No-Arbitrage Bounds Linking Implied and Instantaneous Volatility

Summary

The document explores whether implied volatility in a diffusion model must be bounded by the model’s instantaneous volatility. It assumes the asset follows a stochastic-volatility diffusion under the pricing measure and considers a setting where instantaneous volatility stays between lower and upper bounds. The proposed argument uses the terminal profit and loss from delta hedging a vanilla option with a fixed Black–Scholes implied volatility. If the hedge volatility lies above the assumed upper bound, the displayed expression suggests a positive gamma-weighted payoff in every scenario, which would conflict with no-arbitrage; the author argues the reverse bound follows similarly.

The author then highlights a gap in the converse direction: bounded implied volatility for an option does not immediately establish that instantaneous volatility stays within those bounds at every time. The text proposes a stronger condition involving implied volatilities through time for a fixed strike and maturity, while noting that such a condition may not be available from the vanilla smile. The exchange provides a conjectural proof outline and caveats, not a settled proof or a general result.

Key ideas

• The proposed upper-bound argument relies on the gamma-weighted P&L of delta hedging at a fixed implied volatility.
• If hedge volatility exceeds a uniform upper bound on instantaneous volatility, the author argues the hedge produces an arbitrage-like sure profit.
• The reverse implication is more difficult because a bounded implied volatility may constrain volatility only at some time rather than throughout the interval.
• The suggested converse requires a time-uniform bound on implied volatility for a fixed option, a condition the author says may not follow from the vanilla smile.

Tags

# Implied vol bounded if and only if instantaneous vol bounded


# Implied vol bounded if and only if instantaneous vol bounded












I'd like to show that in diffusion models IV is bounded iff instantaneous vol is bounded if there is to be no arbitrage. So, assume a model under the pricing measure of the form $$ dS_u = \sigma_u S_u dW_u, $$ where the notation $\sigma_u$ means any local stochastic volatility.

I am going to assume first that $\forall{u}\in[0,T] \;m\leq\sigma_u \leq M$. Suppose then that $I_0(K,T)$ is the IV at $t=0$ for the vanilla option with strike $K$ and maturity $T$. Then delta hedging the option to maturity always using the initial IV $I_0(K,T)$ gives the following (well-known) terminal P/L at maturity date $T$: $$ P/L (T) = \frac12 \int_0^T \Gamma^{BS}_u(I_0(K,T),K)S_u^2 \left[I_0^2(K,T) - \sigma^2_u \right] \,du $$ with $\Gamma^{BS}_u(I_0(K,T),K)$ the BS gamma at time $u$ with fixed IV $I_0(K,T)$. Assume that $I_0(K,T) > M$. Then a sure profit will be made under all scenarios, which contradicts no arbitrage. Hence $I_0(K,T) \leq M$.

All the other proofs follow exactly the same reasoning.

My question: is the proof above correct, have I overlooked something? Is there another/shorter/better proof?

EDIT: Showing that instantaneous vol is bounded if IV is bounded ($m \leq I_0(K,T) \leq M$) is actually less straightforward. Invoking the delta hedging argument above will only show that $\exists u \in [0,T]$ such that $m\leq\sigma_u\leq M$, but not necessarily $\forall u\in [0,T]$.

I think a further restriction needs to be placed on the IVs. The correct statement is I believe $$ \exists K\; \forall u\in [0,T]\; m \leq I_u(K,T) \leq M \Rightarrow \forall u\in [0,T]\; m \leq\sigma_u \leq M $$

But this is a condition that cannot I think readily be obtained from the vanilla 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.