Recovering Black-Scholes Implied Volatility from a Characteristic Function

Summary

The document asks how to obtain Black-Scholes implied volatility from a model’s characteristic function using an integral relation attributed to Gatheral’s treatment of Lewis inversion. It focuses on the contour used to value a covered call, why the payoff transform is restricted to a strip in the complex plane, and why the integration line is chosen midway through that strip.

It also questions whether the resulting relation is limited to at-the-money implied volatility or can be applied across log-moneyness to recover a full volatility smile. For numerical work, it considers solving the integral equation for volatility versus first computing option prices from the characteristic function at each strike and then inverting each price to implied volatility. The document presents these as questions rather than resolved conclusions and gives no implementation, numerical comparison, or validation; practical accuracy will depend on the model, integration method, and inversion procedure.

Key ideas

• A characteristic function can be used with an inversion formula to connect a model to Black-Scholes implied volatility.
• The covered-call transform is valid only within a specified strip of complex arguments.
• The document asks why the integration contour is placed at the center of that strip.
• It raises whether the method applies across strikes or only at the money.
• Two numerical routes are considered: solve the volatility integral equation or invert prices strike by strike.

Tags

# Black-Scholes IV from Characteristic Function


# Black-Scholes IV from Characteristic Function












I'm trying to follow Gatheral 2006 on his derivation of the BSIV from a characteristic function. The most relevant formula is (5.7) page 60.

$$\int_0^\infty\frac{du}{u^2+(1/4)}\Re[e^{-iuk}\left(\phi_T(u-1/2)-e^{-1/2(u^2+1/4)\sigma_{BS}^2T}\right)]=0$$

He starts with a Proof of Lewis 2000 (5.6) by inversion. In the second part of his proof, he integrates a covered call position along the $Im[u] = 0.5$-line. Why does the covered call value only exist in the region $0<Im[u]<1$? Why did he choose to integrate along 0.5?

Furthermore, he substitutes the BS characteristic function and thereby finds a relationship between the ATM BS IV and the underlying process.

a.) Since the expression is dependent on log moneyness, why would he consider it only ATM? It would be my understanding, that this relationship should hold for the entire vol smile (given the same drawbacks of any other BS Vol Smile, and just being a more efficient way of quoting option prices across k)

b.) How would I numerically implement a.)? My approach so far is to minimize the left hand side of (5.7), but that includes multiple evaluations of the same integral, since my variant is sigma. Another way of thinking about it could be that, given a characteristic function, one would calculate option prices along k, and individually compute their BS IV (e.g. using Brent-Dekker), to receive a simulated vol smile.

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