Determination of Forward and Futures Prices Chapter 5

Determination of Forward and Futures Prices Chapter 5

Consumption vs Investment Assets: That is held for investment purposes by significant numbers of investors. (Examples: stocks, bonds, gold, silver) n Consumption Assets: That is held by primarily for consumption. (Examples: copper, oil, pork) n

Short Selling n n Short selling involves selling securities you do not own Your broker borrows the securities from another client and sells them in the market in the usual way Required to maintain a margin account with the broker You must pay dividends and other benefits the owner of the securities receives

Cash flows form short sale and purchase of shares Purchase of shares April : Purchase 500 shares for $120 -$60, 000 May : Receive dividend +$500 July : Sell 500 shares for $100 per share +$50, 000 Net profit= -$9, 500 --------------------------------------Short sale of shares April : Borrow 500 shares and sell them for $120 +$60, 000 May : Pay dividend - $500 July : Buy 500 shares for $100 per share -$50, 000 Replace borrowed shared to short position Net profit= +$9, 500

Assumption and Notation n Assumption: 1. No transaction costs when they trade. 2. The same tax rate on all net trading profits. 3. Borrow money at the same risk-free rate of as they can lend money. 4. Take advantage of arbitrage opportunities as they occur.

Assumption and Notation ﹡NOTATION: S 0: Price of the asset underlying the forward or futures contract today F 0: Futures or forward price today T: Time until delivery date r: Risk-free interest rate for maturity T

Forward Price For an Investment Asset Assume : S 0 = $40, r = 5%, t = 3 months (a)If F 0 =$43 > S 0 ert 1. Borrow $40 at risk-free interest rate of 5% per annum. 2. Short a forward contract to sell one share in 3 -months. $40 e 0. 05 x 3/12 = $40. 5 $43 - $40. 5 = $2. 5

(b)If F 0 =$39 < S 0 ert 1. Short one share, invest the proceeds of the short sale at 5% per annum for 3 months. 2. Take a long position in a 3 -months forward contract. $40 e 0. 05 x 3/12 = $40. 5 - $39 = $1. 5 ∴We deduce that for there to be no arbitrage the forward price must be exactly $40. 5.

F 0 = S 0 er. T This equation relates the forward price and the spot price for any investment asset that provides no income

What If Short Sale Are Not Possible? (a)If F 0 > S 0 ert 1. Borrow S 0 dollars at an interest rate r for T years. 2. Buy 1 ounce of gold. 3. Short a forward contract on 1 ounce of gold. The investor make a profit of F 0 - S 0 ert.

(b)If F 0 < S 0 ert 1. Sell the gold for S 0. 2. Invest the proceeds at interest rate r for time T. 3. Take a long position in a forward contract on 1 ounce of gold. The investor make a profit of S 0 ert - F 0.

When an Investment Asset Provides a Known Dollar Income F 0 = (S 0 – I )er. T where I is the present value of the income during life of forward contract

Known Income Assume : S 0 = $900 I = 40 e-0. 03 x 4/12 = $39. 6 r =0. 04 T = 0. 75(9/12) I: ? 0 $40 4 9 F 0 = (900. 00 – 39. 6)e 0. 04 x 0. 75 = $886. 60

(a)If F 0 = $910 > (S 0 - I)ert = $886. 60 1. Borrow $900 to buy the bond. 2. Short a forward contract. → → → 900. 00 - 39. 6 = $860. 40 e 0. 04 x 0. 75 = $886. 60 910. 00 -886. 60 = $23. 40

(b)If F 0 = $870 < (S 0 - I)ert = $886. 60 1. Short the bond. 2. Enter into a long forward contract. → 900 - 39. 6 = $ 860. 4 → 860. 40 e 0. 04 x 0. 75 = $886. 60 → 886. 60 - 870 = $16. 60 ∴The forward price must be $886. 60 5. 15 Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005

When an Investment Asset Provides a Known Yield F 0 = S 0 e(r–q )T where q is the average yield during the life of the contract (expressed with continuous compounding)

Known Yield n Assume : S 0 = 25, r = 0. 1, and T = 0. 5, the yield is 4% per annum with semiannual compounding. 1+0. 04 = (1+q/2)2 q = 3. 96% F 0 = 25 e(0. 10 – 0. 0396)x 0. 5 = $25. 77

Valuing a Forward Contract n K is delivery price in a forward contract F 0 is forward price today ƒ : Value of forward contract today The value of a long forward contract, ƒ, is ƒ = (F 0 – K )e–r. T n Similarly, the value of a short forward contract is 　 (K – F 0 )e–r. T n

n The value of a forward contract on an investment asset that provides no income: ƒ = (F 0–K)e-rt Equation shows that F 0 = S 0 ert ƒ = (S 0 ert–K)e-rt ƒ = S 0 – Ke-rt n The value of a long forward contract on an investment asset that provides a known income with present value I: ƒ = S 0 – I – Ke-rt n The value of a long forward contract on an investment asset that provides a known yield at rate q: ƒ = S 0 e-qt – Ke-rt

Forward vs Futures Prices n n A strong positive correlation between interest rates and the asset price implies the futures price is slightly higher than the forward price A strong negative correlation implies the reverse Last only a few months are in most circumstances sufficiently small to be ignored Forward and futures prices are usually assumed to be the same. When interest rates are uncertain they are, in theory, slightly different

Futures Prices Of Stock Index n n Can be viewed as an investment asset paying a dividend yield The futures price and spot price relationship is therefore F 0 = S 0 e(r–q )T where q is the average dividend yield on the portfolio represented by the index during life of contract

Futures Prices Of Stock Index n F 0 = S 0 e(r-q)T q: The dividend yield Example: r = 0. 05 S 0 = 1, 300 T = 3/12 (0. 25) F 0 = 1, 300 e(0. 05 -0. 01)x 0. 25 = $1, 313. 07 q = 0. 01

Index Arbitrage n If F 0 > S 0 e(r-q)T 1. Buying the stocks underlying the index at the spot price 2. Shorting futures contracts By a corporation holding short-term money market investment.

Index Arbitrage n If F 0 < S 0 e(r-q)T 1. Shorting or selling the stocks underlying the index 2. Taking a long position in futures contracts By a pension fund that owns an indexed portfolio of stocks

Index Arbitrage n n Program trading Occasionally (e. g. , on Black Monday) simultaneous trades are not possible and theoretical no-arbitrage relationship between F 0 and S 0 does not hold

Futures and Forwards on Currencies Two ways of converting 1, 000 units of a foreign currency to dollars at time T. Here, S 0 is spot exchange rate, F 0 is forward exchange rate, and rf are the dollar and foreign risk-free rates.

Futures and Forwards on Currencies 1, 000 erf. T F 0 = 1, 000 S 0 er. T F 0 = S 0 er. T / erf. T n The relationship between F 0 and S 0

Futures on Commodities n Income and Storage Costs (a)In the absence of storage costs and income, the forward price of a commodity that is an investment asset is give by: F 0 = S 0 er. T (b)If U is the present value of all the storage costs, net of income, during the life of a forward contract: F 0 = (S 0 + U)er. T

(c)If the storage costs net of income incurred at any time are proportional to the price of the commodity, they can be treated as negative: F 0=S 0 e(r+u)T Where u denotes the storage costs per annum as proportion of the spot price net of any yield earned on the asset.

Futures on Consumption Assets (a) F 0 > (S 0 + U)er. T 1. Borrow an amount S 0 + U at the risk-free rate and use it to purchase one unit of the commodity and to pay storage costs. 2. Short a forward contract on one unit of the commodity.

(b) F 0 < (S 0 + U)er. T 1. Sell the commodity, save the storage costs, and invest the proceeds at the risk-free interest rate. 2. Take a long position in a forward contract.

Futures on Consumption Assets F 0 S 0 e(r+u )T where u is the storage cost per unit time as a percent of the asset value. Alternatively, F 0 (S 0+U )er. T where U is the present value of the storage costs.

Convenience Yield * The benefits from holding the physical asset are sometimes referred to as the convenience yield. If the dollar amount of storage costs is known and has a present value U, that the convenience yield y is defined such that: F 0 ey. T = ( S 0 + U )er. T If the storage costs per unit are a constant proportion, u, of the spot price, then y is defined so that: F 0 ey. T = S 0 e(r+u)T or F 0 = S 0 e(r+u-y)T

The Cost of Carry n n n The cost of carry, c, is the storage cost plus the interest costs less the income earned For a non-dividend-paying stock, it is r. For a stock index, it is r - q. For a currency, it is r - rf. For a commodity that provide income at rate q and require storage costs at rate u, it is r - q + u.

The Cost of Carry n n Define the cost of carry as c. For an investment asset , the futures price is F 0 = S 0 ec. T n For a consumption asset, The convenience yield on the consumption asset, y, is defined so that F 0 = S 0 e(c–y )T

Delivery Options n Form equation (F 0 = S 0 e(c–y )T ) that c > y, the benefits from holding the asset (including convenience yield and net of storage costs) are less than the risk-free rate. n If futures prices are decreasing as time to maturity increase (c < y). It is then usually optimal for the party with the short position to deliver as late as possible, and futures prices should, as a rule, be calculated on this assumption

The Risk in a Futures Position The cash flow to the speculator are as follow : Today : - F 0 e-r. T End of futures contract : +ST The futures prices today : F 0 The prices of the asset at time T : ST The risk-free return on funds invested for time : T The investor's required return : k The expected value : E The PV of this investment : - F 0 e-r. T + E(ST)e-k. T Assume net present value = 0 - F 0 e-r. T + E(ST)e-k. T = 0 F 0 = E(ST)e(r-k)T

The Risk in a Futures Position n If the asset has q no systematic risk, then k = r F 0 = E(ST) and F 0 is an unbiased estimate of ST q positive systematic risk, then k > r and F 0 < E (ST ) q negative systematic risk, then k < r and F 0 > E (ST )

Normal Backwardation and Contango Normal backwardation: When the futures price is below the expected future spot price. Contango: When the futures price is above the expected future spot price.