衍生品代寫-BFC2751

時間：2021-08-28

BFC2751 Derivatives 1

Week 3

Chapter 5, Hull

Determination of Forward and Future Prices

Investment vs. Consumption Assets

? Investment assets: assets held primarily for

investment purposes.

? E.g.: stock, bonds, gold, silver.

? Do not have to be held exclusively for investment.

? Silver has a number of industrial uses.

? Consumption assets: assets held primarily for

consumption purposes.

? E.g.: copper, oil, pork bellies.

? Use arbitrage arguments to determine the forward

and futures prices of an investment asset but not for

consumption assets.

Short Selling

? Short selling involves selling assets you do not own.

? Possible for some but not all investment assets.

? Your broker borrows the securities from another

client and sells them in the market in the usual way.

? Later you must buy the securities back, so they can

be replaced in the account of client.

? The investor takes a profit if the stock price has

declined and a loss if it has risen.

Short Selling

? You must pay dividends and other benefits the owner

of the securities should receive on the shares.

? Margin account is kept with the broker to guarantee

that you do not walk away from your obligations.

Short Selling Example

? Investor sells short 100 shares of Commonwealth

Bank at $100 each in March 2016.

? Total value: 100 * 100 = $10,000 (receive).

? In April, stock pays a dividend of $2 per share.

? Total dividend cost: 2*100 = $200 (pay).

? In May, investor closes the position by buying back

the shares on the market at $110 each.

? Total buy back cost: 110 * 100 = $11,000 (pay).

? Profit/loss = $10,000 - $200 - $11,000 = - $1,200.

Assumption and Notation

? Assumption? market participants:

? subject to no transaction costs when they trade.

? subject to same tax rate on all net trading profits.

? can borrow money at the same risk-free rate of interest

as they can lend money.

? take advantage of arbitrage opportunities as they

occur.

S0: Spot price today.

F0: Futures or forward price today.

T: Time to maturity.

r: Risk-free interest rate.

Arbitrage Opportunity?

? Consider a long forward contract to purchase a non-

dividend-paying stock in 3 months.

? Current stock price is $40.

? 3-months risk-free rate is 5% per annum.

? Is there an arbitrage opportunity?

Arbitrage Opportunity?

? Suppose forward price is relative high at $43.

? Today:

? Borrow $40 @ 5%.

? Buy one share.

? Short one forward to sell one share at $43 in 3 months.

? In 3 months:

? Sell the stock at $43.

? Repay the loan: 40*e(0.05*3/12) = $40.50.

? Profit = 43 - 40.50 = $2.5.

? This arbitrage works when F > $40.50.

Arbitrage Opportunity?

? Suppose forward price is relative low at $39.

? Today:

? Short one share to realize $40.

? Invest $40 @ 5% for 3 months.

? Long one forward to buy one share at $39 in 3 months.

? In 3 months:

? Buy the stock at $39, and close the short position.

? Receive from investment: 40*e(0.05*3/12) = $40.50.

? Profit = 40.5 - 39 = $1.5.

? This arbitrage works when F < $40.50.

? There is no arbitrage only when F = $40.50.

Forward Pricing

? If the spot price is S0 and the forward price for a

contract deliverable in T years is F0, r is the risk-free

interest rate.

? To avoid arbitrage, F0 should be equal to:

F0 = S0erT

? This equation relates the forward price and the spot

price for any investment asset that provides no

income and has no storage costs.

? E.g., non-dividend-paying stocks, zero-coupon bond.

? In our examples, S0 = 40, T = 3/12, and r = 0.05.

F0 =S0erT = 40*e(0.05*3/12) = $40.5.

Forward Pricing: Known Income

? Consider a forward contract on an investment asset

that will provide a perfectly predictable cash income to

the holder.

? E.g., stock paying known dividends and coupon-

bearing bonds.

F0 = (S0 – I)erT

? where I is the present value of the income during life

of forward contract.

Forward Pricing: Known Income

? Consider a long forward contract to purchase a coupon-

bearing bond whose current price is $900.

? The forward contract matures in 9 months.

? A coupon payment of $40 (i.e., income) is expected after

4 months.

? 4- and 9-month rates (continuously compounded) are 3%

and 4% per annum.

Forward Pricing: Known Income

? F=$910: use the income to pay part of the borrowing in 4

month. Take away the PV(I) from the $900 borrowing today.

? F=$870: use part of the investment to pay the income of the

shorted asset in 4 month. Take away the PV(I) from the $900

investment today.

Forward Pricing: Known Yield

? Consider the situation where the asset underlying a

forward contract provides a known yield rather than a

known cash income.

F0 = S0 e(r–q)T

? where q is the average yield during the life of the

contract (expressed with continuous compounding).

Forward Pricing: Known Yield

? Stock price is $25 today.

? 6-month forward contact.

? Dividend yield q=3.96% per annum with continuous

compounding.

? Risk-free interest rate is 10% per annum.

? What is the 6-months forward price of this stock?

F0 = S0 e(r–q)T = 25 e(10%–3.96%)*6/12 = $25.77

Valuing Forward Contracts

? When a forward contract is first entered into, its value

is zero.

? Suppose that K is delivery price in a forward contract

and F0 is forward price that would apply to the

contract today.

? The value of a long forward contract is

? = (F0 – K)e–rT

? Similarly, the value of a short forward contract is

? = (K – F0)e–rT

Valuing Forward Contracts

? One month ago, you entered a long forward contract

on gold with delivery price of $1200 per ounce.

? The contract has two months remaining.

? If the spot price of gold is $1400, and continuously

compounded interest rate is 6%, what is the value of

this contract now?

? First, find the current forward price of gold:

F0 = S0erT = 1400e0.06*2/12 = $1414.07

? Now we can value the contract:

f = (F0 – K)e–rT = (1414.07 - 1200)e-0.06*2/12 = $211.94

Forward vs. Futures Pricing

? Forward and futures prices are the same when short term

interest rates are constant and same maturities.

? When interest rates are unpredictable they are slightly

different.

? If a strong positive correlation between IR and asset price.

Futures price is slightly higher than forward price.

? Consider a long future:

? asset price increases (IR increases) ? immediate gain

(daily settlement) ? the gain is reinvested at a higher

rate.

? asset price decreases (IR decreases) ? immediate

loss (daily settlement) ? the loss is refinanced at a

lower rate.

? Underlying asset: one unit of the foreign currency.

? Holder of the currency can earn risk-free IR prevailing

in foreign country.

? Foreign currency ? a security providing a dividend

yield (i.e., foreign risk-free IR).

Forward and Futures on Currencies

? S0: spot price in local currency in one unit of foreign

currency.

? F0: forward/futures price in local currency in one unit

of foreign currency.

? rf: foreign risk-free IR.

? r: domestic risk-free IR.

Forward and Futures on Currencies

Trr feSF )(00

??

Why the Relation Must Be True

1000 units of

foreign currency

at time zero

units of foreign

currency at time T

Trfe1000

dollars at time T

TrfeF01000

1000S0 dollars

at time zero

dollars at time T

rTeS01000=

? Starts with 1,000

units of the foreign

currency.

? Two ways it can be

converted to dollars

at time T.

1. Investing for T years

at rf. Entering a forward

contract to sell the

proceeds for dollars at

time T.

2. Exchanging foreign

currency for dollars in

the spot market today

and investing for T

years at rate r.

Arbitrage Example 5.6

? 2-years IR are 5% in Australia and 7% in US

(domestic).

? Spot exchange rate = 0.62 USD/AUD.

? 2-years forward exchange rate must

= 0.62e(0.07 – 0.05)*2 = 0.6453.

? If forward rate = 0.63 USD/AUD ? possibility of

arbitrage.

? If forward rate = 0.66 USD/AUD ? possibility of

arbitrage.

If forward rate = 0.63 USD/AUD

? Borrow 1,000 AUD @ 5% (need to repay

1,105.17AUD = 1,000e0.05*2).

? Convert into 620 USD and invest @ 7% for 2 years.

? Enter a 2-years forward contract to buy 1,105.17

AUD for 1,105.17*0.63 = 696.26 USD.

? In 2 years, receive investment 620e0.07*2 = 713.17

USD.

? Pay 696.26 USD to buy 1,105.17 AUD and repay the

loan.

? Profit = 713.17 – 696.26 = 16.91USD.

If forward rate = 0.66 USD/AUD

? Borrow 1,000 USD @ 7%.

? Convert into 1,000/0.62=1612.90 AUD and invest @

5% for 2 years.

? Enter a 2-years forward contract to sell AUD:

1,782.53AUD*0.66 =1,176.47 USD.

? In 2 years, receive investment 1,612.90e0.05*2=

1,782.53 AUD. Then, exchange to 1,176.47 USD.

? Pay US borrowing 1,000e0.07*2=1,150.27.

? Profit = 1,176.47 – 1,150.27 = 26.20USD.

Futures on Investment Commodities

? Gold and silver: can provide income, but have

storage costs.

? U: present value of all storage costs, net of income,

during the life of forward contract.

F0 = (S0+U)erT

? If storage costs u, net of income, are proportional to

commodity price:

F0 = S0 e(r + u)T

? Storage costs can be treated as negative yield.

Futures on Consumption Assets

? Consumption assets: high storage costs (U) and no

income, but consumption value.

? Suppose we have F0 > (S0+U)erT.

? An arbitrageur can borrow (S0+U), buy and store the

commodity, and short a futures contract.

? At expiration, the riskless profit = F0 - (S0+U)erT > 0

? There is a tendency for S0 to increase and F0 to

decrease, thus the relation can not hold for long.

Futures on Consumption Assets

? Suppose we have F0 < (S0+U)erT.

? An arbitrageur can sell the commodity, save the

storage costs, invest in risk-free rate, and long

futures contact.

? At expiration, the riskless profit = (S0+U)erT - F0 > 0

? There is a tendency for S0 to decrease and F0 to

increase, thus the relation can not hold for long.

? Wait a minute, is that true?

? The commodity is used for consumption, the holders

are reluctant to sell. The strategy does not work.

? There is nothing to stop the relation from holding.

Futures on Consumption Assets

? In case of consumption assets we have inequality:

F0 ? (S0+U)erT

? Alternatively,

F0 ? S0 e(r+u)T

? where u is the annual storage cost as a percent of the

asset value.

Convenience Yield

? For consumption commodities users, the ownership

of physical commodities are not obtained by future

contract.

? Ownership of physical asset enables a manufacturer to

keep a production process running and perhaps profit

from temporary local shortages.

? E.g., the crude oil in inventory can be an input to the

refining process, but not for futures contact.

Convenience Yield

? Convenience yield: benefit derived from holding the

physical asset.

? Only for consumption assets.

? Reflects the market’s expectations concerning the

future availability of the commodity.

? If inventories are low, shortages are more likely and the

convenience yield is usually higher.

? Convenience yield y balances out the previous

inequality:

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

The Cost of Carry

? The relationship between futures prices and spot prices

can be summarized in terms of the cost of carry.

? Cost of carry (c) = storage cost + interest costs -

income earned

? Non-dividend-paying stock: c = r.

? Dividend-paying stock: c = r – q.

? Currency: c = r - rf .

? Commodity: c = r - q + u.

? For an investment asset: F0 = S0ecT

? For a consumption asset: F0 = S0e(c–y)T

學霸聯盟

Week 3

Chapter 5, Hull

Determination of Forward and Future Prices

Investment vs. Consumption Assets

? Investment assets: assets held primarily for

investment purposes.

? E.g.: stock, bonds, gold, silver.

? Do not have to be held exclusively for investment.

? Silver has a number of industrial uses.

? Consumption assets: assets held primarily for

consumption purposes.

? E.g.: copper, oil, pork bellies.

? Use arbitrage arguments to determine the forward

and futures prices of an investment asset but not for

consumption assets.

Short Selling

? Short selling involves selling assets you do not own.

? Possible for some but not all investment assets.

? Your broker borrows the securities from another

client and sells them in the market in the usual way.

? Later you must buy the securities back, so they can

be replaced in the account of client.

? The investor takes a profit if the stock price has

declined and a loss if it has risen.

Short Selling

? You must pay dividends and other benefits the owner

of the securities should receive on the shares.

? Margin account is kept with the broker to guarantee

that you do not walk away from your obligations.

Short Selling Example

? Investor sells short 100 shares of Commonwealth

Bank at $100 each in March 2016.

? Total value: 100 * 100 = $10,000 (receive).

? In April, stock pays a dividend of $2 per share.

? Total dividend cost: 2*100 = $200 (pay).

? In May, investor closes the position by buying back

the shares on the market at $110 each.

? Total buy back cost: 110 * 100 = $11,000 (pay).

? Profit/loss = $10,000 - $200 - $11,000 = - $1,200.

Assumption and Notation

? Assumption? market participants:

? subject to no transaction costs when they trade.

? subject to same tax rate on all net trading profits.

? can borrow money at the same risk-free rate of interest

as they can lend money.

? take advantage of arbitrage opportunities as they

occur.

S0: Spot price today.

F0: Futures or forward price today.

T: Time to maturity.

r: Risk-free interest rate.

Arbitrage Opportunity?

? Consider a long forward contract to purchase a non-

dividend-paying stock in 3 months.

? Current stock price is $40.

? 3-months risk-free rate is 5% per annum.

? Is there an arbitrage opportunity?

Arbitrage Opportunity?

? Suppose forward price is relative high at $43.

? Today:

? Borrow $40 @ 5%.

? Buy one share.

? Short one forward to sell one share at $43 in 3 months.

? In 3 months:

? Sell the stock at $43.

? Repay the loan: 40*e(0.05*3/12) = $40.50.

? Profit = 43 - 40.50 = $2.5.

? This arbitrage works when F > $40.50.

Arbitrage Opportunity?

? Suppose forward price is relative low at $39.

? Today:

? Short one share to realize $40.

? Invest $40 @ 5% for 3 months.

? Long one forward to buy one share at $39 in 3 months.

? In 3 months:

? Buy the stock at $39, and close the short position.

? Receive from investment: 40*e(0.05*3/12) = $40.50.

? Profit = 40.5 - 39 = $1.5.

? This arbitrage works when F < $40.50.

? There is no arbitrage only when F = $40.50.

Forward Pricing

? If the spot price is S0 and the forward price for a

contract deliverable in T years is F0, r is the risk-free

interest rate.

? To avoid arbitrage, F0 should be equal to:

F0 = S0erT

? This equation relates the forward price and the spot

price for any investment asset that provides no

income and has no storage costs.

? E.g., non-dividend-paying stocks, zero-coupon bond.

? In our examples, S0 = 40, T = 3/12, and r = 0.05.

F0 =S0erT = 40*e(0.05*3/12) = $40.5.

Forward Pricing: Known Income

? Consider a forward contract on an investment asset

that will provide a perfectly predictable cash income to

the holder.

? E.g., stock paying known dividends and coupon-

bearing bonds.

F0 = (S0 – I)erT

? where I is the present value of the income during life

of forward contract.

Forward Pricing: Known Income

? Consider a long forward contract to purchase a coupon-

bearing bond whose current price is $900.

? The forward contract matures in 9 months.

? A coupon payment of $40 (i.e., income) is expected after

4 months.

? 4- and 9-month rates (continuously compounded) are 3%

and 4% per annum.

Forward Pricing: Known Income

? F=$910: use the income to pay part of the borrowing in 4

month. Take away the PV(I) from the $900 borrowing today.

? F=$870: use part of the investment to pay the income of the

shorted asset in 4 month. Take away the PV(I) from the $900

investment today.

Forward Pricing: Known Yield

? Consider the situation where the asset underlying a

forward contract provides a known yield rather than a

known cash income.

F0 = S0 e(r–q)T

? where q is the average yield during the life of the

contract (expressed with continuous compounding).

Forward Pricing: Known Yield

? Stock price is $25 today.

? 6-month forward contact.

? Dividend yield q=3.96% per annum with continuous

compounding.

? Risk-free interest rate is 10% per annum.

? What is the 6-months forward price of this stock?

F0 = S0 e(r–q)T = 25 e(10%–3.96%)*6/12 = $25.77

Valuing Forward Contracts

? When a forward contract is first entered into, its value

is zero.

? Suppose that K is delivery price in a forward contract

and F0 is forward price that would apply to the

contract today.

? The value of a long forward contract is

? = (F0 – K)e–rT

? Similarly, the value of a short forward contract is

? = (K – F0)e–rT

Valuing Forward Contracts

? One month ago, you entered a long forward contract

on gold with delivery price of $1200 per ounce.

? The contract has two months remaining.

? If the spot price of gold is $1400, and continuously

compounded interest rate is 6%, what is the value of

this contract now?

? First, find the current forward price of gold:

F0 = S0erT = 1400e0.06*2/12 = $1414.07

? Now we can value the contract:

f = (F0 – K)e–rT = (1414.07 - 1200)e-0.06*2/12 = $211.94

Forward vs. Futures Pricing

? Forward and futures prices are the same when short term

interest rates are constant and same maturities.

? When interest rates are unpredictable they are slightly

different.

? If a strong positive correlation between IR and asset price.

Futures price is slightly higher than forward price.

? Consider a long future:

? asset price increases (IR increases) ? immediate gain

(daily settlement) ? the gain is reinvested at a higher

rate.

? asset price decreases (IR decreases) ? immediate

loss (daily settlement) ? the loss is refinanced at a

lower rate.

? Underlying asset: one unit of the foreign currency.

? Holder of the currency can earn risk-free IR prevailing

in foreign country.

? Foreign currency ? a security providing a dividend

yield (i.e., foreign risk-free IR).

Forward and Futures on Currencies

? S0: spot price in local currency in one unit of foreign

currency.

? F0: forward/futures price in local currency in one unit

of foreign currency.

? rf: foreign risk-free IR.

? r: domestic risk-free IR.

Forward and Futures on Currencies

Trr feSF )(00

??

Why the Relation Must Be True

1000 units of

foreign currency

at time zero

units of foreign

currency at time T

Trfe1000

dollars at time T

TrfeF01000

1000S0 dollars

at time zero

dollars at time T

rTeS01000=

? Starts with 1,000

units of the foreign

currency.

? Two ways it can be

converted to dollars

at time T.

1. Investing for T years

at rf. Entering a forward

contract to sell the

proceeds for dollars at

time T.

2. Exchanging foreign

currency for dollars in

the spot market today

and investing for T

years at rate r.

Arbitrage Example 5.6

? 2-years IR are 5% in Australia and 7% in US

(domestic).

? Spot exchange rate = 0.62 USD/AUD.

? 2-years forward exchange rate must

= 0.62e(0.07 – 0.05)*2 = 0.6453.

? If forward rate = 0.63 USD/AUD ? possibility of

arbitrage.

? If forward rate = 0.66 USD/AUD ? possibility of

arbitrage.

If forward rate = 0.63 USD/AUD

? Borrow 1,000 AUD @ 5% (need to repay

1,105.17AUD = 1,000e0.05*2).

? Convert into 620 USD and invest @ 7% for 2 years.

? Enter a 2-years forward contract to buy 1,105.17

AUD for 1,105.17*0.63 = 696.26 USD.

? In 2 years, receive investment 620e0.07*2 = 713.17

USD.

? Pay 696.26 USD to buy 1,105.17 AUD and repay the

loan.

? Profit = 713.17 – 696.26 = 16.91USD.

If forward rate = 0.66 USD/AUD

? Borrow 1,000 USD @ 7%.

? Convert into 1,000/0.62=1612.90 AUD and invest @

5% for 2 years.

? Enter a 2-years forward contract to sell AUD:

1,782.53AUD*0.66 =1,176.47 USD.

? In 2 years, receive investment 1,612.90e0.05*2=

1,782.53 AUD. Then, exchange to 1,176.47 USD.

? Pay US borrowing 1,000e0.07*2=1,150.27.

? Profit = 1,176.47 – 1,150.27 = 26.20USD.

Futures on Investment Commodities

? Gold and silver: can provide income, but have

storage costs.

? U: present value of all storage costs, net of income,

during the life of forward contract.

F0 = (S0+U)erT

? If storage costs u, net of income, are proportional to

commodity price:

F0 = S0 e(r + u)T

? Storage costs can be treated as negative yield.

Futures on Consumption Assets

? Consumption assets: high storage costs (U) and no

income, but consumption value.

? Suppose we have F0 > (S0+U)erT.

? An arbitrageur can borrow (S0+U), buy and store the

commodity, and short a futures contract.

? At expiration, the riskless profit = F0 - (S0+U)erT > 0

? There is a tendency for S0 to increase and F0 to

decrease, thus the relation can not hold for long.

Futures on Consumption Assets

? Suppose we have F0 < (S0+U)erT.

? An arbitrageur can sell the commodity, save the

storage costs, invest in risk-free rate, and long

futures contact.

? At expiration, the riskless profit = (S0+U)erT - F0 > 0

? There is a tendency for S0 to decrease and F0 to

increase, thus the relation can not hold for long.

? Wait a minute, is that true?

? The commodity is used for consumption, the holders

are reluctant to sell. The strategy does not work.

? There is nothing to stop the relation from holding.

Futures on Consumption Assets

? In case of consumption assets we have inequality:

F0 ? (S0+U)erT

? Alternatively,

F0 ? S0 e(r+u)T

? where u is the annual storage cost as a percent of the

asset value.

Convenience Yield

? For consumption commodities users, the ownership

of physical commodities are not obtained by future

contract.

? Ownership of physical asset enables a manufacturer to

keep a production process running and perhaps profit

from temporary local shortages.

? E.g., the crude oil in inventory can be an input to the

refining process, but not for futures contact.

Convenience Yield

? Convenience yield: benefit derived from holding the

physical asset.

? Only for consumption assets.

? Reflects the market’s expectations concerning the

future availability of the commodity.

? If inventories are low, shortages are more likely and the

convenience yield is usually higher.

? Convenience yield y balances out the previous

inequality:

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

The Cost of Carry

? The relationship between futures prices and spot prices

can be summarized in terms of the cost of carry.

? Cost of carry (c) = storage cost + interest costs -

income earned

? Non-dividend-paying stock: c = r.

? Dividend-paying stock: c = r – q.

? Currency: c = r - rf .

? Commodity: c = r - q + u.

? For an investment asset: F0 = S0ecT

? For a consumption asset: F0 = S0e(c–y)T

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