Assignment 3
Note: For problems that require you to source information online. Please write clearly the website link,
and the date/time you accessed the website. Points will be deducted if no source is given.
Chapter 19
Data
Go to
this data to answer problems 19.1, 19.2, and 20.1. Please take note of the open interest and volume on
any options that you use, if the option is not traded much, then the price and implied volatility may not
be very accurate. If you would like to use options from another exchange you are free to do so, but please
provide a link.
Problem 19.1
Let’s revisit the hedging simulation from Chapter 19 that we covered in class. Choose an option from
your sample above with a maturity of approximately one month. Assume you are a bank that sold the
option at the market price. Use the implied volatility on the ATM call option to simulate the price
movements on the underlying stock for the next month. I want you to delta-hedge the option once a day.
a) Graph the price of the stock, the delta of the option, and the gamma of the option over the life of
the option.
b) After the option expires compare the price you received for the option and the cost of delta
hedging to maturity.
c) Comment on the difference (if any) between the two price you received for selling the option and
the cost of hedging the option.
d) Repeat the simulation in (b) multiple times and graph the distribution of outcomes. How would
you deal with this as an option seller?
Problem 19.2
From 19.1 pick the option with the highest gamma from your option sample.
a) Describe what ”
gamma” captures, in words.
b) What is the impact of a jump of 3% in the price of the underlying on the value of your chosen
option in dollars?
c) Imagine you are short 1000 of these options, your manager comes to your office and says you
need to hedge out the gamma risk. To hedge you are only allowed to use an option with delta of
0.1 and gamma 0.20, what position in this option contract do you need to take?
d) Your manager now comes back and realizes he forgot to hedge delta, what can you do to hedge
out the delta without changing the gamma?
e) Why does your method for (d) work?
Chapter 20
Problem 20.1
Use the data on your chosen underlying asset above.
a) Graph the implied volatility as a function of strike price for short maturity (less than 4 months)
and long maturity options (maturity greater than one year).
b) Comment on the implied volatility function for each maturity.
c) Describe the implied volatility surface implied by the long and short options and give a
description of the shape and how it relates to the risk-neutral distribution.
Problem 20.2
On Tuesday 3rd November 2020 Alibaba Holding Group Limited had a closing price of 299.80 Hong
Kong Dollars. After the market had closed, news broke that the planned IPO for Ant financial had been
suspended. On Wednesday 4th November at 9:30am when HKEX opened for trading Alibaba was trading
at 273.00 HKD, a price drop of 8.9%.
Imagine you were working for a bank that had sold put options on Alibaba with a strike price of 300.00
HKD, with expiration on the 6th of November. The bank sold the put options using the price suggested by
Black-Scholes. They were hedging the option as if the BSM model was exactly true.
a) What is the status of your hedge portfolio on Wednesday morning?
b) What parts of this trading situation violate the assumptions of the BSM model.
c) How would you recommend they change their pricing and/or hedging arrangements in the
future.
Chapter 22
Problem 22.1.
Consider a position consisting of a $100,000 investment in asset X and a $100,000 investment in asset
Y. Assume that the daily volatilities of both assets are 1% and that the coefficient of correlation between
their returns is 0.45.
a) What is the 5-day 99% VaR for the portfolio?
b) What is the 5-day 99% VaR for a $200,000 investment in asset X?
c) Is your answer to a) different from your answer to b), why?
Problem 22.2
Suppose that the daily change in the value of a portfolio is, to a good approximation, linearly dependent
on three factors, calculated from a principal components analysis. The delta of a portfolio with respect
to the first factor is 6, the delta with respect to the second factor is –4, and the delta with respect to the
third factor is 5. The standard deviations of the factors are 8, 20, and 3 respectively.
a) What is the 5-day 95% VaR?
b) If you could only choose two of the three factors in the PCA, which would you choose and why?
c) Recalculate your answer to part a) given your choice in b).
Chapter 24
Problem 24.1
A company has a one-year bond outstanding, which provides a coupon of 8% per year payable
annually. The yield on the bonds (expressed with continuous compounding) is 6.0% . Risk-free rates are
4.5% for all maturities. The recovery rate is 35%. Defaults can take place halfway through each year.
a) Estimate the risk-neutral default rate for one year
b) Explain carefully the distinction between real-world and risk-neutral default probabilities.
Which is higher?
