Chapter 5 Valuation of Forwards Futures Paul Koch

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Chapter 5: Valuation of Forwards & Futures © Paul Koch 1 -1 A. Notation

Chapter 5: Valuation of Forwards & Futures © Paul Koch 1 -1 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.

A. Notation & Background © Paul Koch 1 -2 3. Shorting the spot asset

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

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 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 -r. T -F +F. $1, 000. Sum of Cash Flows Fe -r. T 0 $1, 000, 000 . Produce identical cash flows in 151 days; Should have same cost today. Pricing relation 1: F e -r. T = S ; or F* = S e r. T. Point : The forward offers something the spot purchase doesn’t, use of your money during life of forward; so e r. T pushes F higher.

B. Forward Prices for a security that provides no income. © Paul Koch 1

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 r. T. 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 r. T 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 r. T ; Keep diff. , [F - S e r. T]. 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 r. T = $40 e. 05(. 25) = $40. 50 i. Suppose F = $43. F is too high relative to S ( F > S e r. T ). 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

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 r. T. 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 r. T. ) Buy a forward on the security. exp. iii. Proceeds worth $S e r. T. Use proceeds to exercise fwd (buy at $F). Deliver security to close out short sale; Keep diff. , [S e r. T - 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 r. T = $40 e. 05(. 25) = $40. 50 i. Suppose F = $39. F is too low relative to S ( F < S e r. T ). 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

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 Direct: Buy T. Bond Indirect: Buy forward contract Buy 60 -day T. Bill Sum of Cash Flows day 0 60 days S future coupons + face value Fe -r. T -F +F. coupons + face value. Fe -r. T 0 coupons + face value. Produce identical cash flows in future; Should have same cost today. Pricing relation 1: F e -r. T = S ; or F* = S e r. T. ** . 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

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 -r. T -F +F. remaining coupons plus face value. Sum of Cash Flows Fe -r. T 0 . remaining coupons + FV. Pricing relation 2: F e -r. T = S - I ; or F* = (S - I) e r. T. ** 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 r. T 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

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* = S e (r-q) T. If pricing relation does not hold, arbitrage opportunities: a. Buy e -q. T (< 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 -q. T x e q. T = 1 unit of security is held at expiration. Under forward contract, this security is sold at expiration for F. initial outflow = Se -q. T; final inflow = F. Today, initial outflow = PV(final inflow). Thus, S e -q. T = F e -r. T or F* = S e (r-q)T.

D. Forward Prices on a security paying known dividend yield. © Paul Koch 1

D. Forward Prices on a security paying known dividend yield. © Paul Koch 1 -9 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 -q. T and buy e -q. T (< 1) units of the security. At expiration, will owe $S e -q. T x e r. T = $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

D. Forward Prices on a security paying known dividend yield. © Paul Koch 1 -10 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 r. T 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

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 -r. T 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.

E. General Formula for Valuation of Futures © Paul Koch 1 -12 2. Consider

E. General Formula for Valuation of Futures © Paul Koch 1 -12 2. Consider formula in above cases: f = (F - K) e -r. T a. security that provides no income. F* = Se r. T, so that f = (Se r. T - K) e –r. T or f = S - Ke -r. T. b. security that provides a known income. F* = (S-I)e r. T, so f = [(S-I)e r. T - K]e -r. T or f = S - I - Ke -r. T. c. security that pays a known dividend yield. F* = Se (r-q)T, so f = [Se (r-q)T - K] e -r. T or f = Se -q. T - Ke -r. T. 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

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

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* = 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.

G. Forward Prices on Foreign Currency Futures © Paul Koch 1 -15 1. Valuation

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* = S e (r - rf) T. Interest Rate Parity. b. Consider two alternative ways to hold riskless debt: i. U. S. riskless debt: ii. Foreign riskless debt [3 steps]: $1 $1 e r T - $ in one year 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 r. T = [ ($1 / S) e rf T ] F or e r. T = (F / S) e rf T or F* = S e (r - rf) T. today $: $1 one year ________U. S. Riskless Debt________ ( S) FC: (1 / S) FC | | | | Foreign Riskless Debt 1 e r. T $ { [ (1 / S) e rf T ] F } $ (x F) [ (1 / S) e rf T ] FC

G. Forward Prices on Foreign Currency Futures © Paul Koch 1 -16 Pricing Relation

G. Forward Prices on Foreign Currency Futures © Paul Koch 1 -16 Pricing Relation 4: F* = S e (r - rf) T. Interest Rate Parity. Intuition: Fisher Equation: → → rus = r. Real + Pe. US True in US. rf = r. Real + Pef True in foreign country. r. US - rf = Pe. US - Pef. If Pe. US > Pef then r. US > rf. then expect S to ↑ (i. e. , cost more $ to buy FC) ( by (r. US - rf ) ).

H. Commodity Futures © Paul Koch 1 -17 Distinguish between commodities held solely for

H. Commodity Futures © Paul Koch 1 -17 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 er. T. b. If storage costs 0, costs can be considered as: Pricing Relation 5: i. Negative income. Let U = PV(storage costs); F* = (S + U) er. T. 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; er. T 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 -18 2. Consumption commodities (not held for

H. Commodity Futures © Paul Koch 1 -18 2. Consumption commodities (not held for investmt purposes). a. Suppose F > (S+U) e r. T. F too high. i. Borrow (S+U), buy 1 unit of commod. for S; pay storage costs; owe (S+U) e r. T at mat. ii. Short futures on 1 unit of commodity. Will give profit of [ F - (S+U) e r. T ]. Can do this for any commodity. Arbitrage will force F down until equal (upper bound). b. Suppose F < (S+U) e r. T. F too low. i. Short 1 unit of comm. , invest proceeds; save storage costs. Will have (S+U) e r. T at mat. ii. Buy futures on 1 unit of commodity. Will give profit of [ (S+U) e r. T - 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 r. T, or F S e (r+u)T. Only have upper bounds for F on consumption commodities.

H. Commodity Futures © Paul Koch 1 -19 Pricing Relation 6: F (S+U) e

H. Commodity Futures © Paul Koch 1 -19 Pricing Relation 6: F (S+U) e r. T, 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 y. T = (S+U) e r. T. c. If storage costs are constant prop. [u] of S, Pricing Relation 7: then convenience yield, y, is defined so that: F e y. T = 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).

I. Cost of Carry © Paul Koch 1 -20 1. Definition: c = r

I. Cost of Carry © Paul Koch 1 -20 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. 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 -21 1. Futures contracts

J. Implied Delivery Options Complicate Things © Paul Koch 1 -21 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)!