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
      • 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 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

Binomial tree for futures option

  • 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

Binomial tree for futures option

  • 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

Binomial tree for futures option

  • The portfolio is riskless when

Binomial tree for futures option

  • 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

probability for option on futures

  • 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

Black's equations

  • 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

 

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