Questions:
Q1
a) Find the 3 month US treasury rate and the closing price of the S&P 500 for the day your option is traded.
b) Using the implied volatility value from the “XYZ Company”file for your option as the volatility input to the Black-Scholes model (including a dividend yield), price your option and compare to the mid-point of the “XYZ Company” bid/ask spread. Assume dividend yield is zero for this part.
c) Can you find a dividend yield (using e.g. excel solver/fmincon in Matlab) for which the BS-M formula price matches the midpoint of the “XYZ Company” Bid/Ask spread exactly?
Q2
Use the data obtained in question 1 (interest rate, dividend yield, implied volatility, along with the option strike and time to expiry) to produce graphs to show:
a) The sensitivity of the Black-Scholes option price to changes in σ (from σ =5% to σ =100% in steps of 5%).
b) The sensitivity of the Black-Scholes option price to changes in the time to maturity T (T= 1 week, 1 month, 1 quarter, 6 months, 1 year, 5 years). Briefly discuss your results in relation to option theory.
c) The sensitivity of the Black-Scholes option price to changes in the interest rate r. (r in the range 0% to 14%, in steps of 0.25%) Discuss your results in relation to option theory.
Briefly discuss your results in relation to option theory and the Black Scholes formula. (To produce the graphs hold all other inputs to the option price formula constant - except for the parameter you are testing.)
Q3
Using the option parameters from Q1, produce a graph comparing a call’s intrinsic value [defined as max(S-X,0)] and its Black-Scholes price at each point for a range of possible index values.
i.e. you are changing S in the formula above across a range, X is the strike from your original option in Q1., and for each S in your range also compute the B-S price using that S as the spot and all other inputs to B-S from your original option in Q1. You then plot the function above and the B-S price varying across S. For example - vary S in the range -60% to + 60% of the index spot on your trade day in 5% steps.
Discuss your results in relation to option theory and whether you would ever exercise your option (or sell it to someone instead for the Black Scholes price).
Likewise, using an at the money put from your date in the options data file produce a graph comparing a put’s intrinsic value [defined as max(X-S,0)] and its Black-Scholes price. Again, discuss your results in relation to option theory and whether you would ever exercise your option (or sell it to someone instead for the Black Scholes price).
Q4
Using a large sample of S&P500 returns up to your option trade date to fit a volatility model and use this model to forecast volatility over the lifetime of the option (the one month from the trade date to the expiry date). Code is provided in Matlab to do this for a GARCH(1,1) model.
In a Table report:
1.your annualised forecast volatility,
2.the annualised realised volatility of the index over the time to expiry of your option (volatility of index returns from the trade date to expiry);.
3.The VIX value on the trade date.
4.The implied volatility of your option from “XYZ Company”.
Compare your volatility forecast to the option implied volatility, based on this comparison or any other rationale that a trader on that date may have applied (rationale to be included in the report):
1.construct an option spread to trade volatility using any subset of all of the options from the full traded set on your date.
2.Plot the payoff function of your option spread
3.In a small Table: Report the cost of the portfolio, the value on expiry of the index and the resulting P&L (using your payoff function at that expiry value).
4.Comment on your results.
Q5
Delta hedge an ATM call option that you assume you are selling on your trade date. You are a market maker and receive the Ask price:
1.Calculate a delta value on each day to expiry from the Black Scholes formula (assuming all parameters including implied volatility stay the same while updating the spot price of the index based on the value at close each day & decrementing the time to expiry of the option by a day each time).
2.Use these values to delta hedge your position (you sold the call option).
3.Assuming transaction costs of 10bps, report the overall P&L from your delta hedged call position aggregated over the daily purchases/sales of the index and your final liability less the final offsetting index position that you sell/.
4.Matlab code is provided for this step.
5.Plot your daily delta value, and the p&l from the index holding.
Repeat the process using your volatility forecast from Q4 instead of the option implied volatility and comment on any differences.
