The pricing of forward and futures contracts Outline Spot and futures prices for non-dividend paying investment assets Spot and futures prices for investment

The pricing of forward and futures contractsThe pricing of forward and futures contracts

Outline

• Spot and futures prices for non-dividend paying investment assets

• Spot and futures prices for investment assets paying a known income

• Spot and futures prices for investment assets paying a known yield/return

• Spot and futures prices for commodities with storage costs

• Spot and futures prices for consumption commodities with storage costs

• The cost of carry

• The valuation of forward contracts

Case 1a: Non-dividend paying investment asset

The forward price of a contract expiring in three months is $43. The three-month annualized interest rate is 5%, and the current price of the underlying asset is $40/share. No dividend is expected

Portfolio Cash flow from portfolio

Today Borrow $40

Buy one share of stock

Short one forward contract

$40

-$40

0

Expiration Take delivery and sell the share

Pay back loan:

$43

- $40.5 = - $40e 0.05(3/12)




Today Borrow $40



$40

-$40

0


Pay back loan:

$43

- $40.5 = - $40e 0.05(3/12)




Today Borrow $40



$40

-$40

0


Pay back loan:

$43

- $40.5 = - $40e 0.05(3/12)




Today Borrow $40



$40

-$40

0


Pay back loan:

$43

- $40.5 = - $40e 0.05(3/12)




Today Borrow $40



$40

-$40

0


Pay back loan:

$43

- $40.5 = - $40e 0.05(3/12)




Today Borrow $40



$40

-$40

0


Pay back loan:

$43

- $40.5 = - $40e 0.05(3/12)

Arbitrage profit at expiration : $2.50

Case 1a: Implications

Eventually, investors would bid up the stock price, and

drive down the forward price

Case 1b: Non-dividend paying investment asset



Today Short/sell one share

Invest proceeds for three months

Buy one forward contract

$40

-$40

0

Expiration Receive proceeds from loan

Take delivery and buy the share

$40.5 = - $40e 0.05(3/12)

- $39







$40

-$40

0



$40.5 = - $40e 0.05(3/12)

- $39







$40

-$40

0



$40.5 = - $40e 0.05(3/12)

- $39







$40

-$40

0



$40.5 = - $40e 0.05(3/12)

- $39


Case 1b: Implications

Eventually, investors would drive down the stock price,

and bid up the forward price

Relationship between spot and forward/futures prices for a non-dividend paying investment asset

F0 = S0erT

F0 = forward/futures price today

S0 = underlying asset spot price today

r = risk-free rate

T = time to expiration

Case 2a: Asset with a known income A bond has the one-year forward price of $930. The current spot price of the bond is $900. Coupon payments are $40 every six months. The six-month risk-free rate is 9%/year and the one-year risk-free rate is 10%.


Today Borrow $38.24 for six months and$861.76 for one year

Buy one bond on the spot

Sell/short one forward contract

$900

-$900

0

In sixmonths

Receive first coupon

Pay back six month loan

$40

-$40

Expiration Receive second couponTake delivery and sell the bond

Pay back the one year loan

$40$930

- $952.39






$900

-$900

0

In sixmonths



$40

-$40



$40$930

- $952.39






$900

-$900

0

In sixmonths



$40

-$40



$40$930

- $952.39






$900

-$900

0

In sixmonths



$40

-$40



$40$930

- $952.39






$900

-$900

0

In sixmonths



$40

-$40



$40$930

- $952.39






$900

-$900

0

In sixmonths



$40

-$40

Expiration Receive second coupon

Take delivery and sell the bond


$40

$930

- $952.39






$900

-$900

0

In sixmonths



$40

-$40



$40$930

- $952.39


Case 2a: ImplicationCase 2a: Implication

Eventually, investors would drive down the forward price, and bid up the spot price of the bond

Case 2b: Asset with a known incomeCase 2b: Asset with a known income A bond has the one-year forward price of $905. The current spot price of the bond is $900. Coupon payments are $40 every six months. The six-month risk-free rate is 9%/year and the one-year risk-free rate is 10%.


Today Sell one bond on the spot

Invest $38.24 for six months and$861.76 for one year


$900

- $900

0

In sixmonths

First investment matures $40

Expiration Second investment matures

Take delivery and buy the bond

$952.39

- $905






$900

- $900

0

In sixmonths




$952.39

- $905






$900

- $900

0

In sixmonths




$952.39

- $905






$900

- $900

0

In sixmonths




$952.39

- $905






$900

- $900

0

In sixmonths




$952.39

- $905






$900

- $900

0

In sixmonths




$952.39

- $905

Arbitrage profit at expiration : $952.39 - $40 - $905 = $7.39

Case 2b: ImplicationCase 2b: Implication

Eventually, investors would drive down the spot price, and bid up the forward price of the bond

Relationship between spot and forward/futures prices for an investment asset providing a known income

F0 = (S0 - PVincome)erT

In our example:

PVincome = $40e-(0.09)(0.5) + $40e-(0.1)

Case 2c: Asset providing a known yield/returnCase 2c: Asset providing a known yield/return

Assume two-year rates in the US and Canada are 7% and 5% respectively. The spot rate of the C$ is US$0.62. The two-year forward rate US$0.63.


Today Borrow C$1,000 at 5%

Buy US$620 on the spot

Invest US$620 at 7%

Buy forward C$1,105.7

C$1,000

-C$,1000 + US$620

-US$620

0

Expiration US$ investment matures

Take delivery and buy C$1,105.7

Repay loan

US$713.17

-US$696.29 + $1,105.7

-C$1,105.7






Invest US$620 at 7%


C$1,000

-C$,1000 + US$620

-US$620

0



Repay loan

US$713.17

-US$696.29 + $1,105.7

-C$1,105.7






Invest US$620 at 7%


C$1,000

-C$,1000 + US$620

-US$620

0



Repay loan

US$713.17

-US$696.29 + $1,105.7

-C$1,105.7






Invest US$620 at 7%


C$1,000

-C$,1000 + US$620

-US$620

0



Repay loan

US$713.17

-US$696.29 + $1,105.7

-C$1,105.7






Invest US$620 at 7%


C$1,000

-C$,1000 + US$620

-US$620

0



Repay loan

US$713.17

-US$696.29 + $1,105.7

-C$1,105.7






Invest US$620 at 7%


C$1,000

-C$,1000 + US$620

-US$620

0



Repay loan

US$713.17

-US$696.29 + $1,105.7

-C$1,105.7






Invest US$620 at 7%


C$1,000

-C$,1000 + US$620

-US$620

0



Repay loan

US$713.17

-US$696.29 + $1,105.7

-C$1,105.7

Arbitrage profit = US$16.91

Case 2c: ImplicationCase 2c: Implication

Eventually, investors would drive down the forward price and bid up the spot price of the US$

Relationship between spot and forward/futures prices for Relationship between spot and forward/futures prices for an investment asset providing a known yield/returnan investment asset providing a known yield/return

F0 = S0e(r-q)T

Where q is the known yield/return provided by the

investment asset

In case 2c, q is the interest rate on the foreign currency.

Case 3a: CommoditiesCase 3a: Commodities

The one-year futures price of gold is $500 per troy once. The spot price is $450 per troy once and the risk-free rate is 7%/year. The storage cost of gold is $2 per troy once per year.


Today Borrow $45,000

Buy 100 ounces of gold

Sell futures on 100 ounces

$45,000

-$45,000

0

Expiration Take delivery and sell 100 ounces

Pay storage costs

Repay loan

$50,000

- $200

- $48,263







$45,000

-$45,000

0


Pay storage costs

Repay loan

$50,000

- $200

- $48,263







$45,000

-$45,000

0


Pay storage costs

Repay loan

$50,000

- $200

- $48,263







$45,000

-$45,000

0


Pay storage costs

Repay loan

$50,000

- $200

- $48,263







$45,000

-$45,000

0


Pay storage costs

Repay loan

$50,000

- $200

- $48,263







$45,000

-$45,000

0


Pay storage costs

Repay loan

$50,000

- $200

- $48,263

Arbitrage profit = $1,537

Case 3a: ImplicationsCase 3a: Implications

In the long run, investors would bid up the spot price of gold and drive down its futures price.

Example 3b: CommoditiesExample 3b: Commodities

The one-year futures price of gold is $470 per troy once. The spot price is $450 per troy once and the risk-free rate is 7%/year. The storage cost of gold is $2 per troy once per year, payable in arrears.


Today Sell 100 ounces of gold

Invest proceeds for one year

Buy futures on 100 ounces

$45,000

-$45,000

0

Expiration Investment matures

Take delivery and buy 100 ounces

Storage costs savings

$48,263

- $47,000

$200







$45,000

-$45,000

0




$48,263

- $47,000

$200







$45,000

-$45,000

0




$48,263

- $47,000

$200







$45,000

-$45,000

0




$48,263

- $47,000

$200







$45,000

-$45,000

0




$48,263

- $47,000

$200







$45,000

-$45,000

0




$48,263

- $47,000

$200

Arbitrage profit = $1,463

Case 3b: ImplicationsCase 3b: Implications

In the long run, investors would bid up the futures price of gold and drive down the spot price

Relationship between spot and forward/futures Relationship between spot and forward/futures prices for an investment commodity with storage prices for an investment commodity with storage

costscosts

F0 = (S0 + U)erT

Where U = PV of storage costs (negative income).

If the storage cost is proportional to the price of the asset, storage costs can be viewed as a negative yield:

F0 = S0e(r + u)T

What if the commodity is held for What if the commodity is held for consumption only?consumption only?

In example 3b, one would might not want to sell the gold and engage in arbitrage.

Hence,

F0 = < (S0 + U)erT

F0 = < S0e(r + u)T

Relationship between spot and forward/futures Relationship between spot and forward/futures prices for a consumption commodity with storage prices for a consumption commodity with storage

costscosts

F0 = (S0 + U)e(r - y)T

or

F0 = S0e(r + u - y)T

Where y is a fudge factor called convenience yield

The cost of carryThe cost of carry

It measures the interest paid to finance the asset plus storage costs less income earned on the asset.

For an investment asset paying no dividend

cc = r

For a stock index

cc = q

For a foreign currency

cc = rf

For a commodity with storage costs proportional to price

cc = r + u

The valuation of forward contractsThe valuation of forward contracts

What is the value of a forward contract at inception?

Zero

The valuation of forward contracts: Investment asset paying The valuation of forward contracts: Investment asset paying no dividend/incomeno dividend/income

What is the value of a forward contract between inception and maturity?

Long contract

f = (F0 - F)e-rT

f = S0 - Fe-rT

Short contract

f = (F - F0)e-rT

f = Fe-rT - S0

Where F is the current forward price of the contract.

The valuation of futures contractsThe valuation of futures contracts

What is the value of a futures contract between inception and maturity?

At the end of each trading day, the value of futures is set back to zero as a result of marking-to-market.

Spot and forward/futures prices: A summary

Asset Forward/futures price Value of long forward

No income F0 = S0erT f = S0 - Fe-rT

Known $income F0 = (S0 - PVincome)erT f = S0 - PVincome– FerT

Known yield/return F0 = S0e(r-q)T f = S0e

-qT - Fe-rT

Commodity withstorage costs

F0 = (S0 + U)erT

F0 = S0e(r + u)T

f = S0 + U – FerT

f = S0euT - Fe-rT

Consumptioncommodity withstorage costs

F0 = < (S0 + U)erT

F0 = < S0e(r + u)T ?






-qT - Fe-rT


F0 = (S0 + U)erT

F0 = S0e(r + u)T

f = S0 + U – FerT

f = S0euT - Fe-rT


F0 = < (S0 + U)erT

F0 = < S0e(r + u)T ?






-qT - Fe-rT


F0 = (S0 + U)erT

F0 = S0e(r + u)T

f = S0 + U – FerT

f = S0euT - Fe-rT


F0 = < (S0 + U)erT

F0 = < S0e(r + u)T ?






-qT - Fe-rT


F0 = (S0 + U)erT

F0 = S0e(r + u)T

f = S0 + U – FerT

f = S0euT - Fe-rT


F0 = < (S0 + U)erT

F0 = < S0e(r + u)T ?






-qT - Fe-rT


F0 = (S0 + U)erT

F0 = S0e(r + u)T

f = S0 + U – FerT

f = S0euT - Fe-rT


F0 = < (S0 + U)erT

F0 = < S0e(r + u)T ?






-qT - Fe-rT


F0 = (S0 + U)erT

F0 = S0e(r + u)T

f = S0 + U – FerT

f = S0euT - Fe-rT


F0 = < (S0 + U)erT

F0 = < S0e(r + u)T ?

Documents

The pricing of forward and futures contracts Outline Spot and futures prices for non-dividend paying investment assets Spot and futures prices for investment