The pricing of forward and futures contracts Outline Spot and futures prices for non-dividend paying...
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Transcript of The pricing of forward and futures contracts Outline Spot and futures prices for non-dividend paying...
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)
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)
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)
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)
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)
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)
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
The forward price of a contract expiring in three months is $39. 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 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
Case 1b: Non-dividend paying investment asset
The forward price of a contract expiring in three months is $39. 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 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
Case 1b: Non-dividend paying investment asset
The forward price of a contract expiring in three months is $39. 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 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
Case 1b: Non-dividend paying investment asset
The forward price of a contract expiring in three months is $39. 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 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
Arbitrage profit at expiration : $1.50
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%.
Portfolio Cash flow from portfolio
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
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%.
Portfolio Cash flow from portfolio
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
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%.
Portfolio Cash flow from portfolio
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
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%.
Portfolio Cash flow from portfolio
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
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%.
Portfolio Cash flow from portfolio
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
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%.
Portfolio Cash flow from portfolio
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 coupon
Take delivery and sell the bond
Pay back the one year loan
$40
$930
- $952.39
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%.
Portfolio Cash flow from portfolio
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
Arbitrage profit at expiration : $17.61
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%.
Portfolio Cash flow from portfolio
Today Sell one bond on the spot
Invest $38.24 for six months and$861.76 for one year
Buy one forward contract
$900
- $900
0
In sixmonths
First investment matures $40
Expiration Second investment matures
Take delivery and buy the bond
$952.39
- $905
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%.
Portfolio Cash flow from portfolio
Today Sell one bond on the spot
Invest $38.24 for six months and$861.76 for one year
Buy one forward contract
$900
- $900
0
In sixmonths
First investment matures $40
Expiration Second investment matures
Take delivery and buy the bond
$952.39
- $905
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%.
Portfolio Cash flow from portfolio
Today Sell one bond on the spot
Invest $38.24 for six months and$861.76 for one year
Buy one forward contract
$900
- $900
0
In sixmonths
First investment matures $40
Expiration Second investment matures
Take delivery and buy the bond
$952.39
- $905
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%.
Portfolio Cash flow from portfolio
Today Sell one bond on the spot
Invest $38.24 for six months and$861.76 for one year
Buy one forward contract
$900
- $900
0
In sixmonths
First investment matures $40
Expiration Second investment matures
Take delivery and buy the bond
$952.39
- $905
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%.
Portfolio Cash flow from portfolio
Today Sell one bond on the spot
Invest $38.24 for six months and$861.76 for one year
Buy one forward contract
$900
- $900
0
In sixmonths
First investment matures $40
Expiration Second investment matures
Take delivery and buy the bond
$952.39
- $905
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%.
Portfolio Cash flow from portfolio
Today Sell one bond on the spot
Invest $38.24 for six months and$861.76 for one year
Buy one forward contract
$900
- $900
0
In sixmonths
First investment matures $40
Expiration Second investment matures
Take delivery and buy the bond
$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.
Portfolio Cash flow from portfolio
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$1,105.7
Case 2c: Asset providing a known yield/returnCase 2c: Asset providing a known yield/return
Assume two-year rates in the US and Canada are 7% and 5% respectively. The spot rate of the C$ is US$0.62. The two-year forward rate US$0.63.
Portfolio Cash flow from portfolio
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$1,105.7
Case 2c: Asset providing a known yield/returnCase 2c: Asset providing a known yield/return
Assume two-year rates in the US and Canada are 7% and 5% respectively. The spot rate of the C$ is US$0.62. The two-year forward rate US$0.63.
Portfolio Cash flow from portfolio
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$1,105.7
Case 2c: Asset providing a known yield/returnCase 2c: Asset providing a known yield/return
Assume two-year rates in the US and Canada are 7% and 5% respectively. The spot rate of the C$ is US$0.62. The two-year forward rate US$0.63.
Portfolio Cash flow from portfolio
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$1,105.7
Case 2c: Asset providing a known yield/returnCase 2c: Asset providing a known yield/return
Assume two-year rates in the US and Canada are 7% and 5% respectively. The spot rate of the C$ is US$0.62. The two-year forward rate US$0.63.
Portfolio Cash flow from portfolio
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$1,105.7
Case 2c: Asset providing a known yield/returnCase 2c: Asset providing a known yield/return
Assume two-year rates in the US and Canada are 7% and 5% respectively. The spot rate of the C$ is US$0.62. The two-year forward rate US$0.63.
Portfolio Cash flow from portfolio
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$1,105.7
Case 2c: Asset providing a known yield/returnCase 2c: Asset providing a known yield/return
Assume two-year rates in the US and Canada are 7% and 5% respectively. The spot rate of the C$ is US$0.62. The two-year forward rate US$0.63.
Portfolio Cash flow from portfolio
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
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.
Portfolio Cash flow from portfolio
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
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.
Portfolio Cash flow from portfolio
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
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.
Portfolio Cash flow from portfolio
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
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.
Portfolio Cash flow from portfolio
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
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.
Portfolio Cash flow from portfolio
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
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.
Portfolio Cash flow from portfolio
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
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.
Portfolio Cash flow from portfolio
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
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.
Portfolio Cash flow from portfolio
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
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.
Portfolio Cash flow from portfolio
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
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.
Portfolio Cash flow from portfolio
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
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.
Portfolio Cash flow from portfolio
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
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.
Portfolio Cash flow from portfolio
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
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 ?
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 ?
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 ?
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 ?
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 ?
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 ?