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.
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) l{the client chose the call at time t1} + max(XP - S(t2), 0) l{the client chose the call at time t1}
where S(t2) is the underlying asset price at time t2, and l{·} 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)
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.
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 risk- neutral 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
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