衍生品代写-BFC2751
时间：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
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