In the last class, we developed formula on how to calculate price of PUT or CALL under uniform distribution.

Mean absolute deviation of a random variable is the expected value of the distance from the mean:

MAD = E[ S_T - E(S_T) ]

The mean of S_T or E(S_T) is (U+L) / 2.

The max distance from the mean is 1/2 of the full range, ie 1/2 * (U-L).

The min distance from the mean is 0 when the random variable falls right at the mean.

The mean distance from the mean, ie the MAD, is 1/2 of the max distance, or 1/4 of the full range for a uniform distribution:

MAD = (U-L) / 4.

If underlying price is S0 (assuming rf=0%), and has a mean absolute deviation (MAD) of M, then the range of underlying distribution at expiration is [L U], where

L is the lower bound L = S0 - 2 *M, and U is the upper bound U = S0 + 2*M.

If X is the strike, then we can derive a price for the PUT or CALL as a function of S0 and M.

A PUT has:

i. probability of expiring ITM of (X-L) / (U-L) = (X-S0+2*M) / (4*M)

ii avg Option PMT of (X-L) / 2 = (X-S0+2*M)/2

iii. PUT price = (X-S0+2*M)^2 / (8*M)

A CALL has:

i. probability of expiring ITM of (U-X) / (U-L) = (S0-X+2*M) / (4*M)

ii avg Option PMT of (U-X) / 2 = (S0-X+2*M)/2

iii. CALL price = (S0-X+2*M)^2 / (8*M)

**********************************************************************************************

Q1. Option Price vs Strike Price

Keep S0=100 and MAD = 25

Create a spreadsheet to:

Q1a. calculate PUT price, PUT intrinsic vs time value, while varying X from 50 to 150 with $1 increment.

Q1b. calculate CALL price, CALL intrinsic vs time value, while varying X from 50 to 150 with $1 increment.

Q1c. Graph PUT price and intrinsic value from Q1a. x-axis is X.

Q1d. Graph CALL price and intrinsic value from Q1b. x-axis is X.

Q2. How option will react to underlying price movement.

Keep X=100 and MAD = 25.

Create a spreadsheet to:

Q2a. calculate PUT price, PUT intrinsic vs time value, while varying S0 from 50 to 150 with $1 increment.

Q2b. calculate CALL price, CALL intrinsic vs time value, while varying S0 from 50 to 150 with $1 increment.

Q2c. Graph PUT price and intrinsic value from Q2a in the same chart. x-axis is S0.

Q2d. Graph CALL price and intrinsic value from Q2b in the same chart. x-axis is S0.

Q3. How option will react to change in the RANGE of underlying price movement

Keep X=100

Create a spreadsheet to:

Q3a. Assume MAD=10. Calculate PUT price, while varying S0 from 80 to 120 with $1 increment.

Q3b. Assume MAD=20. Calculate PUT price, while varying S0 from 80 to 120 with $1 increment.

Q3c. Assume MAD=40. Calculate PUT price, while varying S0 from 80 to 120 with $1 increment.

Q3d. Graph all three PUT prices AND the intrinsic put value under different MAD assumption from Q3a-c in the same chart. x-axis is S0.

(note: intrinsic value is the same for all three MAD scenarios)

Q4. Redo Q3 for the case of CALL

Keep X=100

Create a spreadsheet to:

Q4a. Assume MAD=10. Calculate CALL price, while varying S0 from 80 to 120 with $1 increment.

Q4b. Assume MAD=20. Calculate CALL price, while varying S0 from 80 to 120 with $1 increment.

Q4c. Assume MAD=40. Calculate CALL price, while varying S0 from 80 to 120 with $1 increment.

Q4d. Graph all three CALL prices AND the intrinsic call values under different MAD assumption from Q4a-c in the same chart. x-axis is S0.

(note: intrinsic value is the same for all three MAD scenarios)

**********************************************************************************************