Pinned for reference
B-S Formula For Non-Div Paying European Exercise
ATF Straddle Approximation
S = forward price
σ = annualized volatility
t = fraction of a year until expiry
- S = $100
- r = 0%
- t = 1 year
- Forward Price = Seʳᵗ = $100
- ATF strike (K) = $100
I used Black-Scholes and the approximation side by side to compare straddle prices as for ascending levels of volatility.
- 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%