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# Testing the straddle approximation

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Pinned for reference

B-S Formula For Non-Div Paying European Exercise

where:
S = forward price
σ = annualized volatility
t = fraction of a year until expiry

#### 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:

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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:

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%