For sake of simplicity, let us suppose that interest rate is zero, stock price is 1, and time to expiry is 1. I am interested in implied volatility that gives the following put price. $$P(k, \sigma(k))= max(c(k-k_0), intrinsic(k))$$ Where $0 < c<1$ and $k$ is the strike, $intrinsic(k) = \max(k - 1, 0)$. Since there is zero extrinsic value for puts with strike $k \leq k_0$, I know that $$lim_{k\to k_0^+} \sigma(k) = 0$$ However, I would like to know the higher order term for $\sigma(k)$ around $k=k_0$.
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$\begingroup$ Is $k$ in your notation strike or time to maturity? $\endgroup$– FridoCommented Jun 28, 2023 at 9:50
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1$\begingroup$ I mean what kind of implied vol goes to zero as strike approaches ATM? So I am assuming $k$ is a weird notation for time to maturity, but then your question still does not make much sense to me. $\endgroup$– FridoCommented Jun 28, 2023 at 10:23
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2$\begingroup$ I agree with Frido, you need to clarify this a bit $\endgroup$– KT8Commented Jun 28, 2023 at 10:24
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