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Published on October 4, 2007

Author: Callia

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Chapter 14: Forward & Futures Prices:  Chapter 14: Forward & Futures Prices Copyright © Prentice Hall Inc. 2000. Author: Nick Bagley, bdellaSoft, Inc. Objective How to price forward and futures Storage of commodities Cost of carry Understanding financial futures Chapter 14: Contents:  Chapter 14: Contents 1 Distinction Between Forward & Futures Contracts 2 The Economic Function of Futures Markets 3 The Role of Speculators 4 Relationship Between Commodity Spot & Futures Prices 5 Extracting Information from Commodity Futures Prices 6 Spot-Futures Price Parity for Gold 7 Financial Futures 8 The “Implied” Risk-Free Rate 9 The Forward Price is not a Forecast of the Spot Price 10 Forward-Spot Parity with Cash Payouts 11 “Implied” Dividends 12 The Foreign Exchange Parity Relation Features of Forward Contract:  Features of Forward Contract Two parties agree to make an exchange in the future at a pre-determined price The forward price is the delivery price such that the current market value of the forward contract is zero No money or services are exchanged when the contract is agreed to Face value = quantity * forward price Long position: buyer; short position: seller Futures Contracts:  Futures Contracts Futures contracts are standardized, traded forward contracts Disadvantage of forward: because it is unique, it cannot be traded, so both parties must commit; if it were a futures contract, a party could “exit” the contract by selling it Exiting a contract: called “terminating” With futures, exchange determines commodity, size of trade, and delivery time & place Futures Listings:  Futures Listings Have list of futures contracts traded on Chicago Board of Trade (CBT) of same quantity and commodity; only difference is delivery month Open (opening price), High/Low (that day’s high and low), Settle (the average settlement price near end of the day), Change (from previous settlement price), Lifetime High/Low, Open Interest (no. of contracts outstanding at end of day) Futures Contracts:  Futures Contracts Parties have contracts with CBT, not each other CBT matches long and short positions Orders are executed by brokers on the floor of the CBT (who paid for their seat) To ensure that parties do not default, all exchanges require that there be some collateral in each account to cover losses - called margin requirement All account are marked to market at end of day based on average settlement price Characteristics of Futures:  Characteristics of Futures Futures are: standard contracts immune from the credit worthiness of buyer and seller because exchange stands between traders contracts marked to market daily margin requirements Futures Example:  Futures Example Take a long position in futures contract of 5000 bushels of wheat to be delivered in Sep Broker requires you to deposit some money in the account (e.g. $2000) as collateral The day after purchase of contract, the futures price falls 10 cents per bushel, so have lost $0.10*5000 = $500 Broker takes $500 from your account and gives it to exchange, who transfers it to a party with a short position Futures Contracts:  Futures Contracts If collateral in your account falls below some level (determined by broker and/or exchange), then will receive a margin call from broker asking for money If do not respond that day, then broker liquidates your position at market price and returns remaining collateral (if any) Because have daily marking to market, market value of all contracts is always zero at beginning of the day (irrespective of face value) This minimizes prob. of default Futures vs. Forward Contracts:  Futures vs. Forward Contracts Anyone can use futures contract since your credit worthiness is not checked - margin requirements guarantee your position Forward contracts are thus used when credit worthiness is easy to verify, such as with banks Forward contracts are not necessarily marked to market each day Storage of Commodities:  Storage of Commodities Futures contracts allow one to hedge against risk associated with storing a commodity Futures contracts allow one to separate these two decisions: Whether to physically store a commodity Whether to be exposed to financial risk via price changes Storage: Example:  Storage: Example Current harvest will come in one month; have one ton left of wheat from last harvest Spot price is $2; one-month futures price is F To eliminate exposure to price risk, can either sell at $2 today, or go short at F Storage: Example:  Storage: Example Consider distributor of wheat Cost of carry = cost of physically storing commodity = interest, warehousing, spoilage costs For distributor, cost of carry = 10 cents/bushel/mon Distributor will go short with contract if F exceeds $2.10 (in which case will store wheat for one month); otherwise, will sell at spot price of $2 Another distributor might have cost of carry=15 cents, so he would sell today at spot price if F is less than $2.15 Storage & Spread:  Storage & Spread Let S be spot price, F futures price (one month), and C cost of carry (one month) Distributor will hold wheat in storage for another month (I.e. buy future) if C < F - S Call F - S the spread: it determines how much wheat will be stored from one month to the next Speculators vs. Hedgers:  Speculators vs. Hedgers A hedger is hedging against risk: includes farmers, bakers, producers, distributors, etc. A speculator buys/sells to make a (short-term) profit: he takes positions based on his expectation about prices in the future Example: speculator expects price to be $2 in one month; if F is less than $2, should go long (so that can buy it one month from now at less than $2) If F is greater than $2, should go short (so that can sell one month from now at more than $2) Spot and Futures Prices:  Spot and Futures Prices Consider following arbitrage Buy wheat at spot price S and store it for one month at cost of carry C Can guarantee today to sell it one month from now at price F via futures contract If F > S + C, then can make risk-free profit No arbitrage opportunities can exist in equilibrium, placing upper bound on spread: Extracting Information:  Extracting Information Theoretically, futures price reflects what investors think price will be in the future Two cases to be considered: (Case I) Stock out: when there is no wheat in storage If no wheat is being stored, means would lose money if do so, implying total cost S + C exceeds payoff F, so F - S < C In this case, forward price provides some info about expected price that is not built into spot price Extracting Information:  Extracting Information If wheat is being stored, then spot price contains all relevant info about expectations: forward price does not give more info This is so because in this case F - S = C (to ensure no arbitrage opportunities exist) Hence, futures price is completely determined by spot price and cost of carry Alternatively, can use spot and futures prices to estimate cost of carry (when commodity is being stored) Forward-Spot Price-Parity for Gold:  Forward-Spot Price-Parity for Gold There are two ways to invest in gold buy an ounce of gold at S0, store it for a year at a storage cost of $h (per unit of $S0) and sell it for S1 Forward-Spot Price-Parity for Gold:  Forward-Spot Price-Parity for Gold Alternative: invest in synthetic gold Invest S0 (the spot price) in a 1-year T-bill with return r, and purchase a gold forward at F (go long) for delivery in 1-year Forward-Spot Price-Parity for Gold:  Forward-Spot Price-Parity for Gold According to Law of One Price, both investments must yield the same return Gold Example:  Gold Example Suppose spot price for one ounce of gold is $300, storage costs are 2% per year, and the risk-free rate is 8% Then one-year Forward price should be (1 + 0.02 + 0.08) * 300 = $330 If forward price > $330, can make arbitrage by borrowing at risk-free rate, buying gold at spot price, and at same time selling it for future delivery at forward price If forward price < $330, sell gold short in the spot market, invest the proceeds in the risk-free asset, and go long (buy) the forward contract Gold Arbitrage Example:  Gold Arbitrage Example Suppose F > (1+r+h)*S Borrow S at risk-free rate: CF today = S, CF in 1 yr = -(1+r)*S Invest proceeds in gold: CF today = -S, CF in 1 yr = S1 - h*S Go short forward contract: CF today = 0, CF in 1 yr = F - S1 Net: CF today = 0, CF in 1 yr = F - (1+r+h)*S > 0 Forward-Spot Price-Parity for Gold:  Forward-Spot Price-Parity for Gold A contract with life T: This is not a causal relationship: the forward and current spot price are jointly determined by the market If we know one, then the Law of One Price determines the other Rule of One Price: No Arbitrage Profits:  Rule of One Price: No Arbitrage Profits Purchase Actual Au Sell T-Bill Sell Au Forward Sell Actual Au Settle T-Bill Settle Au Forward Au = Gold Implied Cost of Carry:  Implied Cost of Carry Because of forward-spot price parity relationship, you can’t extract information about the expected future spot price of gold (unlike wheat case) from futures prices - because gold is always stored However, can infer implied cost of carry from spot and forward prices of gold Implied cost of carry = F - S (spread) The implied storage costs are the implied cost of carry per $spot, so Financial Futures:  Financial Futures Financial futures promise future delivery of stocks, bonds, and currencies Securities can be held at very low cost since they are not physical, so can ignore carrying costs At settlement date, securities are not transferred; rather, exchange cash Example: Forward price of stock is $108/share; if stock price at delivery date turns out to be $109, then party that went long gets $1/share from party that went short Financial Futures:  Financial Futures Consider replicating the cash flow of holding a stock using risk-free asset and forward Current stock price = S, stock price in one year = S1, risk-free rate = r% If buy stock today, cash flow today = -S, and cash flow in one year is S1 Consider a synthetic stock (next slide) Synthetic Stock:  Synthetic Stock Go long a forward contract: cash flow today = $0, cash flow in one year = S1 - F Buy a pure discount bond with face value F: cash flow today = PV of bond = -$F/(1+r), cash flow in one year = F Net: cash flow today = -F/(1+r), cash flow in one year = S1 Stock: cash flow today = -S, cash flow in one year = S1 The cash flow today must be equal in equilibrium: Financial Futures:  Financial Futures In general, if maturity of zero-coupon bond and forward contract is T years, with no storage cost, the relationship between the forward and spot is Any deviation from this will result in an arbitrage opportunity Financial Futures Example:  Financial Futures Example Suppose arbitrage equation is violated: let r=8%, S=$100, F=$109 Sell forward contract: CF today = 0, CF in 1 yr = $109 - S1 Borrow $100 at risk-free rate: CF today = $100, CF in 1 yr = -$108 Buy a share of stock: CF today = -$100, CF in 1 yr = S1 Net: CF today = 0, CF in 1 yr = $1 The “Implied” Risk-Free Rate:  The “Implied” Risk-Free Rate We showed that one can replicate the stock using the risk-free asset and a forward Hence, can replicate a pure-discount bond by buying a share of stock and taking a short position in a forward contract One-year pure discount bond has face value F Buy a share of stock at S, and sell forward with forward price F The “Implied” Risk-Free Rate:  The “Implied” Risk-Free Rate Buying bond: CF today = -F/(1+r), CF in 1 yr = F Synthetic Bond: Buy a share of stock: CF today = -S, CF in 1 yr = S1 Sell forward F: CF today = 0, CF in 1 yr = F - S1 Net: CF today = -S, CF in 1 yr = F To have no arbitrage opportunity, require that CF today be the same in both scenarios: The “Implied” Risk-Free Rate:  The “Implied” Risk-Free Rate Rearranging, the implied interest rate on a forward given the spot is This is reminiscent of the formula for the interest rate on a discount bond The Forward Price is not a Forecast of the Spot Price:  The Forward Price is not a Forecast of the Spot Price Suppose risk premium of a stock is p%, and risk-free rate is r%, and current spot price is S This means expected spot price is (1+p+r)*S However, forward-spot price-parity formula says that forward price should be F = (1+r)*S Hence, F does not equal expected spot price Forward-Spot Parity with Cash Payouts:  Forward-Spot Parity with Cash Payouts Up to now have assumed that stocks do not pay dividends - what happens if they do? Suppose stock is expected to pay dividend D Cannot replicate stock exactly since dividend is expected, not known for certain To replicate this stock using risk-free asset and forward, must now purchase bond with face value of F + D Forward-Spot Parity with Cash Payouts:  Forward-Spot Parity with Cash Payouts Buy stock: CF today = -S, CF in 1 yr = D + S1 Synthetic Stock: Go long futures contract: CF today = 0, CF in 1 yr = S1 - F Buy zero-coupon bond with face value D + F: CF today = - (D + F)/(1+r), CF in 1 yr = D + F Net: CF today = - (D + F)/(1+r), CF in 1 yr = D + S1 To prevent arbitrage, CF today must be equal: Forward-Spot Parity with Cash Payouts:  Forward-Spot Parity with Cash Payouts The S0 - F relationship becomes Forward price > Spot price if D < r S, or D/S < r (recall: D/S = dividend yield) Because D is not known with certainty, this is a quasi-arbitrage situation “Implied” Dividends:  “Implied” Dividends From the last slide, we may obtain the implied dividend This is the dividend the market expects to get when prices are F and S Foreign Exchange Rate Parity:  Foreign Exchange Rate Parity Consider link between forward price of a currency and its spot price Maturity of forward contract is t years Denominate prices in dollars Foreign Exchange Rate Parity:  Foreign Exchange Rate Parity Buy a yen-bond: CF today = -S/(1+rY), CF in 1 yr = S1 Synthetic Yen-Bond: Go long on forward: CF today = 0, CF in 1 yr = S1 - F Buy dollar bond with FV F: CF today = - F/(1+r$), CF in 1 yr = F Net: CF today = - F/(1+r$), CF in 1 yr = S1 To avoid arbitrage, CF today must be equal: Interest Rate Forward Contracts:  Interest Rate Forward Contracts Forward rate agreement (FRA): agreement to exchange cash flows based on reference interest rate (e.g. 3-month T-Bill rate) and principal amount at a single time in the future First party pays the second if interest rate on future date exceeds reference rate, otherwise reverse Size of payment depends on Notional principal: principal amount Spread between actual rate and reference rate

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