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


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