- 106 Views
- Uploaded on

Download Presentation
## Chapter 5: Valuation of Forwards & Futures

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

**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: Riskfreerate 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 vsfutures, if LOP is violated.**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.**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 FlowsFe -rT 0 $1,000,000 . Produce identical cash flows in 151 days; Should have same cost today. Pricing relation 1: Fe-rT = S ; orF* = SerT. Point : The forward offers something the spot purchase doesn’t, use of your money during life of forward; so erT pushes F higher.**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 erT. i. F is too high relative to S; Buy at Sand sell at F. today ii. Borrow $Sand buy security. (Will owe $SerTat expiration.) Short a forward on the security. exp. iii. Exercise forward contract; deliver security for $F. Use part of proceeds to pay back loan, SerT; Keep diff., [F- SerT]. 3. Example: Forward contract on non-dividend paying stock; T = .25 (3 months); S = $40; r = .05 a. What should F be? F* = SerT= $40 e .05(.25) = $40.50 i. Suppose F = $43. F is too high relative to S ( F > S erT). 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.**4.Arbitrage Forces make pricing relation hold, if F is too low. a. Suppose F < S erT. i. Fis 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 SerT.) Buy a forward on the security. exp. iii. Proceeds worth $S erT.Use proceeds to exercise fwd (buy at $F). Deliver security to close out short sale; Keep diff., [SerT- F]. 5. Example: Forward contract on non-dividend paying stock; T = .25 (3 months); S = $40; r = .05 a. What should F be? F* = SerT= $40 e .05(.25) = $40.50 i. Suppose F = $39. F is too low relative to S ( F <S erT). 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.**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 Scoupons + 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: Fe-rT = S ; or F* = SerT. ** Same as B., since this T. Bond pays no income during life of forward.**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 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 -rT0 remaining coupons + FV. Pricing relation 2: Fe -rT= S - I ; or F* = (S - I)erT. ** 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 erT 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. 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 -qTx eqT= 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, Se -qT= Fe -rT or F* = Se (r-q)T.**D. Forward Prices on a security paying known dividend**yield. Pricing Relation 3: F* = Se(r-q) T. 2. Suppose F > Se (r-q) T. a. F is too high relative to S; Buy at S and sell at F. today b. Borrow $Se -qT and buy e -qT (< 1) units of the security. At expiration, will owe $Se -qTx erT= $Se (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 - Se (r-q) T ].**D. Forward Prices on a security paying known dividend**yield. 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 erT 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**General Pricing Relation, true for all assets. The relation between f = (F - K)e -rTcurrent 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**2. Consider formula in above cases: f = (F - K)e -rT a. security that provides no income. F* = SerT, so that f = (SerT- K)e –rT or f = S - Ke -rT. b. security that provides a known income. F* = (S-I)erT, so f = [(S-I)erT- 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 -rTor 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**1. Stock Index Futures. a. Examples of underlying asset - the stock index: i. S&P 500 - 400 industrials, 40 utilities, 20 transpco’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**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**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. 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 erf T - FC in one year c) ($1 / S) x erf Tx F - $ in one year. Give same riskless cash flow in US$ in 1 year. So final $ outcome should be same. $1erT= [ ($1 / S) erf T ] F or erT= (F / S) erf T or F* = S e (r - rf) T. todayone year $:$1 1 erT$ _______________U.S. Riskless Debt_______________{ [ (1 / S) erf T ] F } $ || ( S) ||(x F) | | FC:(1 / S) FC Foreign Riskless Debt[ (1 / S) erf T ] FC**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; erTpushing F higher. ii. Forward offers something else spot doesn’t, no storage costs from holding spot; U pushing F higher.**H. Commodity Futures**2.Consumption commodities (not held for investmtpurposes). a. Suppose F > (S+U) erT.F too high. i. Borrow (S+U), buy 1 unit of commod. for S; pay storage costs; owe (S+U)erT at mat. ii. Short futures on 1 unit of commodity. Will give profit of [ F - (S+U)erT]. Can do this for any commodity. Arbitrage will force F down until equal (upper bound). b. Suppose F < (S+U)erT.F too low. i. Short 1 unit of comm., invest proceeds; save storage costs. Will have (S+U)erTat mat. ii. Buy futures on 1 unit of commodity. Will give profit of [ (S+U)erT - 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) erT, or F S e (r+u)T. Only have upper bounds for F on consumption commodities.**H. Commodity Futures**For consumption commodity, F (S+U)erT, or F Se (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 eyT = (S+U) erT. c. If storage costs are constant prop. [u] of S, Pricing Relation 7: then convenience yield, y, is defined so that: F eyT = S e (r+u)T. d. Note: i. For consumption assets, ymeasures 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, & yshould be larger. If y large enough, backwardation (F < S).**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* = Se 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* = Se (r - rf) T; This income ↓ c. d. For commodity, storage costs are like negative income at rate, u; c = r + u; F* = Se (r + u) T;These costs ↑ c. e.Summarizing: For investment asset, F = Se c T; (F > S by amount reflecting c.) For consumption asset, F = Se (c - y) T ; (F > S by amount reflecting the cost of carry, c, net of the convenience yield, y.)**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= Se (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= Se (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)!

Download Presentation

Connecting to Server..