© Paul Koch 1-1

Chapter 5: Valuation of Forwards & Futures

A. Notation & Background: T: Time until delivery of the forward contract (fraction of year) S: Spot price of underlying asset at time t (today) ST: Spot price of underlying asset at time T (maturity); a random variable

K: Delivery price in forward contract F: Forward price prevailing in market at time t f: Value of a long forward contract at time t r: Riskfree rate per annum at time t, for investment maturing at T (LIBOR)

1. F f.

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 $).

2. Valuation depends on opportunity to arbitrage;buying / selling spot vs futures, if LOP is violated.

© Paul Koch 1-2

A. Notation & Background

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.

© Paul Koch 1-3

B. Forward Prices for a security that provides no income.

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.

|- - - - - - 91 days - - - - - -|

Action day 0 60 days 151 days .

Direct: Buy 151-day T. Bill S $1,000,000

Indirect: Buy forward contract -F $1,000,000

Buy 60-day T. Bill Fe -rT +F . .

Sum of Cash Flows Fe -rT 0 $1,000,000 .

Produce identical cash flows in 151 days; Should have same cost today.

Pricing relation 1: F e -rT = S ; or F* = S e rT.Point : The forward offers something the spot purchase doesn’t,

use of your money during life of forward; so e rT pushes F higher.

© Paul Koch 1-4

B. Forward Prices for a security that provides no income.

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

© Paul Koch 1-5

B. Forward Prices for a security that provides no income.

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

© Paul Koch 1-6

C. Forward Prices on a security paying a known income.

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 60 days future .

Direct: Buy T. Bond S coupons + face value

Indirect: Buy forward contract -F coupons + face value

Buy 60-day T. Bill Fe -rT +F . .

Sum of Cash Flows Fe -rT 0 coupons + face value .

Produce identical cash flows in future; Should have same cost today.

Pricing relation 1: F e -rT = S ; or F* = S e rT.

** Same as B., since this T. Bond pays no income during life of forward.

© Paul Koch 1-7

C. Forward Prices on a security paying a known income.

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, 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 day 0 60 days future .

Direct: Borrow $I and use coupon remaining couponsto help Buy T. Bond S - I pays loan plus face value

Indirect: Buy forward contract -F remaining coupons

Buy 60-day T. Bill Fe -rT +F . plus face value .

Sum of Cash Flows Fe -rT 0 remaining coupons + FV.

Pricing relation 2: F e -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.

© Paul Koch 1-8

D. Forward Prices on a security paying known dividend yield.

1. Let q = annual dividend yield, paid continuously.(e.g., stock indexes, foreign currencies.)

Pricing Relation 3: F* = S e (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.

© Paul Koch 1-9

D. Forward Prices on a security paying known dividend yield.

Pricing Relation 3: F* = S e (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 ].

© Paul Koch 1-10

D. Forward Prices on a security paying known dividend yield.

Pricing Relation 3: F* = S e (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.

© Paul Koch 1-11

E. General Formula for Valuation of Futures

General Pricing Relation, true for all assets.

The relation between

f = (F - K) e -rT current futures price (F) & delivery price (K), in terms of spot price (S) & K.

1. 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.

© Paul Koch 1-12

E. General Formula for Valuation of Futures

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.

© Paul Koch 1-13

F. Applications – Stock Index Futures

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).

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F. Applications – Stock Index Futures

b. Valuation. Consider S&P 500 futures.

Treat as security with known dividend yield.

Pricing Relation 3: F* = S e (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.

© Paul Koch 1-15

G. Forward Prices on Foreign Currency Futures

1. Valuation -- 2 different explanations:

a. Treat FC as security with known dividend yield, q = rf : (American terms – $/FC.)

Pricing Relation 4: F* = S e (r - rf) T. Interest Rate Parity.

b. Consider two alternative ways to hold riskless debt:i. U.S. riskless debt: $1 $1e r T - $ in one year

ii. Foreign riskless debt [3 steps]: a) $1 / S - FC today

b) ($1 / S) x e rf T - FC in one year

c) ($1 / S) x e rf T x F - $ 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 or e rT = (F / S) e rf T  or F* = S e (r - rf) T.

today one year

$: $1 1 e rT $

_______________U.S. Riskless Debt_______________ { [ (1 / S) e rf T ] F } $ | |( S) | | (x F) | |

FC: (1 / S) FC Foreign Riskless Debt [ (1 / S) e rf T ] FC

© Paul Koch 1-16

H. Commodity Futures

Distinguish between commodities held solely for investment,

and commodities held primarily for consumption.

-- Arbitrage arguments used to value F for investment commodities, but only give upper bound for 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:

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.

© Paul Koch 1-17

H. Commodity Futures

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, don't want to do this arbitrage 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, there are no arb. forces to eliminate inequality.

For consumption commodities, F (S+U) e rT, or F S e (r+u)T.

Only have upper bounds for F on consumption commodities.

© Paul Koch 1-18

H. Commodity Futures

For consumption commodity, F (S+U) e rT, or F S e (r+u)T.

3. Convenience yield.

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, Pricing Relation 7:

then convenience yield, y, is defined so that: F e yT = (S+U) e rT.

c. If storage costs are constant prop. [u] of S, Pricing Relation 7:

then convenience yield, y, is defined so that: 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 avout 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).

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I. Cost of Carry

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 earned at rate, q;

c = r - q; F* = S e (r - q) T. This income ↓ c.

c. For foreign currency, storage costs = 0, & income earned at rate, rf;

c = r - rf; F* = S e (r - rf) T; This income ↓ c.

d. For commodity, storage costs are like negative 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 > S by amount reflecting c.)

For consumption asset, F = S e (c - y) T ; (F > S by amount reflecting the

cost of carry, c, net of the convenience yield, y.)

© Paul Koch 1-20

J. Implied Delivery Options Complicate Things

1. Futures contracts specify delivery period. When during delivery period will the short want to deliver?

a. Cost of Carry = c = (r + u - q) = (interest pd + 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 - q + u) > 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) ! Would rather have $F now! Start earning r now!

d. 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 - q + u) < 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)!