Why Flat Option Prices Can Produce Nearly Linear Implied Volatility

Summary

The document examines a fixed-expiry region where European options at different strikes have the same quoted price, as can occur in the wings. It asks why Black–Scholes implied volatility appears nearly linear in log-moneyness and whether the corresponding implied variance must be convex. The stated example uses equally priced puts with a fixed forward, zero rates, no dividends, and a one-year expiry.

The response offers a heuristic for the volatility pattern: as a put strike moves farther out of the money, volatility must rise to preserve its price. Approximately preserving the probability of finishing in the money suggests keeping the strike at a constant number of standard deviations from the forward, which yields a linear relation between log-moneyness and volatility. This is intuition rather than a proof; the document does not establish convexity in general or provide conditions under which it holds.

Key ideas

• Equal option prices across strikes can require higher implied volatility at more distant out-of-the-money strikes.
• Keeping the approximate in-the-money probability constant motivates a constant-standard-deviation heuristic.
• That heuristic leads to an approximately linear relationship between log-moneyness and implied volatility.
• The response does not prove that implied variance is convex in general.

Tags

# Convexity of variance implied from quotes with same price


# Convexity of variance implied from quotes with same price












Suppose that for a given expiry of European options, every strike in a region has the same quoted price (in practice this might be true in the wings, for example).

The Black-Scholes implied volatility from these quotes seems to be nearly linear, and the variance convex, with log-moneyness ($\log(K/F)$).

- Why is the implied volatility close to linear in log-moneyness?

- Is it always true that this implied variance will be convex? If so, is there a proof?

(In the above plot, the vols and variances are implied from a price of 0.01 for puts, with forward 100, risk-free rate of 0, no dividends, expiration time of 1.)

## Answer by dm63 (score 2)

https://quant.stackexchange.com/a/84081

To keep the put price the same as the strike gets further OTM, the implied volatility obviously needs to increase, but by how much. Heuristically, to keep the same option price, the probability of being in the money needs to stay constant during this process. This is achieved if the strike is a constant number of standard deviations out of the money. Thus we have $$ ln(F/K) =C\sigma$$, a linear relationship. Not an exact proof but hope it helps

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