Date	Close
10/12/21	4350.65
10/11/21	4361.19
10/8/21	4391.34
10/7/21	4399.76
10/6/21	4363.55
10/5/21	4345.72
10/4/21	4300.46
10/1/21	4357.04
9/30/21	4307.54
9/29/21	4359.46
9/28/21	4352.63
9/27/21	4443.11
9/24/21	4455.48
9/23/21	4448.98
9/22/21	4395.64
9/21/21	4354.19
9/20/21	4357.73
9/17/21	4432.99
9/16/21	4473.75
9/15/21	4480.7
9/14/21	4443.05
9/13/21	4468.73
9/10/21	4458.58
9/9/21	4493.28
9/8/21	4514.07
9/7/21	4520.03
9/3/21	4535.43
9/2/21	4536.95
9/1/21	4524.09
8/31/21	4522.68
8/30/21	4528.79
8/27/21	4509.37
8/26/21	4470
8/25/21	4496.19
8/24/21	4486.23
8/23/21	4479.53
8/20/21	4441.67
8/19/21	4405.8
8/18/21	4400.27
8/17/21	4448.08
8/16/21	4479.71
8/13/21	4468
8/12/21	4460.83
8/11/21	4447.7
8/10/21	4436.75
8/9/21	4432.35
8/6/21	4436.52
8/5/21	4429.1
8/4/21	4402.66
8/3/21	4423.15
8/2/21	4387.16
7/30/21	4395.26
7/29/21	4419.15
7/28/21	4400.64
7/27/21	4401.46
7/26/21	4422.3
7/23/21	4411.79
7/22/21	4367.48
7/21/21	4358.69
7/20/21	4323.06
7/19/21	4258.49
7/16/21	4327.16
7/15/21	4360.03

1. Use the above of S&P prices for 10/12/21-7/15/21 to estimate the daily volatility using both approaches in Section 10.5 (formulas (10.2) and (10.4)). What are the annualized values of these volatilities?

Using this same Excel data, you are going to calibrate the Power law model in (10.1) as follows:

2. Absolute value the returns, then express each as a multiple of the standard deviation from (10.4).

3. Order these multiples from smallest to largest, and then for each, calculate the probability that the multiples are bigger than that value (just count the number bigger, divided by total number).

4. Now take the log of these multiples (lnx in Hull) and the log of these probabilities (ln(Pr>x) in Hull).

5. Produce a graph like Figure 10.3 (only use data with lnx>0).

6. Use Excels regression (in Data, data analysis) and determine K and alpha (again only use data with lnx>0).