FIN 685 – Derivatives
Summer 2021
Assignment 2 – Part 1
(1) .
(2) .
(3) .
(4) a) Go to Yahoo!Finance , obtain daily Bitcoin (BTC‐USD) returns over the last year (Historical Data tab), and
calculate the annualized volatility of Bitcoin returns over the last year.
b) Find the price of an at‐the‐money European put option on Bitcoin maturing in 3 months. Assume the current
level of Bitcoin is 30,000, and that the Bitcoin volatility over the next 3 months will be equal to the volatility you
calculated in a). The risk‐free rate is 2% per year (continuously compounded).
c) What is the risk‐neutral probability that the Bitcoin put will end up in‐the‐money?
d) Suppose you are really bearish on Bitcoin and have $1,000,000 of capital to support a bearish bet. You could
either the 3‐month ATM put or sell Bitcoin futures. The Bitcoin futures contract that matures in 3‐months
sells for 30,150. Each Bitcoin futures contract involves 5 Bitcoins and requires an initial margin of $60,000. Do
you achieve more leverage ing the put or selling futures? How much more?
(5) An innovative financial institution is launching a new investment product: a TSLA square‐root note (TSLA‐SQRN).
The note matures in one year and pays the square‐root of the value of TSLA stock at maturity. The current level
of TSLA stock is $600. The volatilty of TSLA is 50% per year, and risk‐free rate is 2% per year (continuously
compounded). TSLA stock does not pay dividends. Use risk‐neutral simulation to estimate the fair price of the
SQRN.
Hint: The TSLA‐SQRN is a power option with strike price equal to zero.
(6) .
a)
b) Repeat for a put.
c) You one ATM call and one ATM put (that is, a straddle). Later on the same day the stock
price increases by 1%. Use deltas to estimate by how much the total value of your portfolio
change.
(7) Consider an European call option on a non‐dividend paying the stock with the following characteristics. The
current stock price is $49, the strike price is $50, the risk‐free rate is 5% per year, and the time to maturity is 20
weeks (20/52 of one year). You sold 100,000 calls for $3 each. You bet that the future volatility of the stock will
be 20% per year over the next 20 weeks. Because you have no opinion on whether the stock will go up or down,
you need to delta‐hedge the sold option until its maturity.
a) What is the implied volatility of the call you sold?
b) What is your estimate of the present value of the cost of delta‐hedging the option until its
maturity?
c) Suppose the stock price evolves like in Path 1 of the spreadsheet Delta‐Hedge Question.
1. What was the annualized volatility of the stock during the 20 weeks?
2. What What is your profit or loss at the option’s maturity? Assume you delta‐hedge once
a week.
d) Repeat c.1) and c.2) for Path 2.
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