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An Introduction to Derivatives and Risk Management, 10th ed.
Chapter 8: Principles of Pricing Forwards, Futures, and Options on Futures
Even if we didn’t believe it for a second, there’s an undeniable adrenaline jab that comes from someone telling you that you’re going to make five hundred million dollars.
Doyne Farmer
Quoted in The Predictors, 1999, Page 119.
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Ch. 8: *
An Introduction to Derivatives and Risk Management, 10th ed.
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Chance/Brooks
An Introduction to Derivatives and Risk Management, 10th ed.
Important Concepts in Chapter 8
- Price and value of forward and futures contracts
- Relationship between forward and futures prices
- Determination of the spot price of an asset
- Carry arbitrage model for theoretical fair price
- Contango, backwardation, and convenience yield
- Futures prices and risk premiums
- Pricing options on futures
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Key Assumptions
- Forward contracts are not subject to margin requirements
- Forward contracts are not centrally cleared
- Forward contracts are not otherwise guaranteed by a third party.
- For forward contracts, the risk of default is so small as to be irrelevant.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 10th ed.
Ch. 8: *
© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
An Introduction to Derivatives and Risk Management, 10th ed.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 10th ed.
Generic Carry Arbitrage
- The Concept of Price Versus Value
- Normally in an efficient market, price = value.
- For a futures or forward, price is the contracted rate of future purchase. Value is something different.
- At the beginning of a contract, value = 0 for both futures and forwards.
- Notation
- Vt(0,T), F(0,T), vt(T), ft(T) are values and prices of forward and futures contracts created at time 0 and expiring at time T.
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An Introduction to Derivatives and Risk Management, 10th ed.
Generic Carry Arbitrage (continued)
- The Value of a Forward Contract
- Forward price at expiration:
- F(T,T) = ST.
- That is, the price of an expiring forward contract is the spot price.
- Value of forward contract at expiration:
- VT(0,T) = ST – F(0,T).
- An expiring forward contract allows you to buy the asset, worth ST, at the forward price F(0,T). The value to the short party is (–1) times this.
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An Introduction to Derivatives and Risk Management, 10th ed.
Generic Carry Arbitrage (continued)
- The Value of a Forward Contract (continued)
- The Value of a Forward Contract Prior to Expiration
- A: Go long forward contract at price F(0,T) at time 0.
- B: At time t go long the asset and take out a loan promising to pay F(0,T) at T
At time T, A and B are worth the same, ST – F(0,T). Thus, they must both be worth the same prior to T.
So Vt(0,T) = St – F(0,T)(1 + r)–(T–t)
See Table 8.1.
- Example: Go long 45 day contract at F(0,T) = $100. Risk-free rate = 0.10. 20 days later, the spot price is $102. The value of the forward contract is 102 – 100(1.10)–25/365 = 2.65.
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An Introduction to Derivatives and Risk Management, 10th ed.
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Chance/Brooks
An Introduction to Derivatives and Risk Management, 10th ed.
Generic Carry Arbitrage (continued)
- The Value of a Futures Contract
- Futures price at expiration:
- fT(T) = ST.
- Value during the trading day but before being marked to market:
- vt(T) = ft(T) – ft–1(T).
- Value immediately after being marked to market:
- vt(T) = 0.
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Chance/Brooks
An Introduction to Derivatives and Risk Management, 10th ed.
Generic Carry Arbitrage (continued)
- Forward Versus Futures Prices
- Forward and futures prices will be equal
- One day prior to expiration
- More than one day prior to expiration if
Interest rates are certain
Futures prices and interest rates are uncorrelated
- Futures prices will exceed forward prices if futures prices are positively correlated with interest rates.
- Default risk can also affect the difference between futures and forward prices.
© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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An Introduction to Derivatives and Risk Management, 10th ed.
Carry Arbitrage: Equities
- Forward and Futures Pricing When the Underlying Generates Cash Flows
- For example, dividends on a stock or index
- Assume one dividend DT paid at expiration.
- Buy stock, sell futures guarantees at expiration that you will have DT + f0(T). Present value of this must equal S0, using risk-free rate. Thus,
f0(T) = S0(1 + r)T – DT.
- For multiple dividends, let DT be compound future value of dividends. See Figure 8.1 for two dividends.
- Dividends reduce the cost of carry.
- If D0 represents the present value of the dividends, the model becomes
f0(T) = (S0 – D0)(1 + r)T.
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An Introduction to Derivatives and Risk Management, 10th ed.
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Chance/Brooks
An Introduction to Derivatives and Risk Management, 10th ed.
Carry Arbitrage: Equities (continued)
- Forward and Futures Pricing When the Underlying Generates Cash Flows (continued)
- For dividends paid at a continuously compounded rate of dc,
- Example: S0 = 50, rc = 0.08, dc = 0.06, expiration in 60 days (T = 60/365 = 0.164).
- f0(T) = 50e(0.08 – 0.06)(0.164) = 50.16.
© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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An Introduction to Derivatives and Risk Management, 10th ed.
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Chance/Brooks
An Introduction to Derivatives and Risk Management, 10th ed.
Carry Arbitrage: Equities (continued)
- Valuation of Equity Forward Contracts
- When there are dividends, to determine the value of a forward contract during its life
- Vt(0,T) = St – Dt,T – F(0,T)(1 + r)–(T–t)
- where Dt,T is the value at time t of the future dividends to time T
- Or if dividends are continuous,
© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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An Introduction to Derivatives and Risk Management, 10th ed.
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An Introduction to Derivatives and Risk Management, 10th ed.
Carry Arbitrage: Currencies
- Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity
- Interest Rate Parity: the relationship between futures or forward and spot exchange rates. Same as carry arbitrage model in other forward and futures markets.
- Proves that one cannot convert a currency to another currency, sell a futures, earn the foreign risk-free rate, and convert back without risk, earning a rate higher than the domestic rate.
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An Introduction to Derivatives and Risk Management, 10th ed.
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Chance/Brooks
An Introduction to Derivatives and Risk Management, 10th ed.
Carry Arbitrage: Currencies (continued)
- Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
- S0 = spot rate in domestic currency per foreign currency. Foreign rate is r. Holding period is T. Domestic rate is r.
- Take S0(1 + r)–T units of domestic currency and buy (1 + r)–T units of foreign currency.
- Sell forward contract to deliver one unit of foreign currency at T at price F(0,T).
- Hold foreign currency and earn rate r. At T you will have one unit of the foreign currency.
- Deliver foreign currency and receive F(0,T) units of domestic currency.
© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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An Introduction to Derivatives and Risk Management, 10th ed.
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Chance/Brooks
An Introduction to Derivatives and Risk Management, 10th ed.
Carry Arbitrage: Currencies (continued)
- Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
- So an investment of S0(1 + r)–T units of domestic currency grows to F (0,T) units of domestic currency with no risk. Return should be r. Therefore
F(0,T) = S0(1 + r)–T(1 + r)T
- This is called interest rate parity.
- Sometimes written as
F(0,T) = S0(1 + r)T/(1 + )T
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An Introduction to Derivatives and Risk Management, 10th ed.
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Chance/Brooks
An Introduction to Derivatives and Risk Management, 10th ed.
Carry Arbitrage: Currencies (continued)
- Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
- Example (from a European perspective): S0 = €1.0304.
U. S. rate is 5.84%. Euro rate is 3.59%. Time to expiration is 90/365 = 0.2466. - F(0,T) = €1.0304(1.0584)–0.2466(1.0359)0.2466 = €1.025
- If forward rate is actually €1.03, then it is overpriced.
- Buy (1.0584)–0.2466 = $0.9861 for 0.9861(€1.0304) = €1.0161. Sell one forward contract at €1.03.
- Earn 5.84% on $0.9861. This grows to $1.
- At expiration, deliver $1 and receive €1.03.
- Return is (1.03/1.0161)365/90 – 1 = 0.0566 (> 0.0359)
- This transaction is called covered interest arbitrage.
© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Ch. 8: *
An Introduction to Derivatives and Risk Management, 10th ed.
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Chance/Brooks
An Introduction to Derivatives and Risk Management, 10th ed.
Carry Arbitrage: Currencies (continued)
- Pricing Foreign Currency Forward and Futures Contracts: Interest Rate Parity (continued)
- It is also sometimes written as
- F(0,T) = S0(1 + )T(1 + r)–T
- Here, the spot rate is being quoted in units of the foreign currency.
- Note that the forward discount/premium has nothing to do with expectations of future exchange rates.
- Difference between domestic and foreign rate is analogous to difference between risk-free rate and dividend yield on stock index futures.
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Chance/Brooks
An Introduction to Derivatives and Risk Management, 10th ed.
Pricing Models and Risk Premiums
- Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets
- First assume no uncertainty of future price. Let s be the cost of storing an asset and i be the interest rate for the period of time the asset is owned. Then
- S0 = ST – s – iS0
- If we now allow uncertainty but assume people are risk neutral, we have
- S0 = E(ST) – s – iS0
- If we now allow people to be risk averse, they require a risk premium of E(). Now
- S0 = E(ST) – s – iS0 – E()
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An Introduction to Derivatives and Risk Management, 10th ed.
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Chance/Brooks
An Introduction to Derivatives and Risk Management, 10th ed.
Pricing Models and Risk Premiums
- Spot Prices, Risk Premiums, and the Carry Arbitrage for Generic Assets (continued)
- Let us define iS0 as the net interest, which is the interest foregone minus any cash received.
- Define s + iS0 as the cost of carry.
- Denote cost of carry as .
- Note how cost of carry is a meaningful concept only for storable assets
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An Introduction to Derivatives and Risk Management, 10th ed.
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Chance/Brooks
An Introduction to Derivatives and Risk Management, 10th ed.
Pricing Models and Risk Premiums
- The Theoretical Fair Price (Forward/Futures Pricing Revisited)
- Do the following
- Buy asset in spot market, paying S0; sell futures contract at price f0(T); store and incur costs.
- At expiration, make delivery. Profit:
P = f0(T) – S0 – q
- This must be zero to avoid arbitrage; thus,
f0(T) = S0 + q
- See Figure 8.2.
- Note how arbitrage and quasi-arbitrage make this hold.
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An Introduction to Derivatives and Risk Management, 10th ed.
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Chance/Brooks
An Introduction to Derivatives and Risk Management, 10th ed.
Pricing Models and Risk Premiums
- Forward/Futures Pricing Revisited (continued)
- See Figure 8.3 for an illustration of the determination of futures prices.
- Contango is f0(T) > S0. See Table 8.2.
- When f0(T) < S0, convenience yield is c , an additional return from holding asset when in short supply or a non-pecuniary return. Market is said to be at less than full carry and in backwardation or inverted.
See Table 8.3. Market can be both backwardation and contango. See Table 8.4.
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An Introduction to Derivatives and Risk Management, 10th ed.
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Chance/Brooks
An Introduction to Derivatives and Risk Management, 10th ed.
Pricing Models and Risk Premiums
- Futures Prices and Risk Premia
- The no risk-premium hypothesis
- Market consists of only speculators.
- f0(T) = E(ST). See Figure 8.4.
- The risk-premium hypothesis
- E(fT(T)) > f0(T).
- When hedgers go short futures, they transfer risk premium to speculators who go long futures.
- E(ST) = f0(T) + E(f). See Figure 8.5.
- Normal contango: E(ST) < f0(T)
- Normal backwardation: f0(T) < E(ST)
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An Introduction to Derivatives and Risk Management, 10th ed.
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Chance/Brooks
An Introduction to Derivatives and Risk Management, 10th ed.
Put-Call-Forward/Futures Parity
- Can construct synthetic futures with options.
- See Table 8.5.
- Put-call-forward/futures parity
- Pe(S0,T,X) = Ce(S0,T,X) + [X – f0(T)](1 + r)–T
- Numerical example using S&P 500. On May 14, S&P 500 at 1337.80 and June futures at 1339.30. June 1340 call at 40 and put at 39. Expiration of June 18 so
T = 35/365 = 0.0959. Risk-free rate at 4.56%.
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An Introduction to Derivatives and Risk Management, 10th ed.
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Chance/Brooks
An Introduction to Derivatives and Risk Management, 10th ed.
Put-Call-Forward/Futures Parity (continued)
- So Pe(S0,T,X) = 39
- Ce(S0,T,X) + [X – f0(T)](1 + r)–T
- = 40 + (1340 – 1339.30)(1.0456)–0.0959 = 40.70.
- Buy put and futures for 39, sell call and bond for 40.70 and net 1.70 profit at no risk. Transaction costs would have to be considered.
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An Introduction to Derivatives and Risk Management, 10th ed.
Pricing Options on Futures
- The Intrinsic Value of an American Option on Futures
- Minimum value of American call on futures
- Ca(f0(T),T,X) ³ Max[0, f0(T) – X]
- Minimum value of American put on futures
- Pa(f0(T),T,X) ³ Max[0, X – f0(T)]
- Difference between option price and intrinsic value is time value.
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Chance/Brooks
An Introduction to Derivatives and Risk Management, 10th ed.
Pricing Options on Futures (continued)
- The Lower Bound of a European Option on Futures
- For calls, construct two portfolios.
See Table 8.6. - Portfolio A dominates Portfolio B so
- Ce(f0(T),T,X) ³ Max{0, [f0(T) – X](1 + r)–T}
- Note that lower bound can be less than intrinsic value even for calls.
- For puts, see Table 8.7.
- Portfolio A dominates Portfolio B so
- Pe(f0(T),T,X) ³ Max{0, [X – f0(T)](1 + r)–T}
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An Introduction to Derivatives and Risk Management, 10th ed.
Pricing Options on Futures (continued)
- Put-Call Parity of Options on Futures
- Construct two portfolios, A and B.
- See Table 8.8.
- The portfolios produce equivalent results. Therefore they must have equivalent current values. Thus,
- Pe(f0(T),T,X) = Ce(f0(T),T,X) + [X – f0(T)](1 + r)–T.
- Compare to put-call parity for options on spot:
- Pe(S0,T,X) = Ce(S0,T,X) – S0 + X(1 + r)–T.
- If options on spot and options on futures expire at same time, their values are equal, implying
f0(T) = S0(1 + r)T, which we obtained earlier (no cash flows).
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An Introduction to Derivatives and Risk Management, 10th ed.
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An Introduction to Derivatives and Risk Management, 10th ed.
Pricing Options on Futures (continued)
- Early Exercise of Call and Put Options on Futures
- Deep in-the-money call may be exercised early because
- behaves almost identically to futures
- exercise frees up funds tied up in option but requires no funds to establish futures
- minimum value of European futures call is less than value if it could be exercised
- See Figure 8.6.
- Similar arguments hold for puts
- Compare to the arguments for early exercise of call and put options on spot.
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Chance/Brooks
An Introduction to Derivatives and Risk Management, 10th ed.
Pricing Options on Futures (continued)
- Options on Futures Pricing Models
- Black model for pricing European options on futures
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Chance/Brooks
An Introduction to Derivatives and Risk Management, 10th ed.
Pricing Options on Futures (continued)
- Options on Futures Pricing Models (continued)
- Note that with the same expiration for options on spot as options on futures, this formula gives the same price.
- Example
- See Table 8.9.
- Software for Black-Scholes-Merton can be used by inserting futures price instead of spot price and risk-free rate for dividend yield. Note why this works.
- For puts
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An Introduction to Derivatives and Risk Management, 10th ed.
See Table 8.10 for a summary of equations.
See Figure 8.7 for linkage between forwards/futures, underlying asset and risk-free bond.
Summary
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)
(T)N(d
[f
e
C
1
2
2
0
1
2
1
0
T
r
c
s
s
s
-
=
+
=
-
=
-
)]
N(d
[1
(T)e
f
)]
N(d
[1
Xe
P
1
T
r
0
2
T
r
c
c
-
-
-
=
-
-
