Chapter 5: Valuation of Forwards & Futures
© Paul Koch 1-1
A. Notation & Background:
T:
S:
ST:
K:
F:
f:
r:
Time until delivery of the forward contract (fraction of year)
Spot price of underlying asset at time t (today)
Spot price of underlying asset at time T (maturity); a random variable
Delivery price in forward contract
Forward price prevailing in market at time t
Value of a long forward contract at time t
Riskfree rate per annum at time t, for investment maturing at T (LIBOR)
1.
F  f.
2.
a.
F is the delivery price at any time
that would make the contract have a zero value.
b.
When contract is initiated, set K = F, so f = 0.
c.
As time passes, F changes, so f changes (win or lose $).
Valuation depends on opportunity to arbitrage;
buying / selling spot vs futures, if LOP is violated.
A. Notation & Background
© Paul Koch 1-2
3. Shorting the spot asset is different from shorting futures.
Shorting futures is just like going long futures.
Positions are symmetric. Each is simply a promise - to buy or sell at a price agreed upon today, but deliver sometime in the future.
Besides margin & marking-to-market, no cash is paid today.
Shorting the spot – selling today something you don’t own.
Today: must borrow asset from someone else, and then sell it.
Receive proceeds of the sale now.
This money is your asset; earns interest while you wait.
Your liability is fact that you owe the asset & all its benefits
(like dividends) to the original owner.
Must maintain a margin account to protect against losses.
Later: buy the asset back, and give it back to owner.
If price ↓, make money. If price ↑, lose.
B. Forward Prices for a security that provides no income.
© Paul Koch 1-3
e.g., discount bonds, non-dividend paying stocks, gold, silver
1.
Example: T.Bill - sold at discount; pays $1,000,000 at mat.
Suppose you wish to hold 151-day T. Bill. Two alternatives:
Direct purchase: Buy 151-day T. Bill at S (today's spot price).
Indirect purchase: Buy forward contract (at F) that delivers 91-day T. Bill in 60 days,
and Buy 60-day T. Bill that will pay F in 60 days.
Action
day 0
|- - - - - - 91 days - - - - - -|
60 days
151 days
Direct:
Buy 151-day T. Bill
S
Indirect:
Buy forward contract
Buy 60-day T. Bill
Fe -rT
-F
+F .
$1,000,000
.
Sum of Cash Flows
Fe -rT
0
$1,000,000
.
$1,000,000
.
Produce identical cash flows in 151 days; Should have same cost today.
Pricing relation 1:
Point :
Fe -rT = S ;
or
F* = Se rT.
The forward offers something the spot purchase doesn’t,
use of your money during life of forward; so e rT pushes F higher.
B. Forward Prices for a security that provides no income.
© Paul Koch 1-4
2.
Arbitrage Forces make pricing relation hold, if F is too high.
a. Suppose F > S e rT.
i.
F is too high relative to S; Buy at S and sell at F.
today
ii. Borrow $S and buy security. (Will owe $S e rT at expiration.)
Short a forward on the security.
exp.
iii. Exercise forward contract; deliver security for $F.
Use part of proceeds to pay back loan, S e rT ; Keep diff., [F - S e rT].
3.
Example: Forward contract on non-dividend paying stock;
T = .25 (3 months); S = $40; r = .05
a. What should F be? F* = S e rT = $40 e .05(.25) = $40.50
i.
Suppose F = $43.
F is too high relative to S ( F > S e rT ).
today
ii. Borrow $40 and buy the stock. (Will owe $40.50 at expiration.)
Short a forward on the stock.
exp.
iii. Exercise forward contract; deliver stock for $43.
Use part of proceeds to pay back loan, $40.50; Keep diff., $2.50
B. Forward Prices for a security that provides no income.
© Paul Koch 1-5
4.
Arbitrage Forces make pricing relation hold, if F is too low.
a. Suppose F < S e rT.
i.
F is too low relative to S; Sell at S and buy at F.
today
ii. Short the security, receive $S. Invest proceeds at r. (Will have S e rT.)
Buy a forward on the security.
exp.
iii. Proceeds worth $S e rT. Use proceeds to exercise fwd (buy at $F).
Deliver security to close out short sale; Keep diff., [S e rT - F].
5.
Example: Forward contract on non-dividend paying stock;
T = .25 (3 months); S = $40; r = .05
a. What should F be? F* = S e rT = $40 e .05(.25) = $40.50
i.
Suppose F = $39.
F is too low relative to S ( F < S e rT ).
today
ii. Short the stock, receive $40. Invest proceeds at r.
Buy a forward on the stock.
exp.
iii. Proceeds worth $40.50; Use proceeds to exercise fwd (buy at $39).
Deliver stock to close out short sale;
Keep diff., $1.50
C. Forward Prices on a security paying a known income.
© Paul Koch 1-6
e.g., coupon-bearing bonds, dividend-paying stocks.
1. Example #1: T. Bond (pays coupons + face value at mat.).
**
Assume T. Bond pays no coupon during next 60 days. Two alternatives:
Direct Purchase: Buy T. Bond at S (today's spot price).
Indirect Purchase: Buy forward (at F) that delivers a T. Bond in 60 days,
and Buy 60-day T. Bill that will pay F in 60 days.
Action
day 0
Direct:
Buy T. Bond
Indirect:
Buy forward contract
Buy 60-day T. Bill
Sum of Cash Flows
60 days
S
future
coupons + face value
Fe -rT
-F
+F .
coupons + face value
.
Fe -rT
0
coupons + face value .
Produce identical cash flows in future; Should have same cost today.
Pricing relation 1:
**
.
Fe -rT = S ;
or
F* = Se rT.
Same as B., since this T. Bond pays no income during life of forward.
C. Forward Prices on a security paying a known income.
© Paul Koch 1-7
2. Example #2: T. Bond (pays coupons + face value at mat.).
**
Now assume T. Bond pays coupon during next 60 days. Two alternatives:
Direct Purchase:
Borrow $I = NPV(coupon), and use this to help buy T. Bond.
Must put up ($S - $I) today. Then coupon pays off loan.
Indirect Purchase: Buy forward (at F) that delivers a T. Bond in 60 days,
and Buy 60-day T. Bill that will pay F in 60 days.
Action
Direct:
Indirect:
day 0
60 days
future
Borrow $I and use
to help Buy T. Bond
S-I
coupon
pays loan
remaining coupons
plus face value
Buy forward contract
Buy 60-day T. Bill
Fe -rT
-F
+F .
remaining coupons
plus face value .
Sum of Cash Flows
Fe -rT
0
remaining coupons + FV.
Pricing relation 2: Fe -rT = S - I ; or F* = (S - I)e rT.
**
.
Point: Now two forces at work:
1. The forward offers something the spot purchase doesn’t,
the use of your money during life of forward;
so e rT pushes F higher.
2. The spot purchase offers something the forward doesn’t,
the first coupon;
so $I pushes F lower.
D. Forward Prices on a security paying known dividend yield.
© Paul Koch 1-8
1. Let q = annual dividend yield, paid continuously.
(e.g., stock indexes, foreign currencies.)
Pricing Relation 3:
F* = Se (r-q) T.
If pricing relation does not hold, arbitrage opportunities:
a. Buy e -qT (< 1) units of security today.
b. Reinvest dividend income into more of security.
c. Short a forward contract.
This amount of the security grows at rate q;
therefore, e -qT x e qT = 1 unit of security is held at expiration.
Under forward contract, this security is sold at expiration for F.
initial outflow = Se -qT;
final inflow = F.
Today, initial outflow = PV(final inflow).
Thus,
S e -qT = F e -rT
or
F* = S e (r-q)T.
D. Forward Prices on a security paying known dividend yield.
© Paul Koch 1-9
Pricing Relation 3:
F* = Se (r-q) T.
2. Suppose F > S e (r-q) T.
a.
F is too high relative to S; Buy at S and sell at F.
today
b.
Borrow $S e -qT and buy e -qT (< 1) units of the security.
At expiration, will owe $S e -qT x e rT = $S e (r-q) T.
Short a forward on the security (promise to sell for F).
then
c.
Security will provide dividend income at rate, q;
Reinvest the dividend income into more of the security.
exp.
d.
Now hold one unit of the security.
Exercise forward contract; deliver security for $F.
Use proceeds to pay off the loan. Keep diff., [ F - S e (r-q) T ].
D. Forward Prices on a security paying known dividend yield.
© Paul Koch 1-10
Pricing Relation 3:
F* = Se (r-q) T.
3. Similar formula to C.
Two forces at work:
a.
The forward offers something the spot purchase doesn’t,
use of your money during the life of the forward;
so e rT is pushing F higher.
b.
The spot purchase offers something the forward doesn’t,
continuous stream of dividends at rate, q;
so e -qt is pushing F lower.
E. General Formula for Valuation of Futures
© Paul Koch 1-11
General Pricing Relation, true for all assets.
The relation between
f = (F - K)e -rT
1.
current futures price (F)
& delivery price (K),
in terms of spot price (S) & K.
Explanations.
a. Expl #1: If not, then arbitrage opportunities.
b. Expl #2: When forward contract is entered, set F = K; f = 0.
Later, as S changes, the appropriate value of F changes
and f will become positive or negative.
As F moves away from K, value (f) moves away from 0.
E. General Formula for Valuation of Futures
© Paul Koch 1-12
2.
Consider formula in above cases:
f = (F - K)e -rT
a. security that provides no income.
F* = Se rT,
so that f = (Se rT - K) e –rT
or f = S - Ke -rT.
b. security that provides a known income.
F* = (S-I)e rT, so
f = [(S-I)e rT - K]e -rT
or f = S - I - Ke -rT.
c. security that pays a known dividend yield.
F* = Se (r-q)T,
so
f = [Se (r-q)T - K] e -rT
or f = Se -qT - Ke -rT.
d. Note: in each case, the forward price at the current time (F)
is the value of K that makes f = 0.
F. Applications – Stock Index Futures
© Paul Koch 1-13
1. Stock Index Futures.
a. Examples of underlying asset - the stock index:
i. S&P 500 - 400 industrials, 40 utilities, 20 transp co’s, and 40 banks.
Companies amount to  80% of total mkt cap on NYSE.
Two contracts traded on CME: i. $250 x index; ii. $50 x index.
ii. S&P Midcap 400 - composed of middle-sized companies.
Futures traded on CME. One contract is on $500 x index.
iii. Nikkei 225 - largest stocks on TSE.
Traded on CME. One contract is on $5 x index.
iv. NYSE Composite Index - all stocks listed on NYSE.
Traded on NYFE. One contract is on $250 x index.
v. Nasdaq 100 - 100 Nasdaq stocks. Two contracts traded on CME:
One is on $100 x index; Other (mini-Nasdaq) is on $20 x index.
vi. International - CAC-40 (Euro stocks), DJ Euro Stoxx 50 (Euro stocks),
DAX-30 (German stocks), FT-SE 100 (UK stocks).
F. Applications – Stock Index Futures
© Paul Koch 1-14
b. Valuation. Consider S&P 500 futures.
Treat as security with known dividend yield.
Pricing Relation 3:
F* = Se (r-q)T
where q = average dividend yield.
Problem 5.10.
The risk-free rate of interest is 7% per annum with continuous
compounding, and the avg dividend yield on a stock index is 3.2% p.a.
The current value of the index is 150. What is the six-month futures price?
Using the above equation, the six-month futures price is
F* = 150 e (.07 - .032) x 0.5 = $152.88.
G. Forward Prices on Foreign Currency Futures
© Paul Koch 1-15
1. Valuation -- 2 different explanations:
a. Treat FC as security with known dividend yield, q = rf : (American terms – $/FC.)
Pricing Relation 4: F* = Se (r - rf) T. Interest Rate Parity.
b. Consider two alternative ways to hold riskless debt:
i.
ii.
U.S. riskless debt:
Foreign riskless debt [3 steps]:
$1  $1e r T
a) $1 / S
b) ($1 / S) x e rf T
c) ($1 / S) x e rf T x F
-
$ in one year
FC today
FC in one year
$ in one year.
Give same riskless cash flow in US$ in 1 year. So final $ outcome should be same.
$1 e rT = [ ($1 / S) e rf T ] F
e rT = (F / S) e rf T
or
or
today
$:
$1
( S) 
FC:
(1 / S) FC
F* = S e (r - rf) T.
one year



_______________U.S. Riskless Debt_______________
|
|
|
|
|
|

Foreign Riskless Debt

1 e rT $
{ [ (1 / S) e rf T ] F } $
 (x F)
[ (1 / S) e rf T ] FC
H. Commodity Futures
© Paul Koch 1-16
Distinguish between commodities held solely for investment,
and commodities held primarily for consumption.
-- Arbitrage arguments are used to value F for investment commodities,
but only give an upper bound on F for consumption commodities.
1.
Gold and Silver (held primarily for investment).
a. If storage costs = 0, like security paying no income:
F* = S erT.
b. If storage costs  0, costs can be considered as:
Pricing Relation 5:
i. Negative income. Let U = PV(storage costs); F* = (S + U) erT.
ii. Negative div yield. Let u = % cost per annum; F* = S e(r + u) T.
c. Point: Now two forces at work in same direction:
i. Forward offers something spot purchase doesn’t,
use of your money during life of forward; erT pushing F higher.
ii. Forward offers something else spot doesn’t,
no storage costs from holding spot;
U pushing F higher.
H. Commodity Futures
© Paul Koch 1-17
2. Consumption commodities (not held for investmt purposes).
a. Suppose F > (S+U) e rT.
F too high.
i. Borrow (S+U), buy 1 unit of commod. for S; pay storage costs; owe (S+U) e rT at mat.
ii. Short futures on 1 unit of commodity. Will give profit of [ F - (S+U) e rT ].
Can do this for any commodity. Arbitrage will force F down until equal (upper bound).
b. Suppose F < (S+U)e rT.
F too low.
i. Short 1 unit of comm., invest proceeds; save storage costs. Will have (S+U) e rT at mat.
ii. Buy futures on 1 unit of commodity.
Will give profit of [ (S+U) e rT - F ].
Can do this for gold and silver - held for investment. Arb. will force S down and F up.
** However, may not want (or be able) to do this arb for consumption commodities (if F too low).
Commodity is kept in inventory because of its consumption value, not for investment.
Cannot consume a futures contract! Thus, may be no arb. forces to eliminate inequality.
Pricing Relation 6: F  (S+U) e rT, or F  S e (r+u)T.
Only have upper bounds for F on consumption commodities.
H. Commodity Futures
© Paul Koch 1-18
Pricing Relation 6: F  (S+U) e rT,
or F  S e (r+u)T.
3. Convenience yield on consumption commodities.
a. Benefits from ownership of commodity not obtained with futures contract:
(i) Ability to profit from temporary shortages.
(ii) Ability to keep a production process going.
b. If PV of storage costs (U) are known,
then convenience yield, y, is defined so that:
Pricing Relation 7:
F e yT = (S+U) e rT.
c. If storage costs are constant prop. [u] of S,
then convenience yield, y, is defined so that:
Pricing Relation 7:
F e yT = S e (r+u)T.
d. Note: i. For consumption assets, y measures extent to which lhs < rhs.
ii. For investment assets, y = 0, since arb. forces work both directions.
iii. y reflects market's expectation about future availability of commodity.
If users have high inventories, shortages less likely & y should be smaller.
If users have low inventories, shortages more likely, & y should be larger.
If y large enough, backwardation (F < S).
I. Cost of Carry
© Paul Koch 1-19
1. Definition: c = r + u - q =
interest paid to finance asset + storage cost - income earned.
a.
For non-dividend paying stock, storage costs = income earned = 0;
c = r;
F* = S e r T;
Cost of carry (c) = r.
b.
For stock index, storage costs = 0, & income is earned at rate, q;
c = r - q;
F* = S e (r - q) T.
This income ↓ c.
c.
For foreign currency, storage costs = 0, & income is earned at rate, rf;
c = r - rf ;
F* = S e (r - rf) T;
This income ↓ c.
d.
For commodity, storage costs are like negative div. income at rate, u;
c = r + u;
F* = S e (r + u) T;
These costs ↑ c.
e.
Summarizing:
For investment asset, F = S e c T;
(F > or < S by amount reflecting c.)
For consumption asset, F = S e (c - y) T; (F > or < S by amount reflecting the
cost of carry, c, net of the convenience yield, y.)
J. Implied Delivery Options Complicate Things
© Paul Koch 1-20
1. Futures contracts specify a delivery period.
When during delivery period will the short want to deliver?
a.
Cost of Carry
= c = (r + u - q) = (interest paid + storage costs - income).
b.
Benefits from holding asset = (y + q - u) = (conv. yield + income - storage costs).
c.
If F is an increasing function of time, (F > S: contango),
then r > (y + q - u).
[Then F = S e (c - y) T; c - y > 0; c > y; (r + u - q) > y; r > y + q - u ].
Then it is usually optimal for short position to deliver early,
since interest earned on cash (r) outweighs benefits of holding asset longer (y + q - u).
** Deliver early! Sell @ F (> S) !
d.
Would rather have $F now!
Start earning r now!
If F is a decreasing function of time, (F < S: backwardation), then r < (y + q - u).
[Then F = S e (c - y) T; c - y < 0; c < y; (r + u - q) < y;
r < y + q - u ].
Then it is usually optimal for short position to deliver late,
since benefits of holding asset longer (y + q - u) outweigh interest earned on cash (r).
** Deliver late!
Sell @ F (< S) !
Would rather hold onto asset! Keep getting (y + q - u)!