Overview of forwards, Overview of forwards, futures: Pricing & Value futures: Pricing & Value

Selected discussion from Chapters 8 Selected discussion from Chapters 8 and 9and 9

FIN 441FIN 441Spring 2012Spring 2012

Basics of forward and futures Basics of forward and futures contractscontracts

• Forward contract definition:– Agreement between 2 parties calling for delivery of a specific

asset at a specified future date with a price fixed at contract signing.

• Futures contract definition:– Agreement in which 1 party agrees to sell an asset at a price

fixed at contract inception, and another party agrees to buy the asset at the fixed price. Each party deals with the futures exchange (rather than with each other).

• Forwards and futures are, in many ways, the same “type” of derivative.– Linear payoff structure.– Locks in purchase price of asset for buyer of derivative.– Locks in selling price of asset for seller of derivative.

Differences between forwards Differences between forwards and futuresand futures

• Forward contracts are bilateral (i.e., negotiated directly between buyer and seller) while buyers and sellers of futures contracts do not negotiate with one another.

• Forwards are traded OTC (unregulated) while futures are exchange-traded (regulated).

• Forward contracts can be customized to the needs of the transacting parties while futures contracts are standardized by the exchange.

• Forward contracts are settled at contract maturity while futures are settled daily.

• Forward contracts are typically much less “liquid” than are futures contracts.

• Default risk is managed by exchange (clearinghouse) for futures contracts. This risk must be managed directly by forward contract participants.

• Futures contracts will typically expose “hedgers” to “basis risk” (because of underlying standardization) while basis risk should be minimal for hedgers using forward contracts.

Futures exchangesFutures exchanges

• Historically, non-profits, but evolving into for-profit, publicly-held corporations.

• Constantly looking for assets (or more esoteric underlyings) that could generate sufficient trading interest as futures.

• Many exchanges around the world, trading wide array of products.– See Table 8.1 “Exchanges on Which Futures Trade,

November 2008” in Chance & Brooks.– See Chicago Mercantile Exchange (www.cme.com)

for examples

Some basic mechanics of futures Some basic mechanics of futures tradingtrading

• Market order– Implies “buy @ ask” or “sell @ bid.”

• Limit order– Specifies specific price at which to buy or sell.

• Stop order– Useful in preventing catastrophic loss if market makes sudden

turn.• Day order vs. good-till-canceled order• Figure 8.2 outlines process (NOTE: exchanged-traded

option trading process is almost identical…see Figure 2.2).– Difference between futures and options is that both futures

contract buyer and futures contract seller must deposit margin while only option seller deposits margin.

Daily settlement & margin Daily settlement & margin requirementsrequirements

• Credit risk is a major distinction between OTC forwards and exchange-traded futures.– This distinction is also true for OTC vs. exchange-

traded options

• Exchanges address credit risk by requiring “margin” deposits from buyer & seller.– Exchange “clearinghouse” administers margin.– A lot of potential players in this process (see Figure

8.2 in Chance and Brooks).

Mechanics of futures trading – Mechanics of futures trading – order processorder process

• Order is placed by customer.– Buy (long) futures contracts.– Sell (short) futures contracts.– Market, limit, day order, good-till-canceled, etc.

• Broker calls trading desk on exchange floor.• Order is “run” to the trading floor.

– This process has become mostly “electronic” over the last decade.

• When order is filled, details are relayed back to customer.

Mechanics of future trading – Mechanics of future trading – clearing processclearing process

• After order is filled, customer’s initial margin must be deposited (with clearinghouse).– Margin money reflects good-faith deposit that

customer will satisfy obligation.

• At end of each trading day, “settlement price” of futures contract is established.– Customers’ futures contracts are “marked-to-market.”– Is customer’s margin account greater than

maintenance margin?• If “No,” then customer needs to deposit additional funds into

margin account (“variation margin”).

Examples of daily settlementExamples of daily settlement

• Treasury bond futures example– Example similar to Table 8.2 in textbook– Each 1/32 point = $31.25– Aug 1st: Sell one CBOT T-bond futures contract @ 97-27/32– Aug 18th: Buy one CBOT T-bond futures contract @ 100-

16/32– Initial margin = $2,500, and maintenance margin = $2,000.

• Class example– Crude oil spreadsheet

• Students: do problem 11 in Chance & Brooks– Stock index futures

What happens if customer does not What happens if customer does not place “offsetting” order?place “offsetting” order?

• One of the great advantages of futures over forwards is the ease of entering into an “offsetting” trade (because of differences in liquidity).– Example: Buy October futures on August 5, Sell

October futures before expiration of contract.• If original trade is not offset, then the “long” futures

position must take delivery, and “short” futures position must make delivery.– Exchange matches longs and shorts.– One exception: Exchange for Physical (EFP)

Who are the players in futures Who are the players in futures markets?markets?

• Also applies to option markets.• Commission brokers

– Fee-based• Hedgers (i.e., risk managers)

– Use futures contracts to offset correlated business risk.• Speculators

– Scalpers (similar to floor brokers)– Day traders– Position traders

• Spread traders– Could be a hedger or arbitrage trader

• Arbitrage traders– Might often be hedge funds

Detailed examples of a couple of Detailed examples of a couple of futures contractsfutures contracts

• CME Group – http://www.cmegroup.com– Light, sweet crude oil– #2 Heating oil

Light sweet crude oilLight sweet crude oil• Underlying asset: light sweet crude oil

– Delivery in Cushing, OK.• Exchange• Contract size

– 1,000 US barrels (i.e., 42,000 US gallons)• Tick size

– $0.01 per barrel (i.e., $10 per contract)• Price limits

– ???

• Available delivery dates– See “Product Calendar”

• Expiration date– 3rd business day prior to the 25th calendar day of the month.

• Margin– See “Performance Bonds/Margins.”

• Trading volume & Open Interest– See “Settlements”

Heating oilHeating oil• Underlying asset: #2 heating oil

– Delivery point is New York Harbor.• Exchange

– New York, NY @ New York Mercantile Exchange (NYMEX). Part of the CME Group• Contract size

– 1,000 US barrels (i.e., 42,000 US gallons)• Tick size

– $0.0001 per gallon (i.e., $4.20 per contract)• Price limits

– ??????

• Available delivery dates– See “Product Calendar”

• Expiration date– Last business day of the month preceding the delivery month.

• Margin– See See “Performance Bonds/Margins.”

• Trading volume & Open Interest– See “Settlements.”

Cost of carry (carry abitrage) model Cost of carry (carry abitrage) model - Introduction- Introduction

• What does “futures price” or “forward price” mean?

• What does “value” mean when discussing futures or forward contracts?

• Unlike stocks, bonds, options, etc., price and value are entirely different concepts for futures/forward contracts.

Cost of carry model: Initial value of Cost of carry model: Initial value of futures or forwardfutures or forward

• Price of futures or forward is merely the agreed-upon price at which the future delivery will be made.

• Value refers to how much is paid by buyer to enter into contract.

• At inception of futures or forward contract, the value is always “zero!”

Cost of carry model: NotationCost of carry model: Notation

• Vt(0,T) = Value of forward contract created at time 0, as of time t, with expiration at time T.

• Vt(T) = Value of corresponding futures contract.

• F(0,T) = price at time t of forward contract with expiration at time T.

• ft(T) = price at time t of corresponding futures contract.

Cost of carry model: Value of Cost of carry model: Value of forward contractforward contract

• F(T,T) = ST

– Forward price of forward created at time of expiration.– Forward price = spot price of asset (S).– Trivial case, but HAS to be true (or arbitrage).

• VT(0,T) = ST – F(0,T)– Value of forward contract at expiration.– Example: entered into “long” forward to buy asset for $500 in 1 month.

One month later, spot price of asset is $550. How much (opportunity) profit did you earn on the forward contract?

• Vt(0,T) = St – F(0,T)(1+r)-(T-t)

– Value of forward contract prior to expiration.– Use the same example, but suppose it’s 10 days into contract and spot

price is at $540 and annual interest rate is 10% (assume 365 days per year).

– Value = $42.60 (SHOW IT!!!!)

Cost of carry model: Forward price Cost of carry model: Forward price (relative to spot)(relative to spot)

• Based on last equation and fact that forward contract has zero value at creation,– V0(0,T) = S0 – F(0,T)(1+r)-T = 0

• Solving last equation for F(0,T),– F(0,T) = S0(1+r)T

– Forward price is the spot price compounded to the contract’s expiration.

– If not true, then an arbitrage opportunity exists.• Example

– What must the spot price of gold be if 2-year forward contracts for gold are priced at $850? Assume 2-year T-bills yield 1.5% per year? (see kitco.com)

Forward currency priceForward currency price• Illustration of interest rate parity.

– Fundamental relation between spot and forward prices and interest rates in 2 countries.

• F(0,T) = S0(1 +ρ)-T(1+r)T

– ρ = foreign currency interest rate

– r = domestic currency interest rate

– S0 = spot rate (in domestic currency/unit of foreign currency)

– NOTE: can “reverse” the notation!

• Suppose you believe that you can earn higher interest rate in another county.– Example: What should be the forward price (in US$) of NZ$100,000 to

be delivered in 1 year if 1-year US T-bills yield 3.25% per year, the NZ 1-year rate is 7.5%, and the spot rate is NZ$1.3333/US$1 (or US$0.75/NZ$1)?

An example of a forward An example of a forward currency strategycurrency strategy

• Buy NZ$133,333 for US$100,000.• Enters into forward contract to buy US$ with NZ$ in 1 year.• After 1 year, treasurer holds NZ$143,333 (i.e., 133,333 x 1.075).• Convert back to US$. What forward rate at time of original

investment would guarantee that treasurer could convert NZ dollars into $103,250 (i.e., 100,000 x 1.0325)?

• F(0,T) = NZ$1.3882/US$1 (or US$0.7204/NZ$1) = 1.33333(1.075)/(1.0325) = 0.75(1.0325)/(1.075). The answer to the question posed previously is US$72,035 (see forward rate above).

• Moral: even though I can earn higher interest rate in another currency, the forward rate is more expensive (US$0.72 vs. spot of US$0.75).

• See example on page 299 for similar example.

Cost of carry model: Value of Cost of carry model: Value of futures contractfutures contract

• fT(T) = ST

– Futures price at expiration of contract.– Same result as with forward price.

• vt(T) = ft(T) – ft-1(T) before contract is marked-to-market.– Suppose I bought November crude oil at $99.00 at market open

on Oct 1, but at 1 PM, November crude is trading at $98.50.– The contract is worth negative 50 cents per barrel to me.

• vt(T) = 0 as soon as the contract is marked-to-market.– Suppose settlement price for November crude is $99.30.– My account is credited with the $0.30 per barrel gain, so the

futures contract itself has zero value once this happens.

Cost of carry model: Futures price Cost of carry model: Futures price relative to spotrelative to spot

• ft(T) = S0(1+r)T

• This fact is true immediately after each daily settlement (i.e., after marking to market).

• Thus, futures price = forward price.

Cost of carry for underlying that Cost of carry for underlying that generates cash flowsgenerates cash flows

• Standard example: single-stock futures or stock index futures.– Cash flow on underlying = dividends.

• Assume dividend (DT) on stock paid at expiration of futures contract.

• f0(T) = S0(1+r)T – DT– Futures price is equal to the compounded spot price of stock

minus the amount of the dividend.– If dividends are paid at multiple times between time 0 and time

T, DT represents the future value of all expected dividend payments.

Cost of carry with cash flows on Cost of carry with cash flows on underlying (cont’d)underlying (cont’d)

• f0(T) = (S0 – D0)(1+r)T

– D0 is present value of expected dividends.

• f0(T) = S0e(r – δ)T

– If dividends are paid continuously (like on large index), use continuous compounding with δ as the continuous dividend yield.

• Stock futures pricing examples.• Value of forward contract on underlying

with cash flows (see page 297).

Additional issues in futures/forward Additional issues in futures/forward pricingpricing

• Storage costs– Denoted by “s” (page 300).– Futures price equals compounded spot price plus storage costs.

• Risk premium– Under certainty or risk neutrality, price of an asset (today) equals

expected price of asset at future date minus storage costs and incurred interest costs.

– If investors are risk averse AND there is uncertainty about future asset prices, today’s price reflects a risk premium, so the spot price also includes a discount to reflect risk premium.

• Final definition of “cost of carry” (page 302):– Denoted by θ.– Combination of storage costs and “net interest.”– Cost of carry applies to most assets traded on futures exchanges.

Future/forward pricing equilibriumFuture/forward pricing equilibrium

• Futures price equals spot price plus cost of carry.– f0(T) = S0 + θ

• Explain what happens if this equality does not hold.• Pricing Implications:

– Positive cost of carry (true for most commodities)• Contango (futures price > spot price)

– When can cost of carry be negative?– Can futures price < spot price?

• Convenience yield = Premium earned by those who hold inventories of a commodity that is in short supply.

• Backwardation (inverted market)• More common for assets with negligible storage costs (financials).• Occasional occurrence in commodity markets (see first point).

Futures prices and risk premiaFutures prices and risk premia

• We’ve considered the idea of risk premiums for investing in “spot” assets.

• Are there risk premiums for investing in futures contracts?– No risk premium hypothesis

• f0(T) = E(ST)• Futures price is an unbiased expectation of future spot price.

– Risk premium hypothesis• f0(T) < E(ST) = f0(T) + E(φ) = E(fT(T))• Futures prices are (downward) biased expectations of future spot

prices.• Suggested by economists arguing that spot and futures markets are

dominated by those who are “naturally long” in the underlying (i.e., farmers who own wheat, corn, etc.).

• Risk premium could be negative if hedgers are predominantly buyers of futures contracts.

Next 3 classesNext 3 classes

• Hedging with futures (Chapter 11)– Price risk– Short vs. long hedge– Basis and basis risk– Hedge ratio– Liquidating a hedge– Hedging a “spot” transaction vs. hedging an

“ongoing” transaction.

• Introduction to class project