<aside> 📌 Pinned for reference

B-S Formula For Non-Div Paying European Exercise

ATF Straddle Approximation

where:

S = forward price

σ = annualized volatility

t = fraction of a year until expiry

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Assume:

• 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:

Graphically:

<aside> 🪟 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:

   

2. 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.

3. 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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