Futures / Forward Options

Read Hull, Chapter 16

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
    • 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
    • Holder exercises 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 August 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 incur 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
  • We regard European spot options and European futures options as the same when we value them 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 F ₀ = 50
    • You do not put money down for the futures (ignoring margin)
  • Up branch gains 3 on futures
  • Lower branch loses 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
  • 3 Δ – 5 = –3 Δ – 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 = 3 Δ – 5 = 3(0.8333) – 5 = –2.5001
  • The value of the portfolio today is
    • PV = 0 Δ – 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 ₀u 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 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
    • S ₀ = current futures price, F ₀
    • 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
  • 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 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 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 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
    • F ₀ N(d ₁) – K N(d ₂)
    • Note: payoff = F ₀ – K
  • The futures price for a put futures-style option is
    • K N(d ₂) – F ₀ N(d ₁)
    • Note: payoff = K – F ₀

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 F ₀e ^-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 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 + 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 ^{-r _fT}

• For future options, set q = r and S ₀ = F ₀

c + Ke ^-rT = p + F ₀ e ^-rT

3. Other Relations

• American Inequality

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