Breeden–Litzenberger Density, No-Arbitrage, and the Risk-Neutral Mean

Summary

The document asks whether an arbitrage-free call-price slice necessarily implies that the risk-neutral density inferred from its second strike derivative has a mean equal to the current underlying price. It frames the question in terms of a given implied-volatility slice and its corresponding option prices, then wonders whether shifting implied volatility can preserve convexity while moving the inferred distribution’s mean.

The text presents the relationship as a question and does not include an answer, derivation, or example. It also does not specify discounting, dividends, forwards, or boundary conditions, which matter when relating option-price derivatives to a risk-neutral distribution and its mean. The issue is therefore a useful pricing-theory question, but the document alone does not establish what conditions guarantee the proposed equality or whether the suggested volatility shift is arbitrage-free.

Key ideas

• The second strike derivative of call prices is used to infer a risk-neutral density under suitable assumptions.
• The author asks how no-arbitrage restrictions relate to the mean of that density.
• The question does not specify discounting, dividends, or boundary conditions.
• No resolution or worked example is provided.

Tags

# Arbitrage-Free, Breeden-Litzenberger and Risk-Neutral Measure


# Arbitrage-Free, Breeden-Litzenberger and Risk-Neutral Measure












Let $\sigma_{\text{BS}}(K, T)$ a given IV slice at $T$, which is implied by a price slice $C(K, T)$.

From this price slice, we can infer the Risk Neutral density of the price distribution at $T$ using the Breeden-Litzenberger as : $$f_{\mathbb{Q}}(S_T) = \partial_{KK} C(K, T)$$

If the slice is arbitrage-free, does it mean necessarly that : $$\int S_T f_{\mathbb{Q}}(S_T) dS_T = S_0$$

If so, I can shift the implied volatility and still get a convex price curve but for which the mean of the density will be shifted.

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