FIN 685 – Derivatives
Summer 2021
Assignment 1 – Part 2
(1) Assume the interest rate below is quoted in continuously compounded convention.
(2) The interest rate below is quoted in continuously compounded convention.
(3) Identify the arbitrage open to a trader who faces no transaction costs when ing or short‐selling stocks and
can borrow and lend at the risk‐free rate. Describe exactly which transactions are undertaken and the associated
profit of the arbitrage trade. The interest rate is quote in continuously compounded terms.
a. A European call option and put option on a stock both have a strike price of $20 an expiration date
in 3 months. Both sell for $3. The risk‐free rate is 5% per year (continuously compounded). The
current stock price is $19.
b. Same situation as in a) but now the stock is expected to pay a $1 dividend in one month.
(4) Consider the Principal Protected Note on the slides. Your client is not happy with receiving zero interest in his
$1,000,000 investment in case the stock market goes down over the next five years. Build a different Principal
Protected Note in which your client gets some interest if the market goes down, at the expense of having a
capped upside on the stock market. Assume:
i) the 5‐year risk free rate is 4% per year (continuously compounded)
ii) a 5‐year at‐the‐money call on the S&P500 index trades at a ratio C0/S0 equal to 0.25
iii) a 5‐year out‐of‐the‐money call on the S&P500 index with strike price that is 30% above the current index
level trades at a ratio of C0/S0 equal to 0.13.
Hint: There is no unique answer to this question because the amount of interest to be paid if the S&P500 goes
down can be set at different levels, and that affects how much of the appreciation of the S&P500 the client gets.
Here is a solution strategy. Start by replicating what was done in class, i.e., no cap on the upside. Find the
fraction of the money to be invested in the risk‐free asset, and therefore the fraction remaining to purchase the
ATM call. What slope can you get? If you the ATM call and sell the OTM call (bull spread), what slope can you
get instead? Probably you don’t want a slope greater than 1 … Set the slope equal to 1 FOR EXAMPLE. The
fraction of the money needed to obtain that slope equal to 1 is 0.25‐0.13=0.12. So, there will be 0.88 left to
the risk‐free asset, which is more than what you need to guarantee the principal. What is the (cumulative)
interest associated with investing 0.88 of the capital in the risk‐free asset and using 0.12 to purchase the bull
spread? If you set slope to be 2/3 like the PPN seen in class, what would be the cumulative interest?
a) What strategy creates the Principal Protected Note? Note you must specify exactly how the
clients’ $1,000,000 is going to be invested.
b) Plot the payoff on the Principal Protected Note as a function of the S&P500 return over the
next 5‐years. Note you have to specify how much interest the client gets if the stock market
goes down, how much interest he gets if the stock market appreciates from 0 to 30%, and
how much he gets if the stock market appreciates more than 30%.
(5) Consider the Principal Protected Note on the slides. Assume:
i) the 5‐year risk free rate is 5% per year (continuously compounded)
ii) the dividend yield on the S&P500 index over the next 5 years is 1.5% per year (continuously
compounded)
iii) a 5‐year at‐the‐money call on the S&P500 index trades at a ratio C0/S0 equal to 0.33333
iv) a 5‐year at‐the‐money put on the S&P500 index trades at a ratio P0/S0 equal to 0.20
Hint: Here is a solution strategy. Start by using the same slope as in the example seen in class: 2/3. How much
does it cost, as fraction of the total, to purchase the S&P500 spot and the ATM put (a “protective put” or
“married put” position) that guarantees 2/3 of the amount invested and provides a slope equal to 2/3? Answer,
it would cost 120% of the principal per unit … and we want 2/3 of a unit. So do the math. This is less than 1, so
there is a fraction left, to be invested in the risk‐free asset. How much money do you get for sure after 5 years?
Make sure to consider the dividend yield on the S&P500. Is it more or less than 1,000,000? If it is less than
1,000,000 you cannot offer 2/3 of the appreciation, it must be less. How much less?
a) Create a Principal Protected Note using an ATM Put (and the underlying asset) instead of the
ATM call.
b) Is it better to build the Principal Protected Note using the Put instead of the Call?
c) Check whether put‐call parity is violated. (Hint: Divide by S …)
Note: The put call parity when the underlying asset pays a yield rate q (rather than a discrete dividend D) is:
P+ Se‐qT = Ke‐rT +C
(6) Separately consider a long forward and a short forward position in the question below.
(7) CFA question
(8) Banania is a country in Latin America with really high interest rates and an exchange rate that is fixed to the US
dollar for at least the next year. You work for Goldman Sachs and would like to borrow in US dollars (at 1% per
year continuously compounded) and lend to Banania’s government in local currency. But Banania’s government
charges an entry tax of 10% upfront for foreign investors who want to Banania’s government bonds. There
is no entry tax for foreign investors to invest in Banania’s stock or option markets. The current level of Banania’s
stock market index is 100. And there are 3‐month European calls and puts on Banania’s stock market index with
following strike prices and premiums:
a) How can you invest in Banania’s “riskfree” asset by trading in options?
b) What annualized interest rate can you lock‐in through the trades in a) ?
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