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)

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Q4. How option will react to change in the RANGE of underlying price movement?

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)