Futures / Forward Options
Read Hull, Chapter 16
Outline
 Mechanics of Futures Options
 Binomial Trees
 Black's Model
 PutCall 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
 Trader receives
 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
 Trader receives
 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 overthecounter markets
Binomial Trees
 Example
 A 2month 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 F_{0} = 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
 Riskfree 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 F_{0} u – F_{0}
 The difference in futures price F_{0} d – F_{0}
 The portfolio is riskless when
 Value of the portfolio at time T is
 FV = F_{0}u Δ –F_{0} Δ – ƒu
 Value of portfolio today is
 PV = – ƒ
 Value of futures is zero at Time 0
 Hence
 PV = FV e^{rT}
 ƒ = – [F_{0} u Δ –F_{0} Δ – ƒ_{u} ]e^{rT}
 Substituting for Δ we obtain
 ƒ = [ p ƒ_{u} + (1 – p )ƒ_{d} ]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 riskneutral 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
 Putcall parity
 Bounds
 Binomial trees
 Valuing European Futures Options
 We can use the formula for an option on a stock paying a dividend yield
 S_{0} = current futures price, F_{0}
 q = domestic riskfree rate, r
 Setting q = r ensures that the expected growth of F in a riskneutral 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
 d_{1} and d_{2} 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 6month European call option on spot gold
 9month futures price is 2,500
 9month riskfree rate is 4%
 Strike price is 2,500
 Volatility of futures price is 25%
 Value of option is given by Black's model with F_{0} = 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 futuresstyle 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 futuresstyle option is
 F_{0} N(d_{1}) – K N(d_{2})
 Note: payoff = F_{0} – K
 The futures price for a put futuresstyle option is
 K N(d_{2}) – F_{0} N(d_{1})
 Note: payoff = K – F_{0}
PutCall Parity
1. Proof: Consider the following two portfolios
 Portfolio A: European call plus Ke^{rT} of cash
 Portfolio C: European put plus long futures plus cash equal to F_{0}e^{rT}
 At maturity, Time T
 F_{T} > K
 Portfolio A
 Exercise call, payout = F_{T}  K
 Cash grows into K
 Total value of Portfolio A = F_{T}
 Portfolio C
 Don't excercise European put, payout = 0
 Value of long futures = F_{T}  F_{0}
 Cash grows into F_{0}
 Total value of Portfolio C = F_{T}
 F_{T} < 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  F_{T}
 Value of long futures = F_{T}  F_{0}
 Cash grows into F_{0}
 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 + F_{0} e^{rT}
2. Generalization of PutCall 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 + S_{0}
 For stock index options that pays q% dividends, leave q alone
c + Ke^{rT} = p + S_{0} e^{qT}
 For currency options, set q = r_{f}
c + Ke^{rT} = p + S_{0} e^{rfT}
 For future options, set q = r and S_{0} = F_{0}
c + Ke^{rT} = p + F_{0} e^{rT}
3. Other Relations
F_{0} e^{rT} – K ≤ C – P ≤ F_{0} – Ke^{rT}
 Lower Bounds for European Call and Put
c ≥ (F_{0} – K)e^{rT}
p ≥ (F_{0} – K)e^{rT}
