I used Black-Scholes and the approximation side by side to compare straddle prices as for ascending levels of volatility.
Table snippet:
Graphically:
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3 Observations
The approximation works well at “reasonable levels” of volatility.
For a one-year option at 50% volatility, the overestimate is still only 1.3%
At 100% vol the overestimate is still under 5%
The approximation always overestimates, again this pic:
Straddle is capped at 200% of S no matter how high volatility goes
The B-S straddle asymptotically approaches $200
The approximation grows linearly and unboundedly as volatility increases
Why?
It’s an old standby — arbitrage bounds:
The maximum value of a call is the stock price itself. If the call traded for more than the stock you could buy the stock and sell the call in a 1-to-1 ratio and make money in 100% of scenarios.
The put is bounded by the strike price. In this case $100.
Both the call and the put independently have maximum values of $100. And they are always worth the same at-the-forward anyway.
If you raise volatility or time to infinity options go to their maximum arbitrage boundary.
Time and volatility work the same way.
For a given level of volatility, increasing the time increases the overestimate.
The straddle approximation will work well for short-dated options even at high volatility!
This is a 1-week option…even at 300% volatility, the error is only 1%
