Knowing that implied volatility represents an annualized +/-1 Standard Deviation range of the stock price, why does the price of an ATM straddle differ from this? Also for simplicity, no rates, no dividends, and the returns of the stock are normally distributed.
Spot = 100 Strike = 100 DTE = 1 year IV = 20% Rates = 0 Dividend = 0
The call price and put price both come out to be 7.97. Meaning the straddle costs: $15.94
To convert the expected 1 Std Dev range of the stock from IV to the 1 Std Dev dollar amount the stock is expected to move:
Implied Vol * √(DTE/252) * Stock Price
In our case, we don't need to do this since our straddle expires in a year (252 days) and IV already represents the annualized Std Dev range of the stock. So it's simply a +/- $20 range. Meaning 68.2% of the time, the stock is expected to stay in the range of ≥80 or 120≤
Then why does the straddle cost \$15.94 instead of \$20.00?
To add to the question, a common formula I've seen traders use to get the price of a straddle if they already know the σ (IV) term is:
Straddle Price = 0.8 * Implied Vol * √(DTE/252) * Stock Price
And if the straddle price is already known then the reverse formula to get the IV is:
Implied Volatility = 1.25 * (Straddle Price/Stock Price) * √(DTE/252) * Stock Price
To summarize my questions are:
Why is the Straddle dollar price different (less) than IV's 1 Standard Deviation dollar range? Does this mean straddles are underpriced because it should cost \$20 but actually costs \$15.94?
In the first formula, why is implied volatility being multiplied by 0.8? If you removed this, then the price of a straddle would be the exact dollar amount of the expected +/- 1 Std Dev range of the stock, which would make sense.
And in the second formula, why are we multiplying by 1.25?
This does not make sense to me as I ought to think an "exact" ATM straddle should cost the expected 1 Std Dev in dollars ("expected move").