Implied Volatility Near a Put Price Boundary

Summary

The document poses an asymptotic option-pricing question. With the underlying price, interest rate, and time to expiry normalized, it defines a put price as the larger of a linear function of strike above a threshold and the put’s intrinsic value. Below or at the threshold, the setup asserts that the put has no extrinsic value; just above it, the implied volatility tends to zero.

The central objective is to find the higher-order behavior of implied volatility as strike approaches that threshold from above. No solution, derivation, numerical evidence, or asymptotic coefficient is supplied, so the document is a problem statement rather than a complete method. Any answer would need to invert the put pricing relation near the boundary and specify which pricing assumptions apply; those details are not developed here.

Key ideas

• The document specifies a put price as the maximum of a linear strike-dependent term and intrinsic value.
• It assumes implied volatility tends to zero as strike approaches a threshold from above.
• The open question is the higher-order asymptotic behavior of implied volatility near that threshold.
• No derivation or evidence is provided, so the asymptotic result remains unresolved in the document.

Tags

# Asymptotic behavior of implied volatility at probability mass


# Asymptotic behavior of implied volatility at probability mass












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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.