT
• ƒ = – [F0 u Δ –F0 Δ – ƒu ]e-rT
• Substituting for Δ we obtain
• ƒ = [ p ƒu + (1 – p )ƒd ]e–rT
• Binomial distribution
• Take probability times the outcome
• Sum over all outcomes
• Where

• Using the previous example
• u = 53 / 50 = 1.06
• d = 47 / 50 = 0.94
• p = ( 1 – 0.94 ) / ( 1.06 – 0.94 ) = 0.5
• 1 – p = 0.5
• discount = e–0.04x2/12 = 0.9934
• f = [ 0.5 (5) + 0.5 (0) ] x 0.9934 = 2.4835
• Growth Rates For Futures Prices
• A futures contract requires no initial investment
• In a risk-neutral world, the expected return should be zero
• The expected growth rate of the futures price is therefore zero
• The futures price can therefore be treated like a stock paying a dividend yield of r
• This is consistent with the results we have presented so far
• Put-call parity
• Bounds
• Binomial trees
• Valuing European Futures Options
• We can use the formula for an option on a stock paying a dividend yield
• S0 = current futures price, F0
• q = domestic risk-free rate, r
• Setting q = r ensures that the expected growth of F in a risk-neutral world is zero
• Referred to as Black's model Fischer Black suggested this model in a paper in 1976

# Black's Model

• The formulas for European options on futures are known as Black's model
• Notice two things
• d1 and d2 have no r terms
• Each term for c and p is discounted by r

• Using Black's Model in practice
• European futures options and spot options are equivalent when future contract matures at the same time as the option.
• Thus, Black's model can value a European option on the spot price of an asset
• One advantage is we do not have to estimate income on the asset explicitly
• Example
• Consider a 6-month European call option on spot gold
• 9-month futures price is 2,500
• 9-month risk-free rate is 4%
• Strike price is 2,500
• Volatility of futures price is 25%
• Value of option is given by Black's model with F0 = 2,500, K = 2,500, r = 0.04, T = 9/12, and σ = 0.25
• c = 209.1435
• p = 209.1435
• American Futures Option Prices vs American Spot Option Prices
• If futures prices exceed spot prices (normal market), an American call on futures is worth more than a similar American call on spot.
• An American put on futures is worth less than a similar American put on spot
• When futures prices are lower than spot prices (inverted market) the reverse is true
• Futures Style Options
• A futures-style option is a futures contract on the option payoff
• Some exchanges trade these in preference to regular futures options
• The futures price for a call futures-style option is
• F0 N(d1) – K N(d2)
• Note: payoff = F0 – K
• The futures price for a put futures-style option is
• K N(d2) – F0 N(d1)
• Note: payoff = K – F0

# Put-Call Parity

1. Proof: Consider the following two portfolios

1. Portfolio A: European call plus Ke-rT of cash
2. Portfolio C: European put plus long futures plus cash equal to F0e-rT
• At maturity, Time T
• FT > K
• Portfolio A
• Exercise call, payout = FT - K
• Cash grows into K
• Total value of Portfolio A = FT
• Portfolio C
• Don't excercise European put, payout = 0
• Value of long futures = FT - F0
• Cash grows into F0
• Total value of Portfolio C = FT
• FT < K
• Portfolio A
• Don't exercise call, payout = 0
• Cash grows into K
• Total value of Portfolio A = K
• Portfolio C
• Exercise put, payout = K - FT
• Value of long futures = FT - F0
• Cash grows into F0
• Total value of Portfolio C = K
• Thus, both portfolios have same value at time T so that
• Futures contract in Portfolio C has a initial value of zero

c + Ke-rT = p + F0 e-rT

2. Generalization of Put-Call Parity

• We can treat stock indices, currencies, & futures like a stock paying a continuous dividend yield of q
• For plain vanilla options, set q = 0

c + Ke-rT = p + S0

• For stock index options that pays q% dividends, leave q alone

c + Ke-rT = p + S0 e-qT

• For currency options, set q = rf

c + Ke-rT = p + S0 e-rfT

• For future options, set q = r and S0 = F0

c + Ke-rT = p + F0 e-rT

3. Other Relations

• American Inequality

F0 e-rT – K ≤ C – P ≤ F0 – Ke-rT

• Lower Bounds for European Call and Put

c ≥ (F0 – K)e-rT

p ≥ (F0 – K)e-rT