Kozminski University
Derivatives Valuation and Advanced Risk
Management
Case Study
PhD Adrian Struciński
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Case study
Total number of points: 30
Deadline: 2025-01-19
Maximum number of group members: 4
Solutions (report in Word or Pdf, estimations in Excel) should be sent on
astrucinski@kozminski.edu.pl.
You work as portfolio manager and one of your crucial clients has just asked you to invest
10 000 000 € in currency American options. He asked you to choose one of the major rate
from the following list:
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EURUSD
GBPUSD
EURGBP
USDJPY
EURJPY
GBPJPY
USDCAD
AUDUSD
Your customer expects increasing volatility on currency markets, thus wishes to open long
strangle strategy (simultaneous positions in out-of-the money call and put options with the
same nominal values and with delta of the whole portfolio close to 0) with 6 months to
maturity. However, your customer wants to earn on increasing volatility and be immune to
slight changes of underlying asset. Hence, you proposed to keep portfolio constantly delta
neutral.
Required:
1. Choose one of the major exchange rates listed above.
2. Determine strikes for both call and put options. You need to open strangle strategy, thus
you have to buy out-of-the money options (both should have the same nominal value).
Assume that options you’re going to buy should have delta between 0,1-0,35 (in
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3.
4.
5.
6.
absolute terms) – you’re free to select delta (which would determine strike). However,
you need to remember that delta of the whole portfolio should be 0.
Calculate the price of both options (determined in point 2.) using 6-step binomial tree.
Hence, you need to divide the whole time to maturity into 6 equal subperiods. Calculate
required parameters with Cox, Ross and Rubinstein approach (discussed during our
classes).
Determine spot exchange rate by reference to currency markets.
Estimate interest rates needed in valuation model (domestic and foreign rates). You may
use LIBOR rates for 6-months period (you may assume, that interest rates term-structure
is flat). However, you need to adjust market quotes, as rates in binomial tree model
should be continuously compounded.
Estimate expected volatility of selected underlying. You may use historic data (historic
continuously compounded percentage changes of the underlying asset) to calculate
expected standard deviation. You should use sufficiently large sample period (at least
30 observations). If you’d like to calculate it on the basis of 6-month changes, then your
sample period would be 15 years (30*0,5). It would cover many observations from
distant past, so it’s better to use shorter subperiod. I would suggest to use monthly
observations (data for last 31 months). However, as volatility needs to be expressed on
the annual basis, you have to adjust the results of your calculations using the following
formula:
𝜎𝐴𝑛𝑛𝑢𝑎𝑙 = 𝜎𝑡 ∗ √𝑡
Where:
𝜎𝐴𝑛𝑛𝑢𝑎𝑙 – volatility on annual basis,
𝜎𝑡 – volatility estimated on the basis of period shorter than 1 year
𝑡 – number of subperiods within 1 year
For example, if volatility is calculated on quarterly data, then it should be multiplied by √4 to
get the volatility on annual basis.
7. Determine the nominal values of acquired options. Both options should have the same
nominal and the total price paid for them should be equal to the amount invested
(10 000 000 EUR).
8. Propose hedging strategy to keep portfolio delta neutral (assume that adjustments are
made at each node in the binomial tree model). To hedge the portfolio you should use
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forward contracts with 9 months to maturity (from the moment of initial purchase of the
options). You need to calculate delta of your whole portfolio, forward prices and delta
of the contract as well as nominal value of contracts you will have to buy/sell at each
node in the binomial tree model. Assume, that interest rates remain constant.
Notes:
A. Your task is to prepare report (in Word with all calculations in Excel) which describes
in details how you would carry out initial strangle and subsequent hedging strategies
discussed above. Your report should include:
a. Chosen exchange rate
b. Detailed description of all steps taken when determining option strikes (e.g.
assumed delta, how you obtained strikes etc.) and valuation model.
c. Explanation how you estimated volatility and how you obtained both spot price
and interest rates. You should report especially: sources of data, sample period
used to calculate volatility, adjustments of volatility and interest rates.
d. Detailed description of hedging strategy applied to keep the portfolio delta
neutral (how you calculated forward prices and delta of forward contracts,
nominal of contracts needed to make portfolio delta neutral at each node, etc.)
B. To calculate volatility and determine interest rates you should use real data from the
chosen currency markets (real interest rates and historic exchange rates). You can use
closing prices for any day up to your deadline (however all data should refer to the same
moment of time). All needed information you can find on many websites related to
financial markets (e.g. www.investing.com) or you can also look for required data in
Bloomberg. In your reports you should put screens of quoted prices, which were basis
of your calculations.
C. You have full discretion on choosing exchange rate, delta and sample period for
volatility calculations. You can work in groups but each group works separately. If
two or more very similar reports will be sent (with similar portfolios and similar
mistakes in your estimations) then number of points is divided by the number of such
alike solutions.
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