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In the standard Black Scholes model, as we take volatility to infinity, the price of call spreads goes to zero and the price of put spreads goes to the difference in strikes.

I ran a simulation using the Bachelier model (underlying normal distribution instead of lognormal) to try and approximate the prices of these spreads. My results seem to indicate that in the limit of infinite vol, both the call and put spreads in the Bachelier model approach 1/2 the distance between strikes for their prices.

Is this true? If so, what is the explanation/intuition? I have some vague intuition about why this should be the case (due to symmetric underlying distribution) but would like input/verification.

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