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程序代写案例-FNCE 30007

时间：2021-04-14

FNCE 30007

Derivative Securities

Lecture – Forward & futures pricing

Outline

Introduction

Arbitrage

Investment v consumption assets

Short selling

Investment assets

Simple relation (no income)

Known income (bond futures)

Known yield (stock index futures, currency futures)

Consumption assets

Valuation

Environmental, social and governance movement

Futures & expected spot prices

Reading: Chapter 5

Introduction: No arbitrage pricing

Lecture 1 considered speculators, lecture 2 considered

hedgers.

We now focus on the other major participant –

arbitrageurs

Arbitrage takes advantage of a price differential between

two or more markets

No up front payment and a positive cash flow later

Arbitrage possible if

Same asset does not trade at the same price on all

markets (the law of one price); or

2 assets with identical cash flows do not trade at the

same price; or

An asset with a known price in the future does not

today trade at its future price discounted at the risk-

free interest rate

Introduction: Forward v Futures prices

Forward and futures prices are usually equal.

We therefore price futures contracts as if they are

forwards

Greatly simplifies the problem

Slight differences may arise from

Taxes

Transaction costs

Margins

Futures more liquid & have no counterparty risk

Introduction: Investment v Consumption assets

Investment assets: held by significant numbers of

people purely for investment purposes (gold, silver,

stocks, bonds).

Can price forwards & futures off spot via

arbitrage.

Consumption assets: held primarily for consumption

& not usually for investment purposes (copper, oil,

pork bellies, soybeans).

Not able to price forwards & futures off spot via

arbitrage.

Introduction: Short selling

Short selling: sell securities you do not own

Your broker borrows the securities from another

client and sells them in the spot market

At some stage you buy the securities back & return

them

If the price falls: sell high, buy low.

You pay dividends & other benefits to the owner of

the securities.

Diagram

Introduction: Assumptions & notation

Assume:

No transaction costs

Same tax rate

Borrowing/lending at risk-free rate

No-arbitrage opportunities (arbitrage opportunities

taken advantage of immediately).

S0: Spot price today

F0: Futures or forward price today

T: Time until delivery date (years)

r: Risk-free interest rate for maturity T

(continuously compounded rate p.a).

Investment assets - Simple pricing

The relationship between F0 and S0 is

If F0 > S0erT, arbitrageurs buy the asset

and short forward contracts

If F0 < S0erT, arbitrageurs short the asset

and buy forward contracts.

rTeSF 00 =

Investment assets - Example 1

Suppose:

The spot price of gold is US$600

The quoted 1-year futures price of gold is

US$650

1-year US$ interest rate is 5% p.a

No income or storage costs for gold

Is there an arbitrage opportunity?

Investment assets - Example 2

Suppose:

The spot price of gold is US$600

The quoted 1-year futures price of gold is

US$590

1-year US$ interest rate is 5% p.a

No income or storage costs for gold

Is there an arbitrage opportunity?

Short sale constraint

What if short sales are not possible for all

investment assets?

Short selling not needed for the forward

price equation.

Only require that at least some people hold

the asset only for investment purposes.

If the forward price is low, they sell the asset &

take a long forward position.

Investment assets - known $ income

Investment asset has income during the life of a

forward contract:

F0 = (S0 – I)erT

where I = present value of the income during life of

forward contract

If F0 > (S0 – I )erT, arbitrageurs buy the asset and

short the forward contract.

If F0 < (S0 – I )erT, arbitrageurs short the asset and

buy the forward contract.

Example: known $ income

Consider a long forward contract to buy a coupon

bond with a current price of $900.

-The forward contract has a 9 month maturity & a

coupon payment of $40 in 4 months.

-The continuously compounded 4 and 9 month risk-

free rates are 3% and 4% p.a

The forward contract price should be $886.60.

To show why we will consider 2 situations

i) where the futures is overvalued at $910.

ii) where the futures is undervalued at $870.

Example: known $ income (i)

The forward price is $910. Arbitrageur will

Today:

Borrow $900. The coupon payment has a present

value of 40e(-0.03)(4/12) = $39.60. So, $39.60 is

borrowed for four months and the rest ($860.40) is

borrowed for nine months (at 4%).

Buy the bond

Enter into a forward contract to sell the asset for $910.

In four months:

Receive $40 coupon payment

Use $40 to repay first loan with interest

In nine months:

Sell bond & receive $910 under the terms of the

forward contract.

Use $886.60 to repay second loan with interest.

Profit: $910 – $886.60 = $23.40

Example: known $ income (ii)

Forward price is $870. Arbitrageur will

Today:

Short the bond ($900)

Enter into a forward contract to buy the bond for $870

in nine months.

Of the $900 realized from shorting the bond, $39.60 is

invested for four months at 3% per annum (grows to

$40). The remaining $860.40 is invested for nine

months at 4% per annum (grows to $886.60)

In four months:

Receive $40 from four-month investment

Use $40 to pay coupon on the bond

In nine months:

Receive $886.60 from nine-month investment

Buy the bond for $870 under the terms of the forward

contract

Close out short position in the bond

Profit: $886.60 – $870 = $16.60

Investment assets - known yield

If the asset underlying a forward contract

has a known yield

F0 = S0e(r–q)T

where q = average yield over the life of the contract

Two examples

Stock index futures

Currency futures

Known yield – stock index futures

Investment asset paying a dividend yield

The price relationship is

F0 = S0 e(r–q )T

where q = dividend yield (over life of contract) on

the portfolio represented by the index

Index must represent an investment asset

changes in the index must correspond to changes

in the value of a tradable portfolio

Known yield – stock index futures

F0 > S0e(r-q)T arbitrageur buys the stocks

underlying the index and sells futures

F0 < S0e(r-q)T an arbitrageur buys futures

and sells the stocks underlying the index

Index arbitrage involves simultaneous

trades in futures & many different stocks

Very often computer used to generate

trades

Known yield – stock index futures

Lets consider the E-mini S&P500 futures contract

S&P500 close Feb 3/2020 3225.52

4 week US T-bill rate 1.53%p.a

Dividend yield 1.79%p.a

Contract specs reveal expires 3rd

Friday of the month. Therefore

34 trading days prior to expiration

T=34/252

[ ]3225.52exp (0.0153 0.0179) 34 252 3224.39oF = − × =

Known yield – currency

Foreign currency provides a continuous

yield - the foreign risk-free interest rate rf

Underlying asset is 1 unit of foreign

currency.

F S e r r Tf0 0=

−( )

Known yield – currency example

2year interest rates in Australia and the U.S. are 5%

and 7%. Spot exchange rate is 0.62 USD per AUD.

Find the two year forward rate.

Given that fx rate is 1 unit of foreign currency,

we consider from perspective of US investor.

Can also redo from the perspective of an

Australian investor.

Consider when 2year forward rate is 0.63 & 0.66.

Consumption assets

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 = present value of the storage

costs.

Consumption assets - example

Consider a 1 year copper futures contract.

Assume no income and that it costs $2 per ounce

per year to store copper, with payment being

made at the end of the year.

The spot price is $600 and the risk-free rate is

5% p.a for all maturities.

Find the futures price.

Consider when futures price is $700 & $610

Consumption assets – convenience yield

Convenience yield is the benefit from holding

the physical asset.

Reflects market expectations of future availability

High inventory expectations low conv yield

Low inventory expectations high conv yield

Convenience yield is

F0eyT = (S0+U )erT

If U is proportional to the spot price

F0eyT = S0e(r + u)T

Example

Cost of carry

Cost of carry, c, is the storage cost plus the

interest costs less the income earned

Investment asset F0 = S0ecT

Consumption asset F0 ≤ S0ecT

Convenience yield on consumption asset, y,

is defined so that

F0 = S0 e(c–y )T

Forward valuation

The value of a futures contract is zero - value reflected in

the margin account.

K is delivery price in a forward contract entered into

previously. F0 is forward price that would apply to the

contract today.

Value of a long forward contract, ƒ, is

ƒ = (F0 – K )e–rT

Value of a short forward contract is

ƒ = (K – F0 )e–rT

Forwards have a value of zero at the time first entered

into because F0=K.

As time passes, the forward price and the value of the

contract change.

Forward valuation: example

Consider a long forward contract on a non-dividend

paying stock entered into some time ago that has

5 months left to maturity. The risk-free rate with

continuous compounding is 9% p.a. The current

stock price is $30 and the delivery price is $28.

The value of the contract is:

F0 = 30e(0.09)(5/12) = $31.15

ƒ = (F0 – K )e–rT = (31.15 – 28)e(-0.09)(5/12) = $3.03

or

ƒ = S0 – Ke–rT = 30 - 28e(-0.09)(5/12) = $3.03

Will use this in the BSM lecture

ESG

Environmental, social and governance (ESG)

movement is now emerging in the derivatives

markets.

ESG companies typically less exposed to

environmental and regulatory tail risks (Value at

Risk lecture)

Eurex and Nasdaq have launched ESG futures

Similar plans for many other markets

China will soon open an exchange solely dedicated

to the trading of carbon finance futures

ESG

Eurex

Futures on the STOXX Europe 600 ESG-X index

Index screens out companies with low ESG rankings

Index enables investors to easily switch portfolio to an ESG

compliant benchmark with low cost and tracking error

Futures on the index now traded

Futures can be used for hedging and speculative purposes

This should add liquidity to the underlying index

Nasdaq

In October 2019 Nasdaq launched a futures based on the

OMXS30 responsible index

Also excludes companies with poor ESG standards

Futures Prices & Expected Spot Prices

So far we have focused on the contemporaneous

relation between F0 and S0.

What about the relation between F0 and E (ST )?

If

F0 = E (ST ) F0 unbiased estimate of ST

F0 F0>E (ST ) contango

Derivative Securities

Lecture – Forward & futures pricing

Outline

Introduction

Arbitrage

Investment v consumption assets

Short selling

Investment assets

Simple relation (no income)

Known income (bond futures)

Known yield (stock index futures, currency futures)

Consumption assets

Valuation

Environmental, social and governance movement

Futures & expected spot prices

Reading: Chapter 5

Introduction: No arbitrage pricing

Lecture 1 considered speculators, lecture 2 considered

hedgers.

We now focus on the other major participant –

arbitrageurs

Arbitrage takes advantage of a price differential between

two or more markets

No up front payment and a positive cash flow later

Arbitrage possible if

Same asset does not trade at the same price on all

markets (the law of one price); or

2 assets with identical cash flows do not trade at the

same price; or

An asset with a known price in the future does not

today trade at its future price discounted at the risk-

free interest rate

Introduction: Forward v Futures prices

Forward and futures prices are usually equal.

We therefore price futures contracts as if they are

forwards

Greatly simplifies the problem

Slight differences may arise from

Taxes

Transaction costs

Margins

Futures more liquid & have no counterparty risk

Introduction: Investment v Consumption assets

Investment assets: held by significant numbers of

people purely for investment purposes (gold, silver,

stocks, bonds).

Can price forwards & futures off spot via

arbitrage.

Consumption assets: held primarily for consumption

& not usually for investment purposes (copper, oil,

pork bellies, soybeans).

Not able to price forwards & futures off spot via

arbitrage.

Introduction: Short selling

Short selling: sell securities you do not own

Your broker borrows the securities from another

client and sells them in the spot market

At some stage you buy the securities back & return

them

If the price falls: sell high, buy low.

You pay dividends & other benefits to the owner of

the securities.

Diagram

Introduction: Assumptions & notation

Assume:

No transaction costs

Same tax rate

Borrowing/lending at risk-free rate

No-arbitrage opportunities (arbitrage opportunities

taken advantage of immediately).

S0: Spot price today

F0: Futures or forward price today

T: Time until delivery date (years)

r: Risk-free interest rate for maturity T

(continuously compounded rate p.a).

Investment assets - Simple pricing

The relationship between F0 and S0 is

If F0 > S0erT, arbitrageurs buy the asset

and short forward contracts

If F0 < S0erT, arbitrageurs short the asset

and buy forward contracts.

rTeSF 00 =

Investment assets - Example 1

Suppose:

The spot price of gold is US$600

The quoted 1-year futures price of gold is

US$650

1-year US$ interest rate is 5% p.a

No income or storage costs for gold

Is there an arbitrage opportunity?

Investment assets - Example 2

Suppose:

The spot price of gold is US$600

The quoted 1-year futures price of gold is

US$590

1-year US$ interest rate is 5% p.a

No income or storage costs for gold

Is there an arbitrage opportunity?

Short sale constraint

What if short sales are not possible for all

investment assets?

Short selling not needed for the forward

price equation.

Only require that at least some people hold

the asset only for investment purposes.

If the forward price is low, they sell the asset &

take a long forward position.

Investment assets - known $ income

Investment asset has income during the life of a

forward contract:

F0 = (S0 – I)erT

where I = present value of the income during life of

forward contract

If F0 > (S0 – I )erT, arbitrageurs buy the asset and

short the forward contract.

If F0 < (S0 – I )erT, arbitrageurs short the asset and

buy the forward contract.

Example: known $ income

Consider a long forward contract to buy a coupon

bond with a current price of $900.

-The forward contract has a 9 month maturity & a

coupon payment of $40 in 4 months.

-The continuously compounded 4 and 9 month risk-

free rates are 3% and 4% p.a

The forward contract price should be $886.60.

To show why we will consider 2 situations

i) where the futures is overvalued at $910.

ii) where the futures is undervalued at $870.

Example: known $ income (i)

The forward price is $910. Arbitrageur will

Today:

Borrow $900. The coupon payment has a present

value of 40e(-0.03)(4/12) = $39.60. So, $39.60 is

borrowed for four months and the rest ($860.40) is

borrowed for nine months (at 4%).

Buy the bond

Enter into a forward contract to sell the asset for $910.

In four months:

Receive $40 coupon payment

Use $40 to repay first loan with interest

In nine months:

Sell bond & receive $910 under the terms of the

forward contract.

Use $886.60 to repay second loan with interest.

Profit: $910 – $886.60 = $23.40

Example: known $ income (ii)

Forward price is $870. Arbitrageur will

Today:

Short the bond ($900)

Enter into a forward contract to buy the bond for $870

in nine months.

Of the $900 realized from shorting the bond, $39.60 is

invested for four months at 3% per annum (grows to

$40). The remaining $860.40 is invested for nine

months at 4% per annum (grows to $886.60)

In four months:

Receive $40 from four-month investment

Use $40 to pay coupon on the bond

In nine months:

Receive $886.60 from nine-month investment

Buy the bond for $870 under the terms of the forward

contract

Close out short position in the bond

Profit: $886.60 – $870 = $16.60

Investment assets - known yield

If the asset underlying a forward contract

has a known yield

F0 = S0e(r–q)T

where q = average yield over the life of the contract

Two examples

Stock index futures

Currency futures

Known yield – stock index futures

Investment asset paying a dividend yield

The price relationship is

F0 = S0 e(r–q )T

where q = dividend yield (over life of contract) on

the portfolio represented by the index

Index must represent an investment asset

changes in the index must correspond to changes

in the value of a tradable portfolio

Known yield – stock index futures

F0 > S0e(r-q)T arbitrageur buys the stocks

underlying the index and sells futures

F0 < S0e(r-q)T an arbitrageur buys futures

and sells the stocks underlying the index

Index arbitrage involves simultaneous

trades in futures & many different stocks

Very often computer used to generate

trades

Known yield – stock index futures

Lets consider the E-mini S&P500 futures contract

S&P500 close Feb 3/2020 3225.52

4 week US T-bill rate 1.53%p.a

Dividend yield 1.79%p.a

Contract specs reveal expires 3rd

Friday of the month. Therefore

34 trading days prior to expiration

T=34/252

[ ]3225.52exp (0.0153 0.0179) 34 252 3224.39oF = − × =

Known yield – currency

Foreign currency provides a continuous

yield - the foreign risk-free interest rate rf

Underlying asset is 1 unit of foreign

currency.

F S e r r Tf0 0=

−( )

Known yield – currency example

2year interest rates in Australia and the U.S. are 5%

and 7%. Spot exchange rate is 0.62 USD per AUD.

Find the two year forward rate.

Given that fx rate is 1 unit of foreign currency,

we consider from perspective of US investor.

Can also redo from the perspective of an

Australian investor.

Consider when 2year forward rate is 0.63 & 0.66.

Consumption assets

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 = present value of the storage

costs.

Consumption assets - example

Consider a 1 year copper futures contract.

Assume no income and that it costs $2 per ounce

per year to store copper, with payment being

made at the end of the year.

The spot price is $600 and the risk-free rate is

5% p.a for all maturities.

Find the futures price.

Consider when futures price is $700 & $610

Consumption assets – convenience yield

Convenience yield is the benefit from holding

the physical asset.

Reflects market expectations of future availability

High inventory expectations low conv yield

Low inventory expectations high conv yield

Convenience yield is

F0eyT = (S0+U )erT

If U is proportional to the spot price

F0eyT = S0e(r + u)T

Example

Cost of carry

Cost of carry, c, is the storage cost plus the

interest costs less the income earned

Investment asset F0 = S0ecT

Consumption asset F0 ≤ S0ecT

Convenience yield on consumption asset, y,

is defined so that

F0 = S0 e(c–y )T

Forward valuation

The value of a futures contract is zero - value reflected in

the margin account.

K is delivery price in a forward contract entered into

previously. F0 is forward price that would apply to the

contract today.

Value of a long forward contract, ƒ, is

ƒ = (F0 – K )e–rT

Value of a short forward contract is

ƒ = (K – F0 )e–rT

Forwards have a value of zero at the time first entered

into because F0=K.

As time passes, the forward price and the value of the

contract change.

Forward valuation: example

Consider a long forward contract on a non-dividend

paying stock entered into some time ago that has

5 months left to maturity. The risk-free rate with

continuous compounding is 9% p.a. The current

stock price is $30 and the delivery price is $28.

The value of the contract is:

F0 = 30e(0.09)(5/12) = $31.15

ƒ = (F0 – K )e–rT = (31.15 – 28)e(-0.09)(5/12) = $3.03

or

ƒ = S0 – Ke–rT = 30 - 28e(-0.09)(5/12) = $3.03

Will use this in the BSM lecture

ESG

Environmental, social and governance (ESG)

movement is now emerging in the derivatives

markets.

ESG companies typically less exposed to

environmental and regulatory tail risks (Value at

Risk lecture)

Eurex and Nasdaq have launched ESG futures

Similar plans for many other markets

China will soon open an exchange solely dedicated

to the trading of carbon finance futures

ESG

Eurex

Futures on the STOXX Europe 600 ESG-X index

Index screens out companies with low ESG rankings

Index enables investors to easily switch portfolio to an ESG

compliant benchmark with low cost and tracking error

Futures on the index now traded

Futures can be used for hedging and speculative purposes

This should add liquidity to the underlying index

Nasdaq

In October 2019 Nasdaq launched a futures based on the

OMXS30 responsible index

Also excludes companies with poor ESG standards

Futures Prices & Expected Spot Prices

So far we have focused on the contemporaneous

relation between F0 and S0.

What about the relation between F0 and E (ST )?

If

F0 = E (ST ) F0 unbiased estimate of ST

F0