Since you are buying platinum anyways, the forward
hedge any risk.
(a) Which future did you pick?
(b) What is the size of the contract? (Contract Unit under
50 troy ounces
(c) How is the price quoted?
U.S. $ per troy ounce
(d) When does trading end?
last bus. day of month
(e) What is the current price of the nearest-to-delivery
future? ("Quotes" tab)
(f) What is the current volume?
(g) What is the current open interest?
(h) Is this market in Contango, Backwardation, or
neither? (Recall how we plotted this in class)
current spot price is lower than the future price,
(i) Given the current open interest, the contract size, and
the price of the nearest-to-delivery future, what is the
approximate total notional value of this commodity?
Total Notional Value = price x contract size x open
$930.4 x 50 x 69,705 = $3.24B
-On September 1st you enter 2 forward contracts to buy
WTI crude oil on December 1st (3 months later) for a
price of $84.42 per barrel. On September 30th, the price
of a December 1st forward contract is $85.56. Before the
end of the day (Sept 30th), you decide to close out your
position by making an offsetting trade.
(a) Describe exactly what trade(s) you must make to
offset your position.
You must enter into 2 short
forward contracts for December 1st Delivery.
(b) What is your profit or loss per barrel?
(St - K) + (F0
- St) = F0 - K $85.56/bbl - $84.42/bbl = $1.14/bbl
(c) What is your total profit or loss?
$1.14/bbl x 2
contracts x 1,000bbl/contract = $2,280
(d) What happens if you forget and do not make any
You are now the proud owner of 2,000
barrels of oil (and get to deal with any additional
-Tesla (TSLA) is currently trading at S0 = $273/share. A
dealer is offering a 3-month forward at a price of F0 =
$275/share. One TSLA futures contract is for 100 shares.
Assume the interest rate is r = 1.5% p.a. You have no
money or financial assets today. However, using your
knowledge of forward pricing, you realize you can make
an arbitrage profit given this dealer's price! (Assume no
margin costs or storage costs.)
(a) Name the strategy we discussed in this class to make a
Cash and Carry
(b) Clearly show each transaction you must make, both
today and at time T, to make a riskless profit
(c) What is your profit per share?
(d) What is your total profit in dollars, assuming you only
use 1 forward contract?
$0.9743/share x 1 contract x
100 shares/contract = $97.43 total profit
-Some time ago, you entered a forward contract to
purchase shares of Advanced Micro Devices, Inc. (AMD)
for $32 per share. Your forward contract ends in 6 months
and the current spot price of AMD is $29.42 per share.
Assume a risk free rate of interest of 1% per year.
(a) What is the current price of a forward which ends in 6
F0=S0erT = 29.42e(0.01*0.5) = $29.5675/share
(b) What is the value of your seasoned forward?
f = (F0 -
K)e-rT = (29.5675 - 32.00)e(-0.01*0.5) = -$2.42/share
Suppose now the current spot price of AMD is $33.42 per
(c) What is the current price of a forward which ends in 6
F0=S0erT = 33.42e(0.01*0.5) = $33.5875/share
(d) What is the value of your seasoned forward?
f = (F0 -
K)e-rT = (33.5875 - 32.00)e(-0.01*0.5)
-You enter a 3-year future contract to purchase gold. The
current spot price is $1,716.42 per ounce. It costs the
bearer of the gold $15 per ounce to store it each year.
Assume an interest rate of 2%.
(a) What is the present value per ounce (U) of the storage
costs that the gold bearer will incur during the life of the
U = $15*(e-r + e2r + e-3r) = $43.24
(b) Using your answer from (a), what is the price on this
-Consider a 6-month forward contract on an asset that
provides a 1% yield annually to the bearer of the asset. If the
current spot price is $25, and assuming a 5.5% annual risk-
free interest rate.
(a) What is the price on this forward contract?
F = S0e(r-q)T
= 25e(0.055 - 0.01)(6/12) = $25.5688
(b) What is the price on a similar contract but with a 1-year
F = S0e(r-q)T = 25e(0.055 - 0.01)(1) =
-The price of a commodity "A" has a standard deviation of
$0.42, while the price of a futures contract "B" on that
commodity has a standard deviation of $0.84. The two have
coefficient of correlation of 0.69.
(a) If you want to hedge commodity A using future B, what is
the optimal hedge ratio that will minimize your risk
? h =
0.69*(0.42/0.84) = 0.345
(b) You decide to hedge this
position by matching quantities. You have 12,000 units of A
to hedge, and a futures contract B is for 2,000 units. Using
your hedge ratio from (a), how many contracts should you
? N = h* 12,000units / 2,000 units = 0.345 * 6 = 2.07
or approximately 2 contracts
-You are a fund manager in charge of a 50-million-dollar
portfolio. Your portfolio has a beta of 1.25 relative to the
S&P500. The S&P500 Index is currently 3,942. One S&P500
futures contract has a value of $250 * the index level. (1.00
(a) Assume you want to remove S&P500 risk from your
portfolio. Do you use long or short contracts?
is naturally long the S&P500, you hedge this by shorting
S&P500 index futures.
(b) What is the optimal number of S&P500 futures contracts
that you need to remove all S&P500 risk from your
N = Beta * Total Value / Forward Contract
Forward Contract Value = $250 * Index Level = $250 *
3,942 = $985,500 N = 1.25 * ($50,000,000/$985,500) = 1.25
* 50.7356 N = 63.4196 or short approximately 63
-Mili Pharmaceuticals is based in New Delhi, India, and has a
4-year contract to produce chemicals at a production facility
located in Delaware, United States of America. Mili received
$80,000 per year and is concerned about the exchange risk
she will bear exchanging US Dollars (USD) to Indian Rupees
(INR) over this period. Luckily, Neel Aeronautics, a US firm
doing business with India, has the exact opposite currency
exposure. Mili and Neel agree to a currency swap in order to
hedge their exchange risk. Assume the US risk free rate is
2%, the Indian risk-free rate is 2.8%, and the exchange rate
between USD ($) and INR (
) is 80.00
/$ (80.00 Rupees can
be exchanged for 1 USD).
rus = 0.02 , rin = 0.028 , S0 = 80.00
a) What is the present value of Mili's 4-year cash flow in
USD, compounded continuously?
PV = 80,000e^(-0.02 x 4)
b) Next, Mili wants to borrow this amount of USD, convert it
to INR at today's spot rate, and lend it to Neel. What is her
loan to Neel in INR?
$304,469.52 x 80 INR/USD =
c) Neel will repay her loan with 4 even payments in INR.
What is the value of one of his yearly payments? (Hint: you
are solving for the coupon)
24,357,566.55 / (e^-0.028 + e^(-
0.028 x 2) + e^(-0.028^3) + e^(-0.028^4)) = 6,527,722 INR
d) What is Mili's net cash flow at the start of the contract,
e) What is Mili's net cash flow during a given year?
f) Explain briefly how this has hedged Mili's foreign
The formula above is similar to a short
forward on the INR to USD exchange rate. Mili has made
a forward position while hedging her exchange risk. Now,
she receives a fixed amount of INR per year rather than
having to rely on the exchange rate.
You spend many sleepless nights building cash flow models
of Facebook Inc. (FB) and are convinced they are
undervalued. The current spot price of FB is $204/share. You
predict that in 6 months they will reach $240/share. Assume a
2% interest rate per annum.
b) Under what circumstances will you make money?
K + Ce^rt = breakeven
$230 + 5.20e^(0.02 x 0.5) = $235.252
If stock prices strike, we will make money because stock
prices will go up
-Despite Tesla's poor performance recently, you believe it
can go lower still. You decide to short a single share of
Tesla stock (TSLA). It is currently valued at $123.22 per
share. In 1 month, TSLA will have their next earnings
announcement, and you are hoping they will not meet
analysts' expectations. Assume you have an interest rate
of r = 0.045 (4.5%) per annum, and that all interest is
(b) On the earnings call, what price does TSLA need to
reach for you to break even?
0 = ST - S0erT = ST -
123.22*e(0.045*1/12) ST = $123.68 breakeven price
(c) Rising lithium prices cause TSLA's earnings to
plummet. Their share price is currently $112.42. What is
your payoff and what is your profit?
Payoff = - ST = -
$112.42/share Profit = S0erT - ST = $123.68 - $112.42
(d) Ellon Musk tweets that Tesla will now accept
Dogecoin as payment and the internet goes wild. TSLA
price jumps to 133.69 per share. What is your total payoff
and total profit?
Payoff = - ST = - $133.69/share Profit
= S0erT - ST = $123.68 - $133.69 = - $10.01/share
-Bed Bath and Beyond (BBBY), once a popular meme
stock, has recently been ignored. Currently trading at
$3.49 per share, you believe the stock is undervalued and
decide to buy 10 shares. One month later, the stock price
rises to $9.42. Assume an interest rate of r = 0.01 (1%)
-(b) What is your breakeven price?
0 = ST - S0erT = ST
- 3.49*e(0.01*1/12) ST = $3.49 breakeven price
(c) What is your payoff and profit per share in this
Payoff = ST = $9.42/share Profit = ST -
S0erT = $9.42 - $3.49 = $5.93/share
-People say we are in a recession, but you don't believe
them. You think the recession is almost over and that the
price of gold will drop soon. You decide to short 2 gold
futures contracts at a price of $1886.50 per ounce,
expiring in 3 months. Assume an interest rate of r = 0.03
(3%). (Hint: 1 gold futures contract covers 100 ounces of
(a) If the price of gold drops to $1833.42 per ounce, what
is your total profit?
Total Profit = (1886.5 - 1833.42) x2
x100 = $10,616.00
(b) If the price of gold rises to $1900.42 per ounce, what
is your total profit?
Total Profit = (1886.5 - 1900.42) x2
x100 = - $2,784.00
You work for Cao Chemical International, and your
company regularly uses platinum as a catalyst for
chemical reactions. To hedge your exposure to the price
of platinum, you convince your boss to let you use
futures contracts. Your company expects that it will need
1,000 troy ounces of platinum in 3 months. The price on
a 3-month platinum future contract is $1084.42 per troy
ounce. A platinum futures contract is for 50 troy ounces-
(a) How many contracts will you need to cover your
exposure, and what will it cost you initially?
1000 / 50 =
20 contracts Initial cost = $0.00 (Futures cost $0 up
(c) In 3 months, the spot price of platinum is $900.42/oz.
What is your total profit assuming you close the position
and do not take delivery?
Total Profit = (900.42 -
1084.42) x 20 x 50 = - $184,000
(d) If the spot price is $800/oz what is your total profit?
Total Profit = (900.42 - 1084.42) x 20 x 50 = -
(e) Does it matter if your futures contracts earn
a profit? Since, you are buying platinum anyways, the
forward will exactly hedge any price risk.