Continuous Time Finance Session 5: Final Project
Plan for the remaining weeks
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The second half of the course is much more practical than the first half, so I do not need another exam as an extra motivation for you to learn the material.
This week: final project.
Week 6: more on derivatives.
• Derivatives sales and trading in practice. • Extensions for the Black-Scholes model.
Homework 2 is mostly based on materials in weeks 4 and 6 and due before class in week 7.
Week 7: application of option theory in corporate finance and real options.
Week 8: project presentation.
Course project
One submission per group. Project report and presentation slides in electronic form due before class in week 8. Project report should be around 4 to 6 pages long with 12pt font size and 1.5 spacing. You can use additional pages for figures and tables.
Project presentation: 12 minutes per group plus 3 minutes for Q&A.
During the presentation, other students and I will challenge you to see if you understand
the intuitions behind your calculation.
A helpful document on preparing presentation and reports and other logistic issues is uploaded on Blackboard.
Project ideas
I will suggest some project ideas. Do NOT view the final project as a homework that you have to follow precisely. My suggestions are NOT detailed step-by-step instructions. Instead, view my suggestions as a blueprint for your own project. Feel free to skip steps that you think are unnecessary and add steps that you think are important.
You do not have to use my suggested projects. In fact, I encourage you to come up with your own projects, but please talk with me first.
If you face obstacles (i.e. do not know how to proceed to next step, what tools to use, what data to look at, where to find data, or how to interpret some results), feel free to ask me. The best way to think about the course project is to think of me as a derivatives trader, and you as an intern on the trading desk.
Project idea 1: Hedging in practice
Current stock price is S.
You hold 1 share of call option and delta
hedge it by shorting ∆ shares of stock.
Suppose stock price changes from S to S′.
You make C′′ − C dollars from call option.
You lose (S′ − S)∆ = C′ − C dollars by shorting ∆ shares of stock.
On net, you make
(C′′ −C)−(C′ −C)=C′′ −C′ dollars.
Project idea 1: Hedging in practice
Suppose the stock price follows Geometric Brownian Motion dSt = μStdt + σStdBt,
and you buy a call option with maturity T , say 60 days.
Simulate X (say 10,000) paths for St.
Compute your terminal portfolio payoff for each path of St under these three scenarios, plot the distribution and compute the mean and variance of terminal portfolio, and assess whether your result makes sense.
• No hedge.
• Delta hedge with the stock with daily rebalancing.
• Delta and Gamma hedge with stock and another option with daily rebalancing.
Project idea 1: Hedging in practice
You want to use the Greek formula in week 4, so that you do the Delta and Gamma hedging correctly.
The key challenge is that because St changes everyday, the Greeks and thus the hedging portfolio (which consists of stocks and another option) will also change everyday. Be extra careful when accounting for the daily P&L.
See if the results you get make sense. Then change to different parameters and specifications and compare across cases. The goal is to build intuitions for hedging.
Project idea 2: Exotic options
Find one different form of exotic options, except for Asian options, which we will discuss in week 6 class.
This can include barrier options, rainbow options, gap options, compound options, lookback options, exchange options, and so on.
For your convenience, do not use exotic options with early exercise features, such as American options, Bermudan options, and shout options.
First, clearly describe the payoff structure and intuitively explain the sign of their Greeks (Delta, Gamma, Theta, Vega). In particular, think about whether those Greeks may change sign for different parameters.
Then use simulation to calculate the option price and Greeks.
Think of situations in which these exotic options can be helpful in real life.
Project idea 3: Identifying embedded optionality in real life
This project asks you to find optionality in real life financial markets.
• Clearly describe the real life setting.
• Analyze the embedded optionality and clearly define option structure and payoffs.
• Use intuitions and numerical methods to analyze the Greeks of the options. (You decide on
the reasonable parameter ranges based on the readings of real life setting.)
• Important: analyze the implication of Greeks to the real-world problem. In other words,
what does the sign of Greeks tell us about people’s behaviors in these situations?
Examples you can use include
• CEO compensation. (An example you cannot use.) • Market order versus limit order.
• Callable or putable bonds.
• IPO greenshoe.
• Mortgage prepayment.
• Hedge fund management fee.
• And many more…
Do not use examples about firm’s debt and equity choices, which we will cover in week 7. 9
2012 2013 2014 2015 2016 2017 2018 2019
Project idea 4: Predicting volatility
45 Implied volatility 40
50 3500 Implied volatility
Volatility (%)
Volatility (%)
2012 2013 2014 2015 2016 2017 2018 2019
Historical volatility
45 S&P 500 40
25 20 15 10
Project idea 4: Predicting volatility
Download longer period of data for S&P 500 indexes (VIX and prices) from Yahoo Finance, and calculate historical volatility using different rolling window length (say 6-month, 3-month, 1-month, or 2-week).
Are the historical volatilities estimated from different window length different from each other? If so, how and why?
Now, use past information on VIX, historical volatility from different rolling window lengths, and past stock returns to see if you can predict the future movement of VIX.
• Use tools that you learn from econometrics and empirical finance classes (regressions, time-series models, ARMA models…)
See if the regression coefficients make sense, and whether you can come up with stories to explain the sign of the coefficients.
• For example, does high stock return in the past predict high or low VIX in the future?
Project idea 4: Predicting volatility
Try to compare across various models or introduce new data sets, in order to come up with the best predictive model for VIX.
• For example, one measure of goodness-of-fit could be adjusted R2. • See if your model can outperform a simple ARMA model.
See if you design a profitable VIX trading strategy based on the predictive model, and calculate Sharpe ratio. (In practice, VIX is not tradable.)
• In each year y, assume you start with $1 dollar and follow your trading strategy.
• If you have x dollars at time t, then the size of long or short position on VIX cannot exceed x.
• If you short VIX, remember sometimes you can go bust. In this case, assume your portfolio is
worth $0.01 for that year.
• At the end of year y, suppose your portfolio is worth Ry dollars.
• Your year y log return is ry = ln(Ry).
• The Sharpe ratio is mean(ry)/std(ry), assuming a zero riskfree rate for simplicity.
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