In this project, assume the continuously compounded risk-free rate is equal to 0.1%.

Question 1:

- Pick a US public company of your choice that has options traded on it o You can rely on yahoo finance to check whether your company has options traded on it. o Motivate the choice of company; why have you picked this company.

Question 2: - Pick a day on which you will download the data needed for this project (stock historical prices as well as the options data) o You need to report this date o You need to obtain the closing stock price on this day

- Obtain the last year daily stock prices (like I have showed you in the session).

- Estimate the daily returns

- Estimate the standard deviation of the returns

Question 3

- Extract option data for your stock from yahoo finance

- In this project, we are interested in options with maturity closest to one month.

o Pick an option maturity that is closest to one month (like I have showed you in the session).

o Estimate the exact maturity of your options

Question 4

- Select the nearest ATM call option in your list of downloaded option.

o What is the option strike price? Compare this strike price to closing stock price you have previously obtained in question 2?

o Relying on the historical volatility you have estimated in question 2, the closing stock price, the strike price, and the given risk-free rate, what is the price of the call option using the Black-Scholes-Merton formula?

o Is the call option price estimated above using the historical volatility equal to the actual call price?

If not, explain in your own words why the calculated price is different from the actual price?

o Relying on the corresponding implied volatility you have downloaded for this option, the closing stock price, the strike price, and the given risk-free rate, what is the price of the call option using the Black-Scholes-Merton formula?

o Is the call option price you have estimated above using the implied volatility closer to the actual call price that the one estimated using historical volatility?

Can you explain why?

Question 5:

- Select one deep OTM call option in your list of downloaded option. o What is the option strike price? Compare this strike price to closing stock price you have previously obtained in question 2?

o Relying on the historical volatility you have estimated in question 2, the closing stock price, the strike price, and the given risk-free rate, what is the price of the call option using the Black-Scholes-Merton formula?

o Relying on the corresponding implied volatility you have downloaded for this option, the closing stock price, the strike price, and the given risk-free rate, what is the price of the call option using the Black-Scholes-Merton formula?

o Compare the deep OTM call option price estimated relying on historical volatility to the deep OTM call option price estimated relying on the corresponding implied volatility. Which one is closest to the actual price? What do you conclude?

Question 6:

- Select one deep ITM call option in your list of downloaded option.

o What is the option strike price? Compare this strike price to closing stock price you have previously obtained in question 2?

o Relying on the historical volatility you have estimated in question 2, the closing stock price, the strike price, and the given risk-free rate, what is the price of the call option using the Black-Scholes-Merton formula?

o Relying on the corresponding implied volatility you have downloaded for this option, the closing stock price, the strike price, and the given risk-free rate, what is the price of the call option using the Black-Scholes-Merton formula?

o Compare the deep ITM call option price estimated relying on historical volatility to the deep ITM call option price estimated relying on the corresponding implied volatility. Which one is closest to the actual price? What do you conclude?

Question 7 (1 pts):

Compare the ITM, ATM, and OTM option prices obtained using implied volatilities? What do you observe? If you notice a trend, can you explain why such a trend exists?

Question 8:

- For the downloaded list of call options data, draw the set of implied volatility as a function of their strike price

- What do you observe, explain?

- For the put options data you have downloaded, draw the set of implied volatility as a function of their strike price

- What do you observe?

- Are the call and put graphs similar?

Question 9 : - Estimate the total Calls Open Interest.

- Estimate the total Puts Open Interest.

- Estimate the total OTM Calls Open Interest.

- Estimate the total OTM Puts Open Interest.

4 - What can you conclude from comparing the OTM Calls and OTM Puts Open Interest?

Question 10

- Relying on the put-call parity relationship, and the nearest ATM call option actual price (that you have downloaded), estimate the corresponding put price (1 pts) - Is the put price equal to the nearest ATM put option price you have downloaded?

- What are the possible factors that might drive the put price estimated using the put-call parity from the actual corresponding put price?

Question 11:

- Relying on the downloaded option data, estimate the below Greeks for your ATM call options and explain what does each of them mean (zero grade will be given if no explanation is provided)? (Hint: you can rely on the excel sheet I provided and build on it in your estimations)

o Delta

o Gamma

o Theta

o Vega

o Rho

Question 12

- Suppose you have written 10 ATM call option on the company you have picked. What shall you do to hedge your position (i.e.: make you position delta-neutral)?

- If the stock price suddenly dropped by 2%, what shall you do to keep the position deltaneutral?

Assume, the drop happens on the same day you have obtained your data and everything else remains the same.

- If the stock price suddenly jumped by 2%, what shall you do to keep the position deltaneutral? Assume, the jump happens on the same day you have obtained your data and everything else remains the same.

Question 13:

5 - Relying on the Implied Volatility of the closest ATM call option you have downloaded, estimate a 1-step binomial tree to price of the corresponding call option? (Stock price, strike price, RF, and maturity are as per the ATM option you have downloaded).

- Relying on the Implied Volatility of the closest ATM call option you have downloaded, estimate a 2-step binomial tree to price of the corresponding call option? (Stock price, strike price, RF, and maturity are as per the ATM option you have downloaded).

- As you increase the number of steps, what do you observe?