Pricing and Application of a Chooser Option

Term Project: Pricing and Application of a Chooser Option
MSF 524, IIT Stuart School of Business
Fall 2017
Professor Sang Baum “Solomon” Kang
In this semester, I will discuss about the term project during our regular classes. Furthermore,
please, feel free to ask any questions during my office hours or via e-mails.
In your term project, you will program Matlab and write white papers in order to perform tasks
related to derivative pricing and financial risk management. Working on the term projects, you
will have hands-on experience of programming Matlab in the context of financial derivatives and
risk management and hone your professional writing skill. For the purpose of this term project,
you should use Matlab for calculation; other math/statistical software is not allowed.
The term project will be intentionally announced at a relatively early part of this semester because
you will have clear expectation of what you can learn from this course. Throughout this semester,
you will learn how to perform tasks set forth in the term project. Submit the term project by Dec
9, 2017 (Sat) 11:59pm. A late submission is not acceptable. Submit both Matlab codes and white
papers as a team of no more than three individuals. When submitting, please, submit your white
paper and a zip file containing your Matlab codes via Blackboard. The evaluation criteria will be
announced in advance. At the end of this semester, you may want to list your term projects in your
resume and use them as a marketing tool for your job search.
Term Project Description
Assume that you work for a (fictitious) large investment bank called Stuart & Partners. Assume
that you are a (quantitative) structuring analyst who is willing to offer a customized derivative
securities to your client. Your clients have different views on the expected stock returns, and the
market volatility – some clients are bullish and others are bearish; some expect high market
volatility and some expect low market volatility. Their investment horizons are two years – a client
may be bullish in the first year but bearish in the second year. So, you are considering to structure
compound option deals for your clients and recommend your structured deals to your clients.
Chooser Option
In a risk neutral world, a non-dividend-paying stock price follows
d𝑆(𝑡) = 𝑟𝑆(𝑡)𝑑𝑡 + 𝜎𝑆(𝑡)𝑑𝑧(𝑡)
where constant r is the annualized continuously compounded risk-free interest rate, 𝜎 is volatility,
and 𝑧(𝑡) is a Brownian motion.
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A “chooser option” is an exotic compound option. An originator in your desk is trying to offer a
“chooser” option to your client. (For the definition of “chooser” option, see Chapter 4.5.4 of
Brandimarte (2006).) The structure of the “chooser option” is as follows:
 At time 0, a client pays the option premium.
 At time 𝑇1, the client chooses either the call option at strike 𝑋𝑐 or the put option at strike
𝑋𝑃.
 At time 𝑇2 , the client who chose the call (put) option at time 𝑇1 has a right but not
obligation to buy (sell) the underlying asset at strike 𝑋𝑐
(𝑋𝑃).
Hence, the payoff of the “chooser option” at time 𝑇2 is
max(𝑆(𝑇2
) − 𝑋𝑐
, 0) 𝐼{𝑇ℎ𝑒 𝑐𝑙𝑖𝑒𝑛𝑡 𝑐ℎ𝑜𝑠𝑒 𝑡ℎ𝑒 𝑐𝑎𝑙𝑙 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑇1}
+ max(𝑋𝑃 − 𝑆(𝑇2
), 0) 𝐼{𝑇ℎ𝑒 𝑐𝑙𝑖𝑒𝑛𝑡 𝑐ℎ𝑜𝑠𝑒 𝑡ℎ𝑒 𝑝𝑢𝑡 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑇1}
where 𝑆(𝑇2
) is the underlying asset price at time 𝑇2, and 𝐼{∙}
is the indicator function which returns
1 when a given statement is true and 0 when a given statement is false.
The motivation of her client is as follows:
 The client was initially interested in a straddle to “buy” volatility. (For the definition of
straddle, see Chapter 11.4 of Hull (2012).)
 Because the client believes the straddle is too expensive, the client is interested in
purchasing the “chooser” option which should be less expensive than the straddle.
You are interested in calculating the fair value of the chooser option using the following parameters:
 The initial underlying price 𝑆(0) = $50;
 Strike price of underlying options 𝑋𝑐 = 𝑋𝑃 = $50;
 Risk-free interest rate r=0.025;
 Expiration of the chooser option 𝑇1=1 year from now;
 Expiration of the underlying options 𝑇2=2 years from now;
Assume that the contract size is 100 shares of the stock.
Heterogeneity in Beliefs
In a physical world, a non-dividend-paying stock price follows:
d𝑆(𝑡) = 𝜇𝑖
(𝑡)𝑆(𝑡)𝑑𝑡 + 𝜎𝑖
(𝑡)𝑆(𝑡)𝑑𝑧(𝑡)
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where 𝜇𝑖
(𝑡) is client 𝑖’s subjective belief on the draft, 𝜎𝑖
(𝑡) is client 𝑖’s subjective belief on
volatility, and 𝑧(𝑡) is a Brownian motion. You have 5 clients who have different subject believes
on the draft and volatility.
As a structuring analyst, you want to propose the strike prices (𝑋𝑐 and 𝑋𝑃) to each client. When
you price a compound option to determine the option premium, you should use the risk-neutral
pricing. However, you also need to do some risk-return analysis for your clients to persuade them.
There is not a single way of risk-return analysis using Monte-Carlo simulation – for example, you
may want to calculate the simulated mean profit, the mean return to the investment, the mean
excess return, the standard deviation of the excess return, the Sharpe’s ratio, and the 95% value at
risk.
For this project use 𝑆(0) = $50, r=0.025, 𝑇1 = 1 year, and 𝑇2 = 2 years. Again, 𝜇𝑖
(𝑡)s and
𝜎𝑖
(𝑡)s are client-specific where 0 ≤ 𝑡 ≤ 𝑇2. The clients’ 𝜇𝑖
(𝑡)s, and 𝜎𝑖
(𝑡) are given as follows:
Client
ID
Client
Name
When 0 ≤ 𝑡 < 𝑇1 When 𝑇1 < 𝑡 ≤ 𝑇2
1 Mrs.
Smith
𝜇𝑖
(𝑡) = 𝑟 + 0.03 & 𝜎𝑖
(𝑡) =0.15 𝜇𝑖
(𝑡) = 𝑟 + 0.005 & 𝜎𝑖
(𝑡) =0.30
2 Mr.
Johnson
𝜇𝑖
(𝑡) = 𝑟 − 0.03 & 𝜎𝑖
(𝑡) =0.20 𝜇𝑖
(𝑡) = 𝑟 − 0.01 & 𝜎𝑖
(𝑡) =0.18
3 Ms.
Williams
𝜇𝑖
(𝑡) = 𝑟 − 0.03 & 𝜎𝑖
(𝑡) =0.18 𝜇𝑖
(𝑡) = 𝑟 + 0.03 & 𝜎𝑖
(𝑡) =0.12
4 Mr.
Jones
𝜇𝑖
(𝑡) = 𝑟 + 0.02 & 𝜎𝑖
(𝑡) =0.35 𝜇𝑖
(𝑡) = 𝑟 + 0.02 & 𝜎𝑖
(𝑡) =0.10
5 Miss
Brown
𝜇𝑖
(𝑡) = 𝑟 + 0.03 & 𝜎𝑖
(𝑡) =0.15 𝜇𝑖
(𝑡) = 𝑟 − 0.05 & 𝜎𝑖
(𝑡) =0.15
With these being said, please, do the following tasks:
a) Because the close-form solutions, if any, cannot incorporate the time-varying 𝜎𝑖
(𝑡), you need
to use numerical methods to use those agents’ time-varying 𝜎𝑖
(𝑡): Programing Matlab, please,
price out the chooser option in the following three methods we learn in this semester:
①. Simple Monte-Carlo simulation: For the simple Monte-Carlo simulation, control the
relative error within plus/minus 0.1 percent.
②. A “smart-lattice” version of CRR binomial tree: For the binomial tree, set the
subinterval to be one calendar day ( 1
365
year).
③. The implicit finite difference method: For the implicit finite difference method, set
the time subinterval to be one calendar day (
1
365
year) and use reasonable parameters of
dS, S_min, and S_max.
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Then, discuss the calculation speeds of these three methods.
b) Use various variance reduction techniques to improve the speed of the MC simulation in a)
and make a recommendation of the variance reduction technique of your choice. (Hint: There
is not a single right answer for this task. Use your creativity. I will measure calculation time
when I grade this task, though.)
c) Use a trinomial to improve the accuracy of the binomial method in a). Set the subinterval to be
one calendar day ( 1
365
year) and use a reasonable “size parameters.” Discuss if the accuracy is
improved related to the binomial method in a).
d) As a structuring analyst, you want to propose the strike prices (𝑋𝑐 and 𝑋𝑃) to each client. If
you believe buying a chooser option is a bad idea for a specific client, you should justify your
claim. (When you recommend strike prices, please, do not recommend a strike price outside
plus/minus 25% of the ATM strike because a deep out-of-the-money option valuation may be
inaccurate.)
When you price a compound option to determine the option premium, you should use the riskneutral pricing. However, you also need to do some risk-return analysis for each of your clients
to persuade them. When you do the risk analysis for your client, you should do Monte-Carlo
simulation using a physical measure against your client’s 𝜇𝑖
(𝑡)s, and 𝜎𝑖
(𝑡). There is not a single
way of risk-return analysis using Monte-Carlo simulation – for example, you may want to
calculate the simulated mean profit, the mean return to the investment, the mean excess return,
the standard deviation of the excess return, the Sharpe’s ratio, and the 95% value at risk.
Through your risk-return analysis, you should make a convincing cases for each of your 5
clients. If you want, you may compare recommendation for each client with an alternative
compound option strategy such as “buying a straddle”. If you want, you may compare your
case for each client with the case using the “average” belief of your 5 clients.
e) For Stuart & Partners, propose another business opportunity directly or indirectly using the
pricing models and risk analysis you programmed in a) – d). Try to relate your proposal to an
academic paper, a magazine/ newspaper article or interview with a real world professional.
Write one white paper to address a) to e) in the above. The white paper should contain the
following sections:
1) Executive Summary
2) Introduction
3) Methodology: Chooser Option, Binomial Tree, Monte-Carlo Simulation, and Finite
Difference Method
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4) Numerical Results and Discussion
5) Recommendation for Improving the Calculation Speed of MC simulation and Binomial
Tree
6) Recommendation for the Strike Prices.
7) Proposal of another Business Opportunity
8) Conclusion
9) Appendix
10) Tables and Figures
11)Reference
Sections 3) and 4) in your white paper correspond to task a); section 5) corresponds to tasks b) and
c); section 6) corresponds to task d); section 7) corresponds to task e). All the tables and figures
should be in section 10).
Reference:
Brandimarte, P. (2006). Numerical Methods in Finance and Economics, a Matlab-based
Introduction, 2nd edition. John Willey and Sons: New Jersey.
Geske, Robert. “The valuation of corporate liabilities as compound options.” Journal of Financial
and Quantitative Analysis 12, no. 04 (1977): 541-552.
Geske, Robert. “The valuation of compound options.” Journal of Financial Economics 7, no. 1
(1979): 63-81.
Haug, E.G., 2007. The complete guide to option pricing formulas. McGraw-Hill Companies.
Hull, J. (2015). Options, Futures and Other Derivatives, 9th edition. Prentice Hall: New Jersey.
Kang, Sang Baum and Hong Luo. “Heterogeneity in beliefs and expensive index options (March
13, 2015).” Available at SSRN: http://ssrn.com/abstract=2578167 or
http://dx.doi.org/10.2139/ssrn.2578167.
Kang, Sang Baum and Pascal Létourneau. “Investors’ reaction to the government credibility
problem: A real option analysis of emission permit policy risk.” Energy Economics 54 (2016):
906-107.
Rubinstein, Mark. “One for another.” Risk 4, no. 7 (1991): 30-32.
Selby, M.J. and Hodges, S.D., 1987. On the evaluation of compound options. Management Science,
33(3), pp.347-355.

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