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Determination of Forward and Futures Prices Chapter 3. Arbitrage: A market situation whereby an investor can make a profit with: no equity and no risk. Efficiency: A market is said to be efficient if prices are such that there exist no arbitrage opportunities.

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slide2
Arbitrage:

A market situation whereby an investor can make a profit with: no equity and no risk.

  • Efficiency:

A market is said to be efficient if prices are such that there exist no arbitrage opportunities.

Alternatively, a market is said to be inefficient if prices present arbitrage opportunities for investors in this market.

slide3
SHORT SELLING STOCKS

An Investor may call a broker and ask to “sell a particular stock short.”

This means that the investor does not own shares of the stock, but wishes to sell it anyway.

The investor speculates that the stock’s share price will fall and money will be made upon buying the shares back at a lower price. Alas, the investor does not own shares of the stock. The broker will lend the investor shares from the broker’s or a client’s account and sell it in the investor’s name. The investor’s obligation is to hand over the shares some time in the future, or upon the broker’s request.

slide4
SHORT SELLING STOCKS

Other conditions:

The proceeds from the short sale cannot be used by the short seller. Instead, they are deposited in an escrow account in the investor’s name until the investor makes good on the promise to bring the shares back.

Moreover, the investor must deposit an additional amount of at least 50% of the short sale’s proceeds in the escrow account.

This additional amount guarantees that there is enough capital to buy back the borrowed shares and hand them over back to the broker, in case the shares price increases.

slide5
SHORT SELLING STOCKS

There are more details associated with short selling stocks. For example, if the stock pays dividend, the short seller must pay the dividend to the lender. Moreover, the short seller does not gain interest on the amount deposited in the escrow account, etc.

We will use stock short sales in many of strategies associated with derivatives.

In terms of cash flows:

St is the cash flow from selling the stock short on date t.

-ST is the cash flow from purchasing the back on date T.

slide6
Risk-Free Asset: is a security of investment whose return carries no risk. Thus, the return on this security is known and guaranteed in advance.
  • Risk-Free Borrowing And Landing:

By purchasing the risk-free asset, investors lend their capital and earn the risk-free rate.

By selling the risk-free asset, investors borrow capital at the risk-free rate.

slide7
The One-Price Law:

There exists only one risk-free rate in an efficient economy.

Continuous Compounding and Discounting:

Calculating the future value of a series of cash flows or, the present value of the cash flows, respectively, in a continuous time framework.

compounded interest
Compounded Interest

Any principal amount, P, invested at an annual interest rate, r, compounded annually, for T years would grow to AT = P(1 + r)T.

If compounded Quarterly:

AT = P(1 +r/4)4T.

In general, with m compounding periods every year, the periodic rate becomes r/m and mT is the number of compounding periods. Thus, P grows to:

AT = P(1 +r/m)mT.

slide9
Monthly compounding becomes:

AT = P(1 +r/12)12T

and daily compounding yields:

AT = P(1 +r/365)365T

Eample: T =10 years; r =12%; P = $100.

1. Simple annual compounding yields:

A10 = $100(1+ .12)10 = $310.58

2. Monthly compounding yields:

A10 = $100(1 + .12/12)120  = $330.03

3. Daily compounding yields:

A10 = $100(1 + .12/365)3,650 = $331.94.

slide10
In the early 1970s, banks came up with the following economic reasoning: Since the bank has depositors’ money all the time, this money should be working for the depositor all the time!

This idea, of course, leads to the concept of continuous compounding.

Observe that continuous time means that the number of compounding periods every year, m, increases without limit. This implies that the length of every compounding time period goes to zero and thus, the periodic interest rate, r/m, becomes smaller and smaller.

slide12
EXAMPLE, continued: First, recall that:

example: xe

1 2

10 2.59374246

1,000 2.71692393

10,000 2.71814592

In the limit e = 2.718281828…

slide13
EXAMPLE, continued: Recall that in

our example: T= 10 years and r = 12% and P=$100. Thus, P=$100 invested at an annual rate of 12%. will grow to by the factor:

CompoundingFactor

Simple 3.105848208

Quarterly 3.262037792

Monthly 3.300386895

Daily 3.319462164

Continuously 3.320116923

continuous compounding page 43
Continuous Compounding(Page 43)
  • In the limit as we compound more and more frequently we obtain continuously compounded interest rates.
  • $100 grows to $100eRT when invested at a continuously compounded rateR for time T.
  • $100 received at time T discounts to

$100e-RT at time zero when the continuously compounded discount rate is R.

conversion formulas page 44
Conversion Formulas (Page 44)

Define

Rc : continuously compounded rate

Rm: same rate with compounding m times per year

futures and spot prices an economics model of demand and supply
FUTURES and SPOT PRICES:AN ECONOMICS MODEL of DEMAND and SUPPLY

SPECULATORS: WILL OPEN RISKY FUTURES POSITIONS FOR EXPECTED PROFITS.

HEDGERS: WILL OPEN FUTURES POSITIONS IN ORDER TO ELIMINATE ALL PRICE RISK.

ARBITRAGERS: WILL OPEN SIMULTANEOUS FUTURES AND CASH POSITIONS IN ORDER TO MAKE ARBITRAGE PROFITS.

slide18
HEDGERS:

HEDGERS TAKE FUTURES POSITIONS IN ORDER

TO ELIMINATE PRICE RISK.

THERE ARE TWO TYPES OF HEDEGES

A LONG HEDGE

TAKE A LONG FUTURES POSITION IN ORDER TO LOCK IN THE PRICE OF AN ANTICIPATED PURCHASE AT A FUTURE TIME

A SHORT HEDGE

TAKE A SHORT FUTURES POSITION IN ORDER TO LOCK IN THE SELLING PRICE OF AN ANTICIPATED SALE AT A FUTURE TIME.

slide19
ARBITRAGE WITH FUTURES:

SPOT MARKETFUTURES MARKET

Contract to buy the product LONG futures

Contract to sell the product SHORT futures

slide20

Ft (k)

Long hedgers want to hedge a decreasing amount of their risk exposure as the premium of the settlement price over the expected future spot price increases.

a

Long hedgers want to hedge all of their risk exposure if the settlement price is less than or equal to the expected future spot price.

Expt [St+k]

b

c

0

Od

Quantity of long positions

Demand for LONG futures positions by long HEDGERS

slide21

Ft (k)

Short hedgers want to hedge all of their risk exposure if the settlement price is greater than or equal to the expected future spot price.

d

Short hedgers want to hedge a decreasing amount of their risk exposure as the discount of the settlement price below the expected future spot price increases.

e

Expt [St + k]

f

0

QS

Quantity of short positions

Supply of SHORT futures positions by short HEDGERS.

slide22

Ft (k)

S

Supply schedule

D

Ft (k)e

Premium

Expt [St + k]

Demand schedule

S

D

0

QS

Qd

Quantity of positions

Equilibrium in a futures market with a preponderance of long hedgers.

slide23

Ft (k)

S

D

Supply schedule

Expt [St + k]

Discount

Ft (k)e

Demand schedule

S

D

0

Qd

QS

Quantity of positions

Equilibrium in a futures market with a preponderance of short hedgers.

slide24

Ft (k)

Speculators will not demand any long positions if the settlement price exceeds the expected future spot price.

a

Speculators demand more long positions the greater the discount of the settlement price below the expected future spot price.

Expt [St + k]

b

c

0

Quantity of long positions

Demand for long positions in futures contracts by speculators.

slide25

Ft (k)

Speculators supply more short positions the greater the premium of the settlement price over the expected future spot price

d

Expt [St + k]

e

Speculators will not supply any short positions if the settlement price is below the the expected future spot price

f

0

Quantity of short positions

Supply of short positions in futures contracts by speculators.

slide26

Ft (k)

S

D

Increased supply from speculators

Expt [St + k]

Discount

Ft (k)e

Increased demand from speculators

S

D

0

Qd QE Qs

Quantity of positions

Equilibrium in a futures market with speculators and a preponderance of short hedgers.

slide27

Ft (k)

S

Increased supply from speculators

D

Ft (k)e

Increased demand from speculators

Premium

Expt [St + k]

S

D

0

QE

Quantity of positions

Equilibrium in a futures market with speculators and a preponderance of long hedgers.

slide28

Ft (k); St

Excess supply of the asset when the spot market price is St

Spot supply

}

Ft (k)e

Premium

Expt [St + k]

Spot demand

0

QE

Quantity of the asset

Equilibrium in the spot market

slide29

Ft (k)

Schedule of excess demand by hedgers and speculators

Expt [St + k]

}

Premium

Ft (k)e

Excess demand for long positions by hedgers and speculators when the settlement price is Ft (k)e

0

Q

Net quantity of long positions held by hedgers and speculators

Equilibrium in the futures market

slide30
ARBITRAGE IN PERFECT MARKETS

CASH -AND-CARRY

DATESPOT MARKETFUTURES MARKET

NOW 1. BORROW CAPITAL. 3. SHORT FUTURES.

2. BUY THE ASSET IN THE

SPOT MARKET AND CARRY

IT TO DELIVERY.

DELIVERY 1. REPAY THE LOAN 3. DELIVER THE STORED

COMMODITY TO CLOSE THE SHORT FUTURES POSITION

slide31
ARBITRAGE IN PERFECT MARKETS

REVERSE CASH -AND-CARRY

DATESPOT MARKETFUTURES MARKET

NOW 1. SHORT SELL ASSET 3. LONG FUTURES

2. INVEST THE PROCEEDS

IN GOV. BOND

DELIVERY: 2. REDEEM THE BOND 3. TAKE DELIVERY ASSET TO CLOSE THE LONG FUTURES POSITION

1. CLOSE THE SPOT SHORT POSITION

gold example from chapter 1
Gold Example (From Chapter 1)
  • For gold

F0 = S0(1 + r )T

(assuming no storage costs)

  • If ris compounded continuously instead of annually

F0 = S0erT

PROOF:

slide34
ARBITRAGE IN PERFECT MARKETS

CASH -AND-CARRY

DATESPOT MARKETFUTURES MARKET

NOW 1. BORROW CAPITAL: S0 3. SHORT FUTURES

2. BUY THE ASSET IN F0,T

THE SPOT MARKET AND CARRY IT TO DELIVERY

DELIVERY 1. REPAY THE LOAN 3. DELIVER THE STORED

COMMODITY TO CLOSE THE SHORT FUTURES POSITION

S0erT F0,T

slide35
ARBITRAGE IN PERFECT MARKETS

REVERSE CASH -AND-CARRY

DATESPOT MARKETFUTURES MARKET

NOW 1. SHORT SELL ASSET: S0 3. LONG FUTURES

2. INVEST THE PROCEEDS F0,T

IN GOV. BOND

DELIVERY: 2. REDEEM THE BOND 3. TAKE DELIVERY ASSET TO CLOSE THE LONG FUTURES POSITION

1. CLOSE THE SPOT SHORT POSITION

S0erT  F0,T

extension of the gold example page 46 equation 3 5
Extension of the Gold Example(Page 46, equation 3.5)
  • For any investment asset that provides no income and has no storage costs

F0 = S0erT

when an investment asset provides a known dollar income page 48 equation 3 6
When an Investment Asset Provides a Known Dollar Income (page 48, equation 3.6)

F0 = (S0 – I )erT

where Iis the present value of the income

when an investment asset provides a known yield page 49 equation 3 7
When an Investment Asset Provides a Known Yield (Page 49, equation 3.7)

F0 = S0e(r–q )T

where q is the average yield during the life

of the contract (expressed with continuous

compounding)

valuing a forward contract page 50
Valuing a Forward ContractPage 50
  • Suppose that K is delivery price in a forward contract, F0,T is forward price today for delivery at T
  • The value of a long forward contract, ƒ, is

ƒ = (F0,T – K )e–rT

  • Similarly, the value of a short forward contract is

(K – F0,T )e–rT

forward vs futures prices
Forward vs Futures Prices
  • Forward and futures prices are usually assumed to be the same. When interest rates are uncertain they are, in theory, slightly different:
  • 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
stock index page 52
Stock Index (Page 52)
  • Can be viewed as an investment asset paying a dividend yield
  • The futures price and spot price relationship is therefore

F0 = S0e(r–q )T

where q is the dividend yield on the portfolio represented by the index

stock index continued
Stock Index (continued)
  • For the formula to be true it is important that the index represent an investment asset
  • In other words, changes in the index must correspond to changes in the value of a tradable portfolio
  • The Nikkei index viewed as a dollar number does not represent an investment asset
index arbitrage
Index Arbitrage
  • When F0>S0e(r-q)T , an arbitrageur buys the stocks underlying the index and sells futures.
  • When F0<S0e(r-q)T , an arbitrageur buys futures and shorts or sells the stocks underlying the index.
index arbitrage continued
Index Arbitrage (continued)
  • Index arbitrage involves simultaneous trades in futures and many different stocks
  • Very often a computer is used to generate the trades
  • Occasionally (e.g., on Black Monday) simultaneous trades are not possible and the theoretical no-arbitrage relationship between F0,T and S0 does not hold
futures and forwards on currencies page 55 58
Futures and Forwards on Currencies (Page 55-58)
  • A foreign currency is analogous to a security providing a dividend yield
  • The continuous dividend yield is the foreign risk-free interest rate
  • It follows that if rf is the foreign risk-free interest rate
slide46

THE INTEREST RATES PARITY

Wherever financial flows are unrestricted, exchange rates, the forward rates and the interest rates in any two countries must maintain a NO- ARBITRAGE relationship:

Interest Rates Parity.

slide47

NO ARBITRAGE: CASH-AND-CARRY

TIMECASHFUTURES

t (1) BORROW $A. rDOM (4) SHORT FOREIGN CURRENCY

(2) BUY FOREIGN CURRENCY FORWARD Ft,T($/FC)

A/S($/FC) [=AS(FC/$)] AMOUNT:

(3) INVEST IN BONDS

DENOMINATED IN THE

FOREIGN CURRENCY rFOR

T (3) REDEEM THE BONDS (4) DELIVER THE CURRENCY TO

EARN CLOSE THE SHORT POSITION

(1) PAY BACK THE LOAN RECEIVE:

IN THE ABSENCE OF ARBITRAGE:

slide48

NO ARBITRAGE:

REVERSE CASH – AND - CARRY

TIMECASHFUTURES

t (1) BORROW FC A. rFOR (4) LONG FOREIGN CURRENCY

(2) BUY DOLLARS FORWARD Ft,T($/FC)

AS($/FC) AMOUNT IN DOLLARS:

(3) INVEST IN T-BILLS

FOR RDOM

T REDEEM THE T-BILLS TAKE DELIVERY TO CLOSE

EARN THE LONG POSITION

PAY BACK THE LOAN RECEIVE

IN THE ABSENCE OF ARBITRAGE:

slide49

FROM THE CASH-AND-CARRY STRATEGY:

FROM THE REVERSE CASH-AND-CARRY STRATEGY:

THE ONLY WAY THE TWO INEQUALITIES HOLD SIMULTANEOUSLY IS BY BEING AN EQUALITY:

slide50

ON MAY 25 AN ARBITRAGER OBSERVES THE FOLLOWING MARKET PRICES:

S(USD/GBP) = 1.5640 <=> S(GBP/USD) = .6393

F(USD/GBP) = 1.5328 <=> F(GBP/USD) = .6524

RUS = 7.85% ; RGB = 12%

CASH AND CARRY

TIMECASHFUTURES

MAY 25 (1) BORROW USD100M AT 7. 85% SHORT GBP 68,477,215 FORWARD

FOR 209 DAYS FOR DEC. 20, FOR USD1.5328/GBP

(2) BUY GBP63,930,000

(3) INVEST THE GBP63,930,000

IN BRITISH BONDS

DEC 20 RECEIVE GBP68,477,215 DELIVER GBP68,477,215

FOR USD104,961,875.2

REPAY YOUR LOAN:

ARBITRAGE PROFIT: USD104,961,875.2 - USD104,597,484.3 = USD364,390.90

futures on consumption assets page 59
Futures on Consumption Assets (Page 59)

F0 S0 e(r+u )T

where u is the storage cost per unit time as a percent of the asset value.

Alternatively,

F0 (S0+U )erT

where U is the present value of the storage costs.

the cost of carry page 60
The Cost of Carry (Page 60)
  • The cost of carry, c, is the storage cost plus the interest costs less the income earned.
  • For an investment asset F0 = S0ecT
  • For a consumption asset F0 = S0ecT
  • The convenience yield on the consumption asset, y, is defined

so that F0 = S0 e(c–y )T

slide53

ARBITRAGE IN THE REAL WORLD

TRANSACTION COSTS

DIFFERENT BORROWING AND LENDING RATES

MARGINS REQUIREMENTS

RESTRICTED SHORT SALES AN USE OF PROCEEDS

STORAGE LIMITATIONS

* BID - ASK SPREADS

** MARKING - TO - MARKET

* BID - THE HIGHEST PRICE ANY ONE IS WILLING TO BUY AT NOW

ASK - THE LOWEST PRICE ANY ONE IS WILLING TO SELL AT NOW.

** MARKING - TO - MARKET: YOU MAY BE FORCED TO CLOSE YOUR POSITION BEFORE ITS MATURITY.

slide54
FOR THE CASH - AND - CARRY:

BORROW AT THE BORROWING RATE: rB

BUY SPOT FOR: SASK

SELL FUTURES AT THE BID PRICE: F(BID).

PAY TRANSACTION COSTS ON:

BORROWING

BUYING SPOT

SELLING FUTURES

PAY CARRYING COST

PAY MARGINS

slide55
THE REVERSE CASH - AND - CARRY

SELL SHORT IN THE SPOT FOR: SBID.

INVEST THE FACTION OF THE PROCEEDS ALLOWED BY LAW: f; 0 ≦ f ≦ 1.

LEND MONEY (INVEST) AT THE LENDING RATE: rL

LONG FUTURES AT THE ASK PRICE: F(ASK).

PAY TRANSACTION COST ON:

SHORT SELLING SPOT

LENDING

BUYING FUTURES

PAY MARGIN

slide56

With these market realities, a new no-arbitrage condition emerges:

BL< F < BU

As long as the futures price fluctuates between the bounds there is no possibility to make arbitrage profits

BU

BU

F

BL

BL

time

slide57
Example

S0,BID (1 - c)[1 + f(rBID )] < F0, t< S0,ASK (1 + c)(1 + rASK)

c is the % of the price which is a transaction cost.

Here, we assume that the futures trades for one price.

In order to understand the LHS of the inequality, remember that the rule in the USA is that you may invest only a fraction, f, of the proceeds from a short sale. So, in the reverse cash and carry, the arbitrager sells the asset short at the bid price. Then (1-f)S0,BID cannot be invested while fS0,BID(1+rBID) is invested. Thus, the inequality becomes:

F0,T  (1-f)S0 + fS0(1+rBID)

F0,T  S0(1 + frBID)

slide58
EXAMPLE 1.

S0,BID (1 - T)[1 + f(rL )] < F0, t< S0,ASK (1 + T)(1 + rB)

S0,ASK = $20.50 / bbl S0,BID = $20.25 / bbl

rASK = 12 % rBID = 8 %

c = 3 %

$20.25(.97)[1+f(.08)]<F0,t< $20.50(1.03)(1.12)

$19.6425 + f($1.57) < F0,t< $23.6488

DEPENDING ON f, ANY FUTURES PRICE BETWEEN THE TWO LIMITS WILL LEAVE NO ARBITRAGE OPPORTUNITIES. THE CASH-AND-CARRY WILL COST $23.6488/bbl. THE REVERSE CASH-AND-CARRY WILL COST 19.6425 + f(1.62). IF f=0.5 THE LOWER BOUND IS $20.45. IN THE REAL MARKET, f = 1, FOR SOME LARGE ARBITAGE FIRMS AND THEIR LOWER BOUND IS $21.26. THUS, IT IS CLEAR THAT THERE ARE DIFFERENT ARBITRAGE BOUNDS APPLICABLE TO DIFFERENT INVESTORS. THE TIGHTER THE BOUNDS, THE GREATER ARE THE ARBITRAGE OPPORTUNITIES.

slide59

Example 2.: THE INTEREST RATES PARITY

In the real markets, buyers pay the ask price while sellers receive the bid price. Moreover, borrowers pay the ask interest rate while lenders only receive the bid interest rate.

Therefore, in the real markets, it is possible for the forward exchange rate to fluctuate within a band of rates without presenting arbitrage opportunities.Only when the market forward exchange rate diverges from this band of rates arbitrage exists.

slide60

NO ARBITRAGE: CASH - AND - CARRY

TIMECASHFUTURES

t (1) BORROW $A. rD,ASK (4) SHORT FOREIGN CURRENCY FORWARD

(2) BUY FOREIGN CURRENCY

A/SASK($/FC) FBID ($/FC)

(3) INVEST IN BONDS

DENOMINATED IN THE

FOREIGN CURRENCY rF,BID

T REDEEM THE BONDS DELIVER THE CURRENCY TO CLOSE THE SHORT POSITION

EARN:

PAY BACK THE LOAN RECEIVE:

IN THE ABSENCE OF ARBITRAGE:

slide61

NO ARBITRAGE:

REVERSE CASH - AND - CARRY

TIMECASHFUTURES

t (1) BORROW FCA . rF,ASK (4) LONG FOREIGN CURRENCY FORWARD FOR FASK(USD/FC)

(2) EXCHANGE FOR

ASBID (USD/FC)

(3) INVEST IN T-BILLS

FOR rD,BID

T REDEEM THE T-BILLS TAKE DELIVERY TO CLOSE THE LONG POSITION

EARN RECEIVE in foreign currency, the amount:

PAY BACK THE LOAN

IN THE ABSENCE OF ARBITRAGE:

slide62

From Cash and Carry:

From reverse cash and Carry

(3) And FASK($/D) > FBID($/D) Always!

Notice that

The RHS(1) > RHS(2)

Define: RHS(1)  BU RHS(2)  BL

slide63

F($/D)

FASK($/D) > FBID($/D).

FASK

BU

BL

BU

BL

FBID

CONCLUSION:

Arbitrage exists only if both ask and bid futures prices are above BU, or both are below BL.

slide64

A numerical example:

Given the following exchange rates:

SpotForwardInterest rates

S(USD/NZ)F(USD/NZ)r(NZ)r(US)

ASK 0.4438 0.4480 6.000% 10.8125%

BID 0.4428 0.4450 5.875% 10.6875%

Clearly, F(ask) > F(bid). (USD0.4480NZ > USD0.4450/NZ)

We will now check whether or not there exists an opportunity for arbitrage profits. This will require comparing these forward exchange rates to: BU and BL

slide65

Inequality (1):

0.4450 < (0.4438)e(0.108125 – 0.05875)/12 = 0.4456 = BU

Inequality (2):

0.4480 > (0.4428)e(0.106875 – 0.06000)/12 = 0.4445 = BL

  • No arbitrage.
  • Lets see the graph
slide66

F

FASK = 0.4480

Clearly:

FASK($/FC) > FBID($/FC).

0.4456

BU

BL

BU

FBID = 0.4450

0.4445

BL

An example of arbitrage: FBID = 0.4465

FASK = 0.4480