Determination of Forward and Futures Prices

Determination of Forward and Futures Prices

Chapter 5

5.1

The Goals of Chapter 5

5.2

Background knowledge– Investment vs. consumption assets, short selling

(賣空 ), and assumptions for market participants Futures prices for investment assets

– Adjustment for known dollar incomes or yields– Futures on stock indices and foreign currencies– Futures vs. forward prices– Valuing forward or futures contracts

Futures prices for consumption assets– Convenience yield (便利殖利率 ) and cost of carry

theory (持有成本理論 ) Futures price vs. expected spot price

5.1 Background Knowledge

5.3

Consumption vs. Investment Assets Investment assets are assets held for

investment purposes, e.g., stock shares, bonds, domestic or foreign currencies, gold

Consumption assets are assets held for consumption, e.g., copper, oil, pork, corn

※ Reason for distinguishing them: The no-arbitrage argument introduced in Ch. 1 can (cannot) be used to fully determine the forward and futures prices of investment (consumption) assets

※ Difficult to cut clearly: Some investment assets, like gold or silver, have a number of industrial uses and thus can be consumed, so they are consumption assets as well 5.4

Short Selling

Short selling (賣空 ) involves selling securities you do not own– Your broker borrows securities from another clients

and sells them in a market on behalf of you– Earn positive payoffs if the security price declines– At some stage you must buy the securities and

return them back to the accounts of the clients who lend you these securities

– You must pay dividends and any incomes that the owners of the securities should receive in this short selling period (The security owners feel as if they continuously held these securities)

– There may be a small fee for borrowing securities 5.5

Assumptions for Market Participants

Four assumptions associated with market participants– They are subject to no transaction costs when they

trade– They are subject to the same tax rate on their net

trading profits– They can borrow or lend money at the risk-free rate

with unlimited amount– They take advantage of any arbitrage opportunity as

it occurs

5.6

5.2 Futures Prices for Investment Assets

5.7

5.8

Theoretical Futures Price for Investment Assets

The effect of the daily settlement of futures is ignored and suppose the interest rate is constant– Under this simplified assumption, the forward and

futures prices are identical and used interchangeably Suppose there is no income or storage costs for

the underlying asset of futures– The spot price today is , and the futures price today

for delivery in years is . Chapter 1 shows,

where is the risk-free interest rate expressed with annual compounding

5.9

Arbitrage Example for Gold Futures

Suppose that– Spot price of gold today is $1000– 1-year gold futures price today is $1100 ($990)– The interest rate is 5% per annum– No income or storage costs for gold

The theoretical value of the futures price on gold is $1,000×(1+5%)=$1,050– Futures price > $1050 Buy the gold spot and take

a short position of the 1-year futures on gold– Futures price < $1050 (Short) sell the gold spot

and take a long position of the 1-year futures on gold

When Interest Rates are Measured with Continuous Compounding

The theoretical futures price expressed with continuous compounding is

,where is the risk-free zero rate, with continuous compounding, for the time to maturity

※ This equation holds for any investment asset that provides no income and has no storage costs, e.g., non-dividend-paying stocks

※ This course always employs the formulae expressed with continuous compounding

5.10

Consider a Known Dollar Income of Investment Assets

When an investment asset provides a known dollar income at time point , then

,where is the present value of the income ※ If there are multiple dollar incomes in , is the sum of the

present values of them

– An intuitive way to understand this formula You can treat as the stock price and as the cash dividend

payment at time It is known that after the payment of the cash dividend at ,

an identical amount is deducted from the stock price To reflect the above situation today, the PV of the cash

dividend payment , i.e., , should be deducted from the current stock price 5.11

Consider a Known Dollar Income of Investment Assets Suppose = $900, an income of $40 occurs at

4 months, and 4-month and 9-month rates are 3% and 4% per annum. If the 9-month futures price is $910 (or $870), is there any arbitrage opportunity?– The PV of the income at 4 months is – The theoretical futures price is

※As long as the futures price deviates from this theoretical price, there is an arbitrage opportunity

5.12

Consider a Known Dollar Income of Investment Assets

For , which is overvalued than its theoretical value– At

Borrow $900: $39.6 for 4 months and $860.4 for 9 months Buy one unit of asset at = $900 Enter into a short position of the 9-month futures ()

– At 4 months Receive $40 of income from the asset Use this $40 (= $39.6) to repay the first loan

– At 9 months Sell the asset through the futures for $910 Use $860.4$886.6 to repay the second loan Profit realized = $910 – $886.6 = $23.4

5.13

Consider a Known Dollar Income of Investment Assets

For , which is undervalued than its theoretical value– At

Short sell one unit of asset at = $900 Invest $39.6 for 4 months and $860.4 for 9 months Enter into a long position of the 9-month futures ()

– At 4 months Receive $39.6$40 from the 4-month investment Use this $40 to pay the lender of the asset

– At 9 months Receive $860.4$886.6 from 9-month investment Buy the asset through the futures for $870 Return the asset to the lender Profit realized = $886.6 – $870 = $16.6 5.14

Consider a Known Yield Income of Investment Assets

When an investment asset provides a known yield income (with continuous compounding) in the period , then

– An intuitive way to understand the formula

A yield income means that the income is expressed as a percent of the asset’s price at the time the income is paid

Similar to the dollar income, whenever the yield income is paid to the asset holder, there is a negative impact on the asset price– Suppose the asset price is today and the asset provides an

annual yield income of 5.15

Consider a Known Yield Income of Investment Assets

– Annual payment frequency for :

– Semiannual payment frequency for :

– When the payment frequency approaches infinity

– Thus, the term reflects the negative impact of the yield income on the asset price for a year If , annual amount of is paid to the asset holder

Based on the original formula , if the negative impact of the yield income is considered, the formula should be adjusted as – Note that the role of is similar to the role of in the futures

price formula on Slide 5.115.16

Consider a Known Yield Income of Investment Assets

The no-arbitrage argument for the formula of – If

Buy units of the asset at today by borrowing dollars– Invest the continuously generated yield income in the same

asset, and thus by time , it is expected to have units of the asset

Enter into a futures to sell units at The final sales proceeds from the futures position, , minus

the repayment amount of the debt, , can generate a positive payoff, i.e.,

5.17

Consider a Known Yield Income of Investment Assets

– If Short sell units of the asset at today and deposit the

proceeds in a bank to earn the interest rate – When continuously generated yield income is paid on the

asset, you owe more on the short position. As a result, the short selling position grows at the rate and thus the arbitrageur needs to return units at time

Enter into a futures to buy units at The interest and principal of the deposit, , minus the fund

to buy units at , , can generate a positive payoff, i.e.,

– To eliminate the above two arbitrage opportunities, we can derive theoretically

5.18

Stock Index Futures

The underlying asset of stock index futures is a stock index level, which can be viewed as an investment asset paying a yield income– A stock index reflects the performance of a portfolio

of stocks– It is infeasible to take the PVs of all cash dividends

of all stocks in this portfolio into account– In practice, the continuous compounding dividend

yield is estimated for this portfolio– The futures and spot prices of a stock index futures

follows , where is the continuous compounding dividend yields on the stock index portfolio during the life of the futures contract 5.19

Index Arbitrage

When , an arbitrageur buys the stock index portfolio and takes a short position of stock index futures

When , an arbitrageur takes a long position of futures and (short) sells the stock index portfolio

※ The above two strategies are known as index arbitrage and the details are similar to the trading strategies introduced on Slides 5.17 and 5.18

5.20

Index Arbitrage

Index arbitrage involves simultaneous trades in futures and many different stocks – Very often a computer program is used to generate

the trades, which is known as program trading – During a financial crisis, simultaneous trades could

be not possible and the no-arbitrage relationship between and does not hold On Oct. 19, 1987 (Black Monday), the S&P 500 index was

225.06 (down 57.88 on that day) and the futures price for the Dec. S&P 500 index futures was 201.5 (down 80.75 on that day)

The overloaded system on exchanges delays the execution of orders and thus the index arbitrage becomes infeasible 5.21

The futures contract on the Nikkei 225 Index in CME views 5 times the Nikkei 225 Index, which is measured in yen, as a dollar number– Suppose you take a long position of the Nikki 225

index futures with to be 1000, and on the delivery date, the Nikki 225 index is 1100 Your payoff is USD$5×(1100 – 1000) = USD$500

– Note that traders cannot trade the stock index portfolio underlying the Nikkei 225 Index in USD The formula of cannot apply to Nikkei 225 index futures,

which is a “quanto” futures (匯率連動期貨 ) where the underlying asset is measured in one currency and the payoff is in another currency 5.22

Stock Index Futures

Futures and Forwards on Currencies A foreign currency is analogous to a security

providing a yield income– The foreign risk-free interest rate is the yield income

an investor can earn if he holds that currency– It follows that if the dividend yield is replaced with

the foreign risk-free interest rate, we can derive the futures price as

() is the spot (futures) price of the foreign currency in terms of the domestic currency, i.e., the current exchange rate (the exchange rate applied on the delivery date)

The and are the domestic and foreign risk-free interest rates, respectively 5.23

Why the Relation Must Be True (US$ is the Domestic Currency)

5.24

※ If , an arbitrage opportunity occurs※ To eliminate all arbitrage opportunities, we can derive

Enter into a foreign currency futures to sell units of foreign currency at

Forward vs. Futures Prices

Forward and futures prices are usually assumed to be the same

When interest rates are stochastic (隨機 ), forward and futures prices could be different due to the enforcement of daily settlement for trading futures– The difference between forward and futures prices

can be significant if there exists a relationship between the interest rate and the underlying variable, e.g., Eurodollar futures introduced in Ch. 6

5.25

Forward vs. Futures Prices

– A positive correlation between interest rates and the price of the asset underlying the futures () With the increase of the asset price (gains for longing

futures), the futures holder can earn doubly from the increase of the balance of the margin account and the higher interest rate

() With the decrease of the asset price (losses for longing futures), the futures holder losses some funds from the margin account, but the opportunity cost for these losses is low due to the lower interest rate

Thus, the futures contract is more attractive and demanded so that futures price > forward price

5.26

Forward vs. Futures Prices

– A negative correlation between interest rates and the asset price () With the increase of the asset price (gains for longing

futures), the benefit of the increase of the balance of the margin account will be offset by the lower interest rate

() With the decrease of the asset price (losses for longing futures), the balance of the margin account decreases such that the futures holder cannot fully enjoy the higher interest rate

Thus, the futures contract is relatively not attractive so that futures price < forward price

5.27

Valuing a Futures or a Forward Contract Suppose that is the delivery price specified in

a futures contract and at time point , is the current futures price that would apply to newly-created futures contracts– The value of a long futures contract at is

,and the value of a short futures contract at is

,where is the risk-free interest rate corresponding to the maturity of

※ Note that the values of the long and short positions are with the same magnitude but with opposite signs 5.28

At , , , , and

※ In practice, when a futures is initiated, the delivery price is set to the current futures price and thus the initial value of the futures equals 0

At (after half a year), , , , and assume

※ With the passage of time, as long as the futures price (determined by the demand and supply of futures) does not equal the delivery price , the value of the futures emerges

5.29

Valuing a Futures or a Forward Contract (for a long position)

5.30

Theoretical Futures/Forward Prices and Futures Values

AssetTheoretical

futures/forward prices ()

Theoretical value of the long position of a futures/forward

()

Without any incomeWith known income whose present value is at With known yield income

※ The current time point is , the maturity time point is , and is the delivery price specified in futures/forwards contracts

※ If is 0 (at the beginning of a contract), the theoretical futures/forward prices () are identical to those introduced on Slides 5.10, 5.11, and 5.15

※ The formulas of the theoretical values of futures/forward contracts in the third column can be obtained by replacing the futures/forward prices with the formulas in the second column

5.3 Futures Prices for Consumption Assets

5.31

Futures Prices for Consumption Assets

Commodities that are consumption assets rather than investment assets usually provide no income, but are subject to significant storage costs– The first way to model the storage cost:

,where is the storage cost per unit time as a percent of the asset value (The arbitrage strategies on gold leading to this formula are on Slides 1.35 to 1.37)

– Alternative way (for the one-time payment of storage costs):

,where is the present value of the storage costs 5.32

Futures Prices for Consumption Assets

– If Borrow at the risk-free rate and use it to purchase one unit of

the commodity and to pay storage costs Short a futures contract on one unit of the commodity Yield a positive payoff of cannot hold

– If Sell the commodity at , save the PV of the storage costs, ,

and invest the proceeds () to earn for years Take a long position in a futures contract Yield a positive payoff of ※Owners of a commodity are reluctant to do so because they

can consume the commodity but cannot consume the long position of a futures or forward contract

5.33

Futures on Consumption Assets

Due to the concern for the need of using or consuming commodities, the relationship between the futures and spot prices of a consumption commodity is

,where is the storage cost per unit time as a percent of the asset value, or is

,where is the present value of the storage costs

5.34

Convenience Yield and Cost of Carry

The benefits from holding the physical asset are measured as the convenience yield (便利殖利率 ), , provided by the commodity

– The convenience yield reflects the concern of the future availability of the commodity The greater the possibility that shortages will occur, the higher the

convenience yield– Use to explain normal and inverted markets on Slides

2.28 and 2.29 If increases with normal market If decreases with inverted market 5.35

Convenience Yield and Cost of Carry The relationship between futures and spot

prices can be unified in terms of cost of carry (持有成本 )– The cost of carry, , is the interest costs plus the

storage cost less the income earned, i.e., – For an investment asset, – For a consumption asset, – The convenience yield on the consumption

asset, , is defined so that ※Note that for investment assets, the convenience yield must

be zero to eliminate arbitrage opportunities5.36

5.4 Futures Price vs. Expected Spot Price

5.37

Relationship Between Futures Prices and Expected Spot Prices

Is the futures price an unbiased estimate of the expected spot prices on the delivery date?1. Keynes and Hicks: Hedgers prepare to lose (accept

unfavorable future prices, but speculators hope to make money (accept only favorable future prices) Hedgers hold short, speculators hold long

– Speculators buy undervalued futures, i.e., , and hedgers would sacrifice (to accept undervalued futures price) in order to reduce risks

Hedgers hold long, speculators hold short – Speculators sell overvalued futures, i.e., , and hedgers would

sacrifice (to accept overvalued futures price) in order to reduce risks

5.38

2. Analysis of the systematic risk based on the CAPM Suppose is the expected return required by investors on

an asset with the price We can invest the amount of now (to earn the risk-free rate

to get back at maturity via a futures contract with the futures price

The present value of the expectation of (discounted by ) should be the initial investment

According to the CAPM, if the asset has– no systematic risk (), then and

is an unbiased estimate of – positive systematic risk (), then and – negative systematic risk (), then and 5.39

Relationship Between Futures Prices and Expected Spot Prices

※ (Normal) backwardation (逆價差 ) is the market condition where the price of a forward or futures contract is trading below the expected spot price, i.e., (Backwardation is the normal case in practice since for most assets, they have positive systematic risk)

※ Contango (正價差 ) is the market condition where the price of a forward or futures contract is trading above the expected spot price, i.e.,

※ Note that in practice, the backwardation and contango are sometimes used to refer to whether the futures prices is below or above the current spot price, i.e., the backwardation is for and the contango is for

5.40

Relationship Between Futures Prices and Expected Spot Prices