Similarity, for a call it is as follows:
S = spot price,
K = strike price,
r = risk-free rate,
T = time to maturity and
sigma is implied volatility.
I want to know what the third and fourth raw moments of a straddle are. A straddle consist of a call and a put
K at maturity. then the call option will have a value of
K, and the put will have no value.
K, the call option will have no value, and the put will be worth
K. This can be written as:
As a result the expected final value is equal to:
This can also be written as:
Which can be simplified to:
Following this logic for the other moments I get:
According to the theory about cumulants if two variables are independent, the
n-th-order cumulant of their sum is equal to the sum of their
n-th-order cumulants. Inspecting the final raw moments of the straddle it looks like this applies. However, a call and a put are not independent. When the value of a call increases/decreases, the value of a put decreases/increases, so the two option types are negatively correlated. This "fact" and the final results make me feel like I used the wrong assumptions.
Question: Are the the defined raw moments for a straddle correct or am I missing something?