Term Project: Pricing and Application of a Chooser Option

MSF 524, IIT Stuart School of Business

Fall 2017

Professor Sang Baum βSolomonβ 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 β 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 β 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

dπ(π‘) = ππ(π‘)ππ‘ + ππ(π‘)ππ§(π‘)

where constant r is the annualized continuously compounded risk-free interest rate, π is volatility,

and π§(π‘) is a Brownian motion.

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A βchooser optionβ is an exotic compound option. An originator in your desk is trying to offer a

βchooserβ option to your client. (For the definition of βchooserβ option, see Chapter 4.5.4 of

Brandimarte (2006).) The structure of the βchooser optionβ is as follows:

ο· At time 0, a client pays the option premium.

ο· At time π1, the client chooses either the call option at strike ππ or the put option at strike

ππ.

ο· At time π2 , the client who chose the call (put) option at time π1 has a right but not

obligation to buy (sell) the underlying asset at strike ππ

(ππ).

Hence, the payoff of the βchooser optionβ at time π2 is

max(π(π2

) β ππ

, 0) πΌ{πβπ ππππππ‘ πβππ π π‘βπ ππππ ππ‘ π‘πππ π1}

+ max(ππ β π(π2

), 0) πΌ{πβπ ππππππ‘ πβππ π π‘βπ ππ’π‘ ππ‘ π‘πππ π1}

where π(π2

) is the underlying asset price at time π2, and πΌ{β}

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:

ο· The client was initially interested in a straddle to βbuyβ volatility. (For the definition of

straddle, see Chapter 11.4 of Hull (2012).)

ο· Because the client believes the straddle is too expensive, the client is interested in

purchasing the βchooserβ 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:

ο· The initial underlying price π(0) = $50;

ο· Strike price of underlying options ππ = ππ = $50;

ο· Risk-free interest rate r=0.025;

ο· Expiration of the chooser option π1=1 year from now;

ο· Expiration of the underlying options π2=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:

dπ(π‘) = ππ

(π‘)π(π‘)ππ‘ + ππ

(π‘)π(π‘)ππ§(π‘)

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where ππ

(π‘) is client πβs subjective belief on the draft, ππ

(π‘) is client πβs subjective belief on

volatility, and π§(π‘) 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 (ππ and ππ) 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 β 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 Sharpeβs ratio, and the 95% value at

risk.

For this project use π(0) = $50, r=0.025, π1 = 1 year, and π2 = 2 years. Again, ππ

(π‘)s and

ππ

(π‘)s are client-specific where 0 β€ π‘ β€ π2. The clientsβ ππ

(π‘)s, and ππ

(π‘) are given as follows:

Client

ID

Client

Name

When 0 β€ π‘ < π1 When π1 < π‘ β€ π2

1 Mrs.

Smith

ππ

(π‘) = π + 0.03 & ππ

(π‘) =0.15 ππ

(π‘) = π + 0.005 & ππ

(π‘) =0.30

2 Mr.

Johnson

ππ

(π‘) = π β 0.03 & ππ

(π‘) =0.20 ππ

(π‘) = π β 0.01 & ππ

(π‘) =0.18

3 Ms.

Williams

ππ

(π‘) = π β 0.03 & ππ

(π‘) =0.18 ππ

(π‘) = π + 0.03 & ππ

(π‘) =0.12

4 Mr.

Jones

ππ

(π‘) = π + 0.02 & ππ

(π‘) =0.35 ππ

(π‘) = π + 0.02 & ππ

(π‘) =0.10

5 Miss

Brown

ππ

(π‘) = π + 0.03 & ππ

(π‘) =0.15 ππ

(π‘) = π β 0.05 & ππ

(π‘) =0.15

With these being said, please, do the following tasks:

a) Because the close-form solutions, if any, cannot incorporate the time-varying ππ

(π‘), you need

to use numerical methods to use those agentsβ time-varying ππ

(π‘): Programing Matlab, please,

price out the chooser option in the following three methods we learn in this semester:

β . Simple Monte-Carlo simulation: For the simple Monte-Carlo simulation, control the

relative error within plus/minus 0.1 percent.

β‘. A βsmart-latticeβ version of CRR binomial tree: For the binomial tree, set the

subinterval to be one calendar day ( 1

365

year).

β’. 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 βsize parameters.β 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 (ππ and ππ) 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 clientβs ππ

(π‘)s, and ππ

(π‘). There is not a single

way of risk-return analysis using Monte-Carlo simulation β 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 Sharpeβs 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 βbuying a straddleβ. If you want, you may compare your

case for each client with the case using the βaverageβ 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) β 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. βHeterogeneity in beliefs and expensive index options (March

13, 2015).β Available at SSRN: http://ssrn.com/abstract=2578167 or

http://dx.doi.org/10.2139/ssrn.2578167.

Kang, Sang Baum and Pascal LΓ©tourneau. βInvestorsβ reaction to the government credibility

problem: A real option analysis of emission permit policy risk.β 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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