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CMSC 5718 Introduction to Computational Finance

Assignment 3: Trading strategies and products (45% of total grade)


Instructions
1) Submit a copy of your report together with supporting programs and/or data files (as
a zipped file) by uploading to Blackboard on or before April 21, 2022, 11:59pm. The
file name of the zipped file or your report should be your student number with the
following format, e.g. 11550xxxxx.zip or 11551yyyyy.zip. [If uploading to
Blackboard is not successful, you may consider sending a email to
kalokchau@cuhk.edu.hk, but submission through Blackboard is preferred.]
2) No late submission is allowed.
3) This is an INDIVIDUAL assignment. Each student should submit one report.
4) Please observe the university’s plagiarism guidelines.


Introduction

In the first part of this assignment, we analyze the risk of a trading portfolio, and make use
of historical prices of some stocks to test the delta hedging strategy. These tests may not be
too realistic as some trading conditions have been ignored. In particular, board lots,
dividends and transaction costs are not included, but our aim is to show the validity of the
theoretical framework. Part two examines some theoretical relationships of derivative
trading strategies and product pricing.


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Part I: Risk Management of a Straddle Position (66%)

1. Choose the stock that you have to work on

For this part, your student number decides which stock you have to use to perform the
analysis. Take the last two digits of your student number, use modulo 50 to obtain the order
number, and look up the stock code from the given data sheet. For example, if your student
number ends with 18, the order number is (18 mod 50 = 18), and the stock is Techtronic
Industries Co. Ltd. (stock code 669). If your student number ends with 84, the order number
is (84 mod 50 = 34), and the stock is BYD Co. Ltd. (stock code 1211). This stock is known
as stock X in the questions below.


2. Option pricing (8%)

With the implied volatility given in the spreadsheet, use the Black-Scholes equations to price
a straddle for stock X, as of December 31, 2020:

1 European call option + 1 European put option, both at the same strike equal to the
stock price as of December 31, 2020, continuously compounded interest rate =
0.65%p.a., maturity = 1 year (December 31, 2021), dividend yield = 0.

Calculate the total price for this option straddle, with N call options and N put options.



3. Value-at-risk of the straddle (28%)

i) Assume that you are short N call options on X and N put options on X as described in
(2) above. Denote VaR_95 as the 5-day, 95% confidence interval Value-at-risk for
this strategy. Using a procedure similar to the one described in Lecture 9, slide 44
and using the model as in Lecture 6, slide 40, calculate VaR_95 with a Monte Carlo
simulation with 500,000 runs. [with t = 5/365]
ii) Denote CVaR_95 as the conditional-VaR at 95% confidence interval, which is given
as the average of the changes in the portfolio values when the changes exceed
VaR_95. Calculate CVaR_95 using the simulation results from 3(i) above.






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4. Test of delta hedging strategy (30%)

i) Assume that you are short N call options on X and N put options on X as described in
(2) above. Using the daily price data given in the spreadsheet, construct a delta
hedging strategy for the position for the period between December 31, 2020 to
December 31, 2021, so that the overall position is delta neutral daily (the format is
given in the spreadsheet). The account balance on each day is calculated by
summing the following components:
• Previous account balance.
• If the account balance is negative, interest has to be paid; if the
account balance is positive, interest will be received. The interest is
calculated daily, so that the total amount = previous day amount
balance x exp(interest_rate x num_days/365), where num_days is the
number of days from the previous date to the current date, so it can be
one day or more than one day, depending on whether there are
holidays.
• Cash required / received from share transaction.

Use the given implied volatility to generate the deltas. In your report, include a few
lines of this table (but no need to include all the dates). On maturity date, either the
call or the put would be exercised. The number of shares in the share account must
be equal to +N or −N, and this position is to be sold to or bought back from the option
holder at the strike price. Denote Fi as the final account balance after taking into
account of the above transactions. Find Fi.

ii) On December 31, 2020, you have deposited the money that you received from
shorting N call options and N put options (as in 2(ii) above) into a deposit account,
earning a continuously compounded interest of 0.65% p.a. The maturity of the
deposit is December 31, 2021, and the total amount received is Pi. Find Pi.
Compare Pi with Fi obtained in 4(i) above (noting that one is positive and the other is
negative). Does the final account balance Fi match the corresponding total amount
in the deposit account Pi? Comment briefly on the result.



Note:
- delta for a European call option with no dividend: N(d1). Delta for a European put option
with no dividend: N(d1) – 1. The total delta for a combination of 1 European call option and 1
European put option = 2N(d1) – 1.
- N refer to the “Number of options traded” field as given in the spreadsheet for the relevant
stock.




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Part II: Problem sets (34%)


1. Arbitrage opportunities (24%)
In each situation below, identify an arbitrage opportunity, suggest trades to be carried out
and calculate the theoretical profit. You should explain how the profit can be realized under
different market conditions. Assume transaction costs are 0.

(a) FX market (10%)

Spot FX level: EUR 1 = USD 1.1345
6-month forward FX rate: EUR 1 = USD 1.1113
EUR 6-month interest rate: 2.15% p.a.
USD 6-month interest rate: 1.33% p.a.

Assume 6 months = 0.5 year and interest is calculated by the convention rt+1 .

(b) Equity options (14%)

Current stock price S0: $100
Option maturity T: 0.5 year
Interest rate r: 2.75% p.a. (erT=1.01384, e−rT=0.98634)
Dividend before option maturity: 0
Price of call option today C0: $9.69
Price of put option today P0: $6.30
Strike of the call and put options K: $98.5


2. Equity Linked Investment (10%)
An investor has bought a capital guaranteed note (CGN) from a bank based on the Hang
Seng index. Some characteristics of the note are:

Notional amount: $500,000
Initial note price: 100% of notional amount
Tenor: 5 years
Reference price S: Hang Seng Index (HSI)
Payoff at maturity: 100%+ ×( − 0, 0)/0

On trade date, Hang Seng Index S0 = 20134, 5-year interest rate is 1.94% p.a. (annually
compounded rate, calculated as (1+r)t), option price is 13.80%, participation rate p is 55%.

i. Calculate the commission rate charged by the bank initially.
ii. After 2 years (i.e. remaining time to maturity = 3 years), Hang Seng index has already
moved up substantially. The mark-to-market option price becomes 41.3%, and the 3-
year interest rate becomes 1.45% p.a. The bank would charge a commission rate of
1.20% for an early termination of the CGN. What is the amount that the investor could
receive if he decides to terminate this trade and redeem the note? What is the
equivalent annualized return for the investor?
iii. If the investor has not terminated the note early and holds the position until maturity,
the level of HSI = ST = 27659. No commission is charged at the final redemption.
Calculate the equivalent annualized return for the investor.
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