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 d ( ) = ( ) + ( ) ( ) where constant r is the annualized continuously compounded risk-free interest rate, is volatility, and ( ) is a Brownian motion. 2 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 1, the client chooses either the call option at strike or the put option at strike .  At time 2 , the client who chose the call (put) option at time 1 has a right but not obligation to buy (sell) the underlying asset at strike ( ). Hence, the payoff of the “chooser option” at time 2 is max( ( 2 ) − , 0) { ℎ ℎ ℎ 1} + max( − ( 2 ), 0) { ℎ ℎ ℎ 1} where ( 2 ) is the underlying asset price at time 2, and {∙} 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 (0) = $50;  Strike price of underlying options = = $50;  Risk-free interest rate r=0.025;  Expiration of the chooser option 1=1 year from now;  Expiration of the underlying options 2=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: d ( ) = ( ) ( ) + ( ) ( ) ( ) 3 where ( ) is client ’s subjective belief on the draft, ( ) is client ’s subjective belief on volatility, and ( ) 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 ( and ) 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 (0) = $50, r=0.025, 1 = 1 year, and 2 = 2 years. Again, ( )s and ( )s are client-specific where 0 ≤ ≤ 2. The clients’ ( )s, and ( ) are given as follows: Client ID Client Name When 0 ≤ < 1 When 1 < ≤ 2 1 Mrs. Smith ( ) = + 0.03 & ( ) =0.15 ( ) = + 0.005 & ( ) =0.30 2 Mr. Johnson ( ) = − 0.03 & ( ) =0.20 ( ) = − 0.01 & ( ) =0.18 3 Ms. Williams ( ) = − 0.03 & ( ) =0.18 ( ) = + 0.03 & ( ) =0.12 4 Mr. Jones ( ) = + 0.02 & ( ) =0.35 ( ) = + 0.02 & ( ) =0.10 5 Miss Brown ( ) = + 0.03 & ( ) =0.15 ( ) = − 0.05 & ( ) =0.15 With these being said, please, do the following tasks: a) Because the close-form solutions, if any, cannot incorporate the time-varying ( ), you need to use numerical methods to use those agents’ time-varying ( ): 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. 4 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 ( and ) 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 ( )s, and ( ). 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 5 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.