Question
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
We developed formula on how to calculate price of PUT or CALL under uniform distribution.
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)
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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.
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