Fundamentals of Finance
Lecture 14: Forwards and Futures I
Vevox Session 154-946-202
IB2660 Fundamentals of Finance, Term 1, 2022
Warwick Business School, University of Warwick
Rory Mullen, Assistant Professor of Finance
Version 1.7
Term Schedule
Week Material
01: Oct 03 Lecture 01: Math Refresher (Optional Self-Study), Lecture 02: Investment Under Certainty, Lecture 03: Risk and Expected Return
Seminar: No Seminars, please read math refresher
Reading: Stitz and Zeager (2013, optional, Chapters 1.3–1.4, 6.1-6.2, and 9.1-9.2), Hillier et al. (2016, optional, Chapters 4.1–4.4 and 5.1–5.3), Hartman et al. (2018, optional, Chapters 1.1–1.3 and
2.1-2.6), Diez et al. (2015, optional, Chapters 2.4–2.6 and 7.1–7.2); Hillier et al. (2016, Appendix 4A); Hillier et al. (2016, Chapters 9.1–9.6), Bodie et al. (2014, Chapters 5.3–5.5 and 18.2)
02: Oct 10 Lecture 04: Risk Aversion and Expected Utility I, Lecture 05: Risk Aversion and Expected Utility II
Seminar: Problem Set 01 (Financial Arithmetic, Investment under Certainty)
Reading: Bodie et al. (2014, Chapter 6.1); Bodie et al. (2014, Chapter 6.1)
03: Oct 17 Lecture 06: Optimal Portfolio Selection I, Lecture 07: Optimal Portfolio Selection II
Seminar: Problem Set 02 (Uncertainty and Risk Aversion)
Reading: Hillier et al. (2016, Chapters 10.1–10.7), Bodie et al. (2014, Chapters 6.2–6.6, 7.1–7.4, and 8.1-8.2); Hillier et al. (2016, Chapters 10.1–10.7), Bodie et al. (2014, Chapters 6.2–6.6, 7.1–7.4, and
8.1-8.2), Solnik (1974, optional), Elton et al. (2011)
04: Oct 24 Lecture 08: Capital Asset Pricing Model I, Lecture 09: Capital Asset Pricing Model II
Seminar: Problem Set 03 (Optimal Portfolio Selection)
Reading: Hillier et al. (2016, Chapters 10.8–10.10, and 12.1–12.3), Bodie et al. (2014, Chapters 9.1–9.4), Fama and French (1992); Hillier et al. (2016, Chapters 10.8–10.10, and 12.1–12.3), Bodie et al.
(2014, Chapters 9.1–9.4)
05: Oct 31 Lecture R1: Review of Part I
Seminar: Problem Set 04 (Capital Asset Pricing Model)
06: Nov 07 Lecture 10: Market Efficiency I, Lecture 11: Market Efficiency II
Seminar: Review Problem Sets 01–04
Reading: Hillier et al. (2016, Chapters 13.1–13.3, and 13.5), Bodie et al. (2014, 11.1 and 11.2); Hillier et al. (2016, Chapters 13.4, and 13.6–13.8), Bodie et al. (2014, pp. 11.3–11.5), Basu (1977), Carhart
(1997), Fama, Fisher, et al. (1969), Jensen (1968), Rendleman et al. (1982), Jegadeesh and Titman (1993)
07: Nov 14 Lecture 12: Bond Pricing I, Lecture 13: Bond Pricing II
Seminar: Problem Set 05 (Market Efficiency)
Reading: Hillier et al. (2016, Chapters 5.1–5.3, and Appendix 5A), Bodie et al. (2014, 5.1, 15.1–15.5, and 16.1); Hillier et al. (2016, Chapters 5.1–5.3, and Appendix 5A), Bodie et al. (2014, 5.1, 15.1–15.5,
and 16.1)
08: Nov 21 Lecture 14: Forwards and Futures I, Lecture 15: Forwards and Futures II
Seminar: Problem Set 06 (Bond Pricing)
Reading: Hillier et al. (2016, Chapters 25.1–25.3), Bodie et al. (2014, Chapters 22.1,22.2, 22.4), Hull (2015, pp. 1.1–1.10, 2.1–2.7, 3.1–3.2, 5.1–5.8); Hillier et al. (2016, Chapters 25.1–25.3), Bodie et al.
(2014, Chapters 20.1, 20.2, 20.4), Hull (2015, 1.1–1.10, 2.1–2.7, 3.1–3.2, and 5.1–5.8)
09: Nov 28 Lecture 16: Options I, Lecture 17: Options II
Seminar: Problem Set 07 (Forwards and Futures)
Reading: Hillier et al. (2016, Chapters 22.1–22.8), Bodie et al. (2014, Chapters 21.1–21.3), Hull (2015, Chapters 10.1–10.5, 11.1–11.7, 13.1–13.6); Hillier et al. (2016, Chapters 22.1–22.8), Bodie et al.
(2014, Chapters 20.1, 20.2, 20.4, and 21.1–21.3), Hull (2015, Chapters 10.1–10.5, 11.1–11.7, 13.1–13.6)
10: Dec 05 Lecture R2: Review of Part II
Seminar: Problem Set 08 (Options)
Fundamentals of Finance
Table of Contents
Lecture 01: Math Refresher (Optional Self-Study) Lecture 10: Market Efficiency I
Lecture 02: Investment Under Certainty Lecture 11: Market Efficiency II
Lecture 03: Risk and Expected Return Lecture 12: Bond Pricing I
Lecture 04: Risk Aversion and Expected Utility I Lecture 13: Bond Pricing II
Lecture 05: Risk Aversion and Expected Utility II Lecture 14: Forwards and Futures I
Lecture 06: Optimal Portfolio Selection I Lecture 15: Forwards and Futures II
Lecture 07: Optimal Portfolio Selection II Lecture 16: Options I
Lecture 08: Capital Asset Pricing Model I Lecture 17: Options II
Lecture 09: Capital Asset Pricing Model II Lecture R2: Review of Part II
Lecture R1: Review of Part I
Lecture 14: Forwards and Futures I
Overview of Topics
14.1. Forwards and futures
14.2. Payoff diagrams
14.3. Spot-forward parity
14.4. Forward price arbitrage*
Reading: Hillier et al. (2016, Chapters 25.1–25.3), Bodie et al. (2014, Chapters 22.1,22.2, 22.4), Hull (2015, pp. 1.1–1.10, 2.1–2.7, 3.1–3.2, 5.1–5.8)
Note: Sections marked by * are covered in prerecorded videos or reserved for self-study. All material is examinable unless otherwise stated.
Forward and future contracts let investors hedge risks
Definition 1
Forward Binding agreements to buy or sell a fixed number of units
and future of an underlying asset at a future expiry date at a price
contracts: specified upfront when the contract is agreed. No money
changes hands at the time the contract is agreed.
Remarks:
I Forwards & futures contracts are derivatives: prices depends on underlying assets
I Notation: payoff depends on forward / future price F0 and underlying spot price ST
I Forwards and futures let investors hedge price, interest-rate, exchange-rate risks
1/15
Forwards versus Futures
Future Contract Forward Contract
Trade: organized exchange over-the-counter
Contract: standardized tailored
Margin: standard negotiated
Risk: clearinghouse guarantee couterparty default risk
Settlement: daily mark-to-market cash flows occur only at expiry
Regulation: NFA and CFTC variable regulation
Remarks:
I A wheat farmer and a miller might negotiate a custom forward contract on wheat price
I Futures markets help to standardize the type of forward arrangement described above
2/15
We will find no-arbitrage prices for forwards and futures
Definition 2
No- The theoretical price of an asset that is implied by the
arbitrage assumption that the asset cannot be combined with a
price: portfolio of other assets to lock in a riskless profit.
Remarks:
I Arbitrage opportunities arise when assets are mispriced, as we saw in bond pricing
I To profit, go long in the under-priced asset, and go short in the over-priced asset
I As a first approximation, we assume that riskless arbitrage profits cannot be found
I This absence of arbitrage assumption will allow us to price many types of derivative
3/15
Long Forward: a numerical example
Question 1 154-946-202
On 1 Sept, the crude oil spot price is S0 = $48.00/barrel, and you agree to buy
1 barrel of crude oil on 1 Dec at forward price F0 = $50.00. On 1 Dec, the spot
price is ST = $52.00, you take physical delivery of 1 barrel of crude oil, and pay
F0 in cash to the seller. What is your payoff?
A. F0 − S0 = $50 − $48 = $2
B. ST − F0 = $52 − $50 = $2
C. F0 − ST = $50 − $52 = −$2
I Payoff equals difference between spot rate at expiry ST and forward rate F0
I Long forwards act as a hedge against the possibility of rising oil spot prices
I Soon we’ll develop a way to finding the forward price F0 using no-arbitrage
4/15
Long Forward: a numerical example
Question 1 154-946-202
On 1 Sept, the crude oil spot price is S0 = $48.00/barrel, and you agree to buy
1 barrel of crude oil on 1 Dec at forward price F0 = $50.00. On 1 Dec, the spot
price is ST = $52.00, you take physical delivery of 1 barrel of crude oil, and pay
F0 in cash to the seller. What is your payoff?
A. F0 − S0 = $50 − $48 = $2
B. ST − F0 = $52 − $50 = $2
C. F0 − ST = $50 − $52 = −$2
I Payoff equals difference between spot rate at expiry ST and forward rate F0
I Long forwards act as a hedge against the possibility of rising oil spot prices
I Soon we’ll develop a way to finding the forward price F0 using no-arbitrage
4/15
Long Forward: a numerical example
Solution 1 154-946-202
By entering a forward agreement to buy an asset (oil) at a forward price F0 = $50
that is agreed today and paid at time T , you are taking a long position.
The payoff to your long forward at time T will equal the value of the underlying
asset ST minus the forward price F0 that you have agreed:
Long Forward Payoff = ST − F0 = $52 − $50 = $2 .
What determines the forward price F0 ? We will show that it should equal a
so-called no-arbitrage price. . .
4/15
Lecture 14: Forwards and Futures I
Overview of Topics
14.1. Forwards and futures
14.2. Payoff diagrams
14.3. Spot-forward parity
14.4. Forward price arbitrage*
Reading: Hillier et al. (2016, Chapters 25.1–25.3), Bodie et al. (2014, Chapters 22.1,22.2, 22.4), Hull (2015, pp. 1.1–1.10, 2.1–2.7, 3.1–3.2, 5.1–5.8)
Note: Sections marked by * are covered in prerecorded videos or reserved for self-study. All material is examinable unless otherwise stated.
Payoff at expiry date T
Long forward (or futures) position:
I holder pays the forward price F0 at expiry in cash
I the holder of long position pays the holder of the short position
Short forward (or futures) position:
I holder either pays underlying spot price ST at expiry in cash (cash-settled)
I or delivers underlying asset (physically settled) to holder of long position
Remarks:
I Payoff to long forward (or futures) position (= buyer of underlying asset)
increases with increases in the spot price ST of the underlying asset
I Payoff to short forward (or futures) position (= seller of underlying asset)
decreases with increases in the spot price ST of the underlying asset
5/15
Payoff diagram for long and short forward positions
I Payoff diagrams show how payoffs
depend on underlying asset prices
Payoff
0
F0 Spot Price ST
6/15
Payoff diagram for long and short forward positions
I Payoff diagrams show how payoffs
Long Forward
ST − F0 depend on underlying asset prices
I The long forward position yields a
payoff of ST − F0
Payoff
0
F0 Spot Price ST
6/15
Payoff diagram for long and short forward positions
I Payoff diagrams show how payoffs
Short Forward Long Forward
F0 − ST ST − F0 depend on underlying asset prices
I The long forward position yields a
payoff of ST − F0
I The short forward position yields a
Payoff
0 payoff of F0 − ST
F0 Spot Price ST
6/15
Payoff diagram for long and short forward positions
I Payoff diagrams show how payoffs
Short Forward Long Forward
F0 − ST ST − F0 depend on underlying asset prices
I The long forward position yields a
payoff of ST − F0
I The short forward position yields a
Payoff
0 payoff of F0 − ST
F0 Spot Price ST
I The payoffs to the holders of long and
short forward positions are symmetric
I If spot price at expiry equals forward
price, the payoff to both parties is zero
6/15
Lecture 14: Forwards and Futures I
Overview of Topics
14.1. Forwards and futures
14.2. Payoff diagrams
14.3. Spot-forward parity
14.4. Forward price arbitrage*
Reading: Hillier et al. (2016, Chapters 25.1–25.3), Bodie et al. (2014, Chapters 22.1,22.2, 22.4), Hull (2015, pp. 1.1–1.10, 2.1–2.7, 3.1–3.2, 5.1–5.8)
Note: Sections marked by * are covered in prerecorded videos or reserved for self-study. All material is examinable unless otherwise stated.
Spot-forward parity and the no-arbitrage condition (1/3)
How should the fair forward price F0 be determined?
Suppose we implement the following arbitrage strategy:
1. Borrow S0 dollars today
2. Buy stock for spot price S0
3. Enter short forward contract with forward price F0
Remarks:
I No-arbitrage condition: the cash flow in t = T must equal zero
I Why? The cash outflow at any time 0 ≤ t < T for this strategy is zero
7/15
Spot-forward parity and the no-arbitrage condition (2/3)
Action Cash Flow at t = 0 Cash Flow at t = T
Remarks:
I Spot-forward parity: $0 net cash flow at t = 0 impies $0 net cash flow at t = T
I Be careful: the current spot price is denoted S0 , the spot price at expiry is ST
I We compound in continuous time—this is common and helpful in derivative pricing
8/15
Spot-forward parity and the no-arbitrage condition (2/3)
Action Cash Flow at t = 0 Cash Flow at t = T
1. Borrow S0 dollars: S0 −S0 eRT
Remarks:
I Spot-forward parity: $0 net cash flow at t = 0 impies $0 net cash flow at t = T
I Be careful: the current spot price is denoted S0 , the spot price at expiry is ST
I We compound in continuous time—this is common and helpful in derivative pricing
8/15
Spot-forward parity and the no-arbitrage condition (2/3)
Action Cash Flow at t = 0 Cash Flow at t = T
1. Borrow S0 dollars: S0 −S0 eRT
2. Buy stock for S0 : −S0 ST
Remarks:
I Spot-forward parity: $0 net cash flow at t = 0 impies $0 net cash flow at t = T
I Be careful: the current spot price is denoted S0 , the spot price at expiry is ST
I We compound in continuous time—this is common and helpful in derivative pricing
8/15
Spot-forward parity and the no-arbitrage condition (2/3)
Action Cash Flow at t = 0 Cash Flow at t = T
1. Borrow S0 dollars: S0 −S0 eRT
2. Buy stock for S0 : −S0 ST
3. Short forward at F0 : 0 F0 − S T
Remarks:
I Spot-forward parity: $0 net cash flow at t = 0 impies $0 net cash flow at t = T
I Be careful: the current spot price is denoted S0 , the spot price at expiry is ST
I We compound in continuous time—this is common and helpful in derivative pricing
8/15
Spot-forward parity and the no-arbitrage condition (2/3)
Action Cash Flow at t = 0 Cash Flow at t = T
1. Borrow S0 dollars: S0 −S0 eRT
2. Buy stock for S0 : −S0 ST
3. Short forward at F0 : 0 F0 − S T
Total cash flows: 0 F0 − S0 eRT
Remarks:
I Spot-forward parity: $0 net cash flow at t = 0 impies $0 net cash flow at t = T
I Be careful: the current spot price is denoted S0 , the spot price at expiry is ST
I We compound in continuous time—this is common and helpful in derivative pricing
8/15
Spot-forward parity and the no-arbitrage condition (3/3)
F0 − S0 eRT = 0 ⇒ F0 = S0 eRT
Remarks:
I Spot-forward parity: $0 net cash flow at t = 0 impies $0 net cash flow at t = T
I A $0 investment today yielding positive cash flow in T would constitute arbitrage
I At no initial cost, you can buy stock with borrowed money and enter a forward
I If this trade is costless today, it cannot yield positive cash flows tomorrow
9/15
Numerical example: spot-forward parity
Question 2 154-946-202
You want to enter a forward agreement to purchase in 6 months at forward price
F0 a single share in a firm with current spot price S0 = $15.50. The risk free rate
is 2%. What is the fair price of the forward contract, and would your position be
long or short?
A. 15.35, short
B. 15.65, long
10/15
Numerical example: spot-forward parity
Question 2 154-946-202
You want to enter a forward agreement to purchase in 6 months at forward price
F0 a single share in a firm with current spot price S0 = $15.50. The risk free rate
is 2%. What is the fair price of the forward contract, and would your position be
long or short?
A. 15.35, short
B. 15.65, long
10/15
Numerical example: spot-forward parity
Solution 2 154-946-202
By entering a forward agreement to purchase an asset at a forward price F0 that
is agreed today and paid at time T , you are taking a long position.
The no-arbitrage forward price satisfies the spot-forward parity, which states that
F0 = ST eRT
= 15.5e0.02×0.5 ≈ 15.65 .
If the forward price violates no-arbitrage condition, you could borrow funds and
buy the underlying to replicate the forward payoff in T and earn arbitrage profit.
10/15
Aside on continuous compounding
Write the T -year compound factor CnT with compound frequency n and yield R as
CnT = (1 + R/n)nT .
For continuous compounding, let 1/m := R/n and take the limit of n → ∞:
lim CnT = lim (1 + R/n)nt
n→∞ n→∞
= lim (1 + 1/m)mRT
m→∞
RT
= lim (1 + 1/m)m = eRT .
m→∞
Remarks:
I Last line of derivation uses the definition of Euler’s number e := limm→∞ (1 + 1/m)m
I Continuous compounding is often mathematically more convenient than discrete
Not Examinable 11/15
Lecture 14: Forwards and Futures I
Overview of Topics
14.1. Forwards and futures
14.2. Payoff diagrams
14.3. Spot-forward parity
14.4. Forward price arbitrage*
Reading: Hillier et al. (2016, Chapters 25.1–25.3), Bodie et al. (2014, Chapters 22.1,22.2, 22.4), Hull (2015, pp. 1.1–1.10, 2.1–2.7, 3.1–3.2, 5.1–5.8)
Note: Sections marked by * are covered in prerecorded videos or reserved for self-study. All material is examinable unless otherwise stated.
Forward over-pricing and arbitrage opportunities (1/3)
Suppose now F0 = $16, but all else as before: S0 = $15.50, T = 0.5, R = 0.02. The forward
is overvalued relative to the fair price of $15.65. Arbitrage strategy:
1. Borrow S0 = $15.50 dollars today
2. Buy stock for spot price S0 = $15.50
3. Enter short forward contract with forward price F0 = $16
Cash flow at t = 0: $0.00
Cash flow at t = T : $0.35 = $16 − $15.50e0.02·0.5
Remarks:
I Note that the cash outflow at time t = 0 for this strategy is zero
I No-arbitrage condition is violated: the cash flow in t = T exceeds zero
12/15
Forward under-pricing and arbitrage opportunities (2/3)
Suppose now F0 = $14, but all else as before: S0 = $15.50, T = 0.5, R = 0.02. The forward
is undervalued relative to the fair price of $15.65. Arbitrage strategy:
1. Lend S0 = $15.50 dollars today
2. Short stock for spot price S0 = $15.50
3. Enter long forward contract with forward price F0 = $14
Cash flow at t = 0: $0.00
Cash flow at t = T : $1.66 = $15.50e0.02·0.5 − $14.00
Remarks:
I Because the forward is underpriced, the arbitrage strategy is exactly reversed
I The principle at work: buy low, sell high; here the forward price is low so you buy
13/15
Forward mis-pricing and arbitrage opportunities (3/3)
Buy low
Sell high
Remarks:
I In the first example, the forward price F0 was too high, so we went short (sold)
I In the second example, the forward price F0 was too low, so we went long (bought)
I If you can perfectly identify mis-priced assets, this strategy yields riskless profit
I Investors exploit it until prices (here, F0 ) adjust to eliminate the arbitrage opportunity
14/15
The spot-forward parity and the expectations hypothesis
Spot-Forward Parity: F0 = S0 eRT
Efficient Markets: F0 = E[ST ]
Remarks:
I How does the forward price compare to the future spot price at expiry of the contract?
I Spot-forward parity: forward price equals future compound value of current spot price
I Efficient market hypothesis: forward price is unbiased predictor of future spot price
I Together: No Arbitrage + Efficient Markets ⇒ E[ST ] = S0 eRT
15/15
Lecture 14: Forwards and Futures I
Revision Checklist
Forwards and futures
Payoff diagrams
Spot-forward parity
Forward price arbitrage*
Reading: Hillier et al. (2016, Chapters 25.1–25.3), Bodie et al. (2014, Chapters 22.1,22.2, 22.4), Hull (2015,
pp. 1.1–1.10, 2.1–2.7, 3.1–3.2, 5.1–5.8)
References I
I Basu, S. (1977). Investment performance of common stocks in relation to their price-earnings ratios: A test
of the efficient market hypothesis. Journal of Finance, 32(3), 663–682.
I Bodie, Z., Kane, A., & Marcus, A. J. (2014). Investments (10th). McGraw Hill.
I Carhart, M. M. (1997). On persistence in mutual fund performance. Journal of Finance, 52(1), 57–82.
I Diez, D., Barr, C., & Çetinkaya-Rundel, M. (2015). Openintro statistics (3rd ed.).
I Elton, E. J., Gruber, M. J., Brown, S. J., & Goetzmann, W. N. (2011). Modern portfolio theory and
investment analysis (8th). John Wiley & Sons.
I Fama, E. F., Fisher, L., Jensen, M. C., & Roll, R. (1969). The adjustment of stock prices to new information.
International Economic Review, 10(1), 1–21.
I Fama, E. F. & French, K. R. (1992). The cross-section of expected stock returns. Journal of Finance,
47(2), 427–465.
I Hartman, G., Siemers, T., Heinold, B., & Chalishajar, D. (2018). Apex calculus (4.0). (J. Bower, Ed.).
I Hillier, D., Ross, S., Westerfield, R., Jaffe, J., & Jordan, B. (2016). Corporate finance (3rd European
Edition). McGraw Hill.
I Hull, J. C. (2015). Options, futures, and other derivatives (9th). Pearson Education.
References II
I Jegadeesh, N. & Titman, S. (1993). Returns to buying winners and selling losers: Implications for stock
market efficiency. Journal of Finance.
I Jensen, M. C. (1968). The performance of mutual funds in the period 1945-1964. Journal of Finance,
23(2), 389–416.
I Rendleman, R. J., Jones, C. P., & Latane, H. A. (1982). Empirical anomalies based on unexpected
earnings and the importance of risk adjustments. Journal of Financial Economics, 10(3), 269–287.
I Solnik, B. H. (1974). Why not diversify internationally rather than domestically?. Financial Analysts
Journal, 30(4), 48–54.
I Stitz, C. & Zeager, J. (2013). Precalculus (3rd Corrected). Stitz Zeager Open Source Mathematics.
