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Chapter 5: Valuation of Forwards & Futures

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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 promised in forward contract at time T 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.

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.

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. | days | Action day days 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 $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.

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 = $ F is too high relative to S ( F > S e rT ). today ii. Borrow $40 and buy the stock. (Will owe $ 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

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 = $ 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

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 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 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.

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 $I = NPV(coupon), and use this to help buy T. Bond. Must put up ($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 day days future Direct: Borrow $I and use coupon remaining coupons to 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 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.

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.

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 ].

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.

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.

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.

13 F. Applications – Stock Index Futures
a. Examples of underlying asset - the stock index: i. S&P 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 composed of middle-sized companies. Futures traded on CME. One contract is on $500 x index. iii. Nikkei 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 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).

14 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 What is the six-month futures price? Using the above equation, the six-month futures price is F* = 150 e ( ) x 0.5 = $

15 G. Forward Prices on Foreign Currency Futures
1. Valuation 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    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

16 G. Forward Prices on Foreign Currency Futures
Pricing Relation 4: F* = S e (r - rf) T. Interest Rate Parity. Intuition: Fisher Equation: rus = rReal + PeUS True in US. rf = rReal + Pef True in foreign country. → rUS - rf = PeUS - Pef . → If PeUS > Pef then rUS > rf . then expect S to ↑ (i.e., cost more $ to buy FC) ( by (rUS - rf ) ).

17 H. Commodity Futures 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.

18 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, 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.

19 Pricing Relation 6: F  (S+U) e rT, or F  S e (r+u)T.
H. Commodity Futures 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, 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 investment assets, y = 0, since arb. forces work both directions. ii. For consumption assets, y measures extent to which lhs < rhs. 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).

20 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 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.)

21 J. Implied Delivery Options Complicate Things
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). [Proof: 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) ! 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). [Proof: 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)!


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