1. Lec 5A APT for Forward and Futures Contracts
Lec 5A: Arbitrage Pricing of Forwards and Futures (Hull, Ch. 5.3 and 5.4)
Suppose the spot price of an asset (e.g., stock) evolves as shown below. The risk-free rate r = 4.88%/yr(c.c.). Consider a Forward contract on 1 share of stock with 1-yr to “expiration”:
Spot Prices Forward Prices
$140 B ? B How to determine
$120 A F = ? forward prices?
$100 C ? C
(A: by trial and error)
Suppose at B forward price = $120. Is this okay?
Arbitrage Strategy:
▸ Buy forward contract, “exercise” immediately, Pay $120
▸ Receive the stock, sell it in the cash market:
CFB = = $20
➟ To preclude arbitrage the forward price must = $140
Using the same logic, the forward price at C must = ?
Lec 5A APT for Forward and Futures Contracts
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2. Lec 5A APT for Forward and Futures Contracts
Forward Price at A:
Spot Price Forward Prices
$140 B $140 B
$120 A F = ? A
$100 C $100 C
Suppose F = $100, Spot Price = $120
➟ Forward price seems low, spot high.
▸ Strategy: {Buy forward contract, buy a Bond} and {Short the Stock}.
▸ Arbitrage Portfolio: {-S, +F, +B(PV = $120)} ➟ CFA = = 0
At Expiration,
If mkt ↑ SB = 140, Bond matures for 120e = $126.
Use $100 and the forward contract to buy the stock and cover the short. ➟ CF = = +$26
If mkt ↓ S C =100, Bond matures for 120(e0.0488) = $126. Use $100 plus the forward to buy stock and cover the short. CF= =+$26
Lec 5A APT for Forward and Futures Contracts
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3. Lec 5A APT for Forward and Futures Contracts
This must be an ARB OPPORTUNITY.
Strategy doesn’t cost anything and yet it pays $26 whether the market goes ↑ or ↓ .
To preclude the arbitrage,
at A Forward price must = $126 = 120(e0.0488)= er S0
Question: How do you create a synthetic forward contract?
Answer: Recall the arbitrage strategy
1. Buy the stock now and sell it forward.
2. Sell a bond also to finance the stock purchase.
{+S, -F, -B } ➟ Net CF0 = $0. Therefore,
{+F}={+S, -B} ➟ Long Forward = Long stock financed with a bond
also,
{+S} = {+F, +B} ➟ Long Stock = Long Forward + Long a Bond
{+S, -F} = {+B} ➟ Buy stock and sell it forward = a risk-free bond
Lec 5A APT for Forward and Futures Contracts
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4. Lec 5A APT for Forward and Futures Contracts
Another Example: Suppose the spot price evolves as shown below. (Assume 1 year to expiration, r = /yr (c.c.),
hence 1 period = 6 months and r = 4.88%/period (c.c.). )
SP500 Price tree Forward Price tree B 140 D B 140 D
A A 126
S0 = E F0 = ? E
C 60 F C 60 F
t = t =
At expiration, the forward price must = Spot (or cash) price
We already know: At B , Forward price must = $126 = SB (er)
At C , Forward PriceC = $84 = Spot PriceC (e0.0488)
What is Arb free price at A ?
Lec 5A APT for Forward and Futures Contracts
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5. Lec 5A APT for Forward and Futures Contracts
SP500 Price tree Forward Price tree B 140 D B 140 D
A A 126
S0 = E F0 = ? E
C 60 F C 60 F
t = t =
At A , does this price make sense: F = 90.25?
It seems low (relative to the spot price) ➟ Buy Forward, Short Stock
{+F contract, -S, +B(PV=$100) }.
➟ CF A = = 0
At Expiration (2 periods later),
if spot price ↑ SD = 140, Bond matures for 100(e2*0.0488) =
Settle forward contract: Pay 90.25, receive one share, then cover short. CF = = $20
Lec 5A APT for Forward and Futures Contracts
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6. Lec 5A APT for Forward and Futures Contracts
if SE = 100,
Bond matures for 100(e2*0.0488) =
Buy stock thru forward contract. Pay 90.25, receive one share,
then cover short. CF = = $20
if SF = 60,
Settle forward contract. Pay and receive one share,
The payoff is always +$20.
To preclude this arbitrage, at point A
Forward price must = = $ = 100 (e2*0.0488)
= Stock PriceA (e2*(r/2))
In general, at t=0, Forward Price F0 must be such that the CF from
{+F, -S, +B(FV=F0)} = 0 ➟ CF = -0 +S0 - F0 (e-r T) = 0 ➟ F0 = (er T)S0 done!
Lec 5A APT for Forward and Futures Contracts
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7. Lec 5A APT for Forward and Futures Contracts
Price Forwards that generate a known Income (Sec 5.5, 5.6)
Suppose the spot price now is S0 = $50, and evolves as shown below.
Assume 1 year to expiration, r = 13.8%/yr (c.c.)
Stock will pay a $15 Dividend in period 1.
Assume the stock price will ↓ by full amount of dividend.
Stock Price tree Forward Price tree
B E E
75→60 B
A F A 40 F
S0 = G F0 = ? G
33.33→ C
C H H
t = t =
Stock price at B and C is ex-Dividend ($15 lower)
Our goal is to find the arb-free forward price at point A: F0
Lec 5A APT for Forward and Futures Contracts
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8. Lec 5A APT for Forward and Futures Contracts
Stock Price tree Forward Price tree
B E E
75→60 B
A F A F
S0 = G F0 = ? G
33.33→ C
C H H
t = t =
Note1: PV($15 Dividend) = 15e-0.138(½) = $14.
Note2: {+S} = {+F, +B(PV=$14), +B(FV = F0 )} <== synthetic stock
Strategy at point A.
t=0 {+S, -B(PV=$14), -F)} ➟ CF0 = = -$36
t= ½ year, receive div = $15, use it to pay back the bond. CF1/2 = 0
t=1 sell the stock thru the forward and receive $F0
Since the strategy has no risk, the ROR must = 0.138
➟ 36e0.138 = F0 ➟ F0 = $41.33
The arb-free forward price at point A is F0 = $41.33
Lec 5A APT for Forward and Futures Contracts
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9. Lec 5A APT for Forward and Futures Contracts
Mark to Market of Forward Contracts
SP500 Price tree Forward Price tree
B E E
A A 126
S0 = G F0 = G
C F H
t = t =
Suppose you go long a forward contract at time 0
Forward Price: F0 = $110.25
At time ½ mkt ↑ to SB = 120. The forward price at B = $126
Long wants to close the forward contract. How?
Just sell a 6-months contract at FB = 126.
Lec 5A APT for Forward and Futures Contracts
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10. Lec 5A APT for Forward and Futures Contracts
Note that at time 1, You will
➀ pay $ and buy 1 share of stock (to satisfy the first contract), and
➁ sell 1 share of stock and receive $126 (from the second contract)
Net CF = = +$15.75 (for sure).
Question: What is the Gain at t = 1/2 ?
Answer: Gain = PV of 15.75
Gain = $15 (=15.75 e )
| ––––––––––––––|–––––––––––––––––––––––––––––|
0 ½
Therefore, the mark-to-market value of the original contract is $15.
Lec 5A APT for Forward and Futures Contracts
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11. Lec 5A APT for Forward and Futures Contracts
Thank You!
(a Favara)
Lec 5A APT for Forward and Futures Contracts
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