Risk-Neutral Skew, Binary Call Payoffs, and Delta

Summary

The document raises questions about an argument linking the skew of an asset distribution, risk-neutral valuation of a binary call, and a call’s delta. It asks whether risk-neutral probabilities compensate for greater payoff potential in one tail, whether a tail integral represents the delta of a vanilla or binary call, and what threshold m denotes in the displayed equation.

It does not provide an answer or supporting derivation. The main learning value is identifying distinctions a reader needs to resolve: the physical distribution versus the risk-neutral pricing measure, payoff-weighted expectations versus event probabilities, and the derivative of an option price with respect to the underlying versus the probability of finishing in the money. The excerpt alone is insufficient to establish which interpretation Taleb intended; resolving the questions requires the surrounding text and precise definitions of the payoff, measure, and threshold.

Key ideas

• The document asks how skewed payoffs relate to risk-neutral valuation.
• It distinguishes a payoff-weighted tail integral from a call's delta.
• It asks whether the option under discussion is vanilla or binary.
• The excerpt does not define the threshold or provide a derivation.

Tags

# Call Option, Delta and Expected Payoff


# Call Option, Delta and Expected Payoff












In Dynamic Hedging by N. Taleb, at pag. 283-284, there is an argument about the relationship between risk neutral evaluation of a binary call and the delta about a call.

The author states that:

> Financial market impose the constraint on every security that the left integral be equal to the right integral plus the risk-neutral drift, which result in a mean of the risk-neutral drift, m $$ \int_{-\infty}^m f(x)p(x)dx = \int_{m}^{\infty} f(x)p(x)dx + \text{risk-neutral drift} $$ [...] The skew by increasing the potential payoff in the left integral needs to be compensated with a shift of the mean to the right to prevent markets from giving the short seller any higher expected return than the long holder.

- What is being compensated? I suppose that the risk-neutral measure will give more probability to the right tail of the distribution of the asset, correct?

- How is possible that the right integral is the delta of a call, as the author states at pag. 284? Or is he referring to a binary call?

- What is "m"?

Note: f() is the payoff, p() the distribution of the asset that has a negative asymmetry.

Please, let me know if more details are needed. Thanks for the help.

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