Testing the straddle approximation

Pinned for reference
B-S Formula For Non-Div Paying European Exercise
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ATF Straddle Approximation
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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.
Table snippet:
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3 Observations
  1. 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:
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  1. 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.
  1. 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%
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