Forward Implied Variance and Delta-Hedged Option P&L
Summary
The document raises questions about the forward implied variance defined in a volatility modeling treatment. The quantity is expressed as a ratio of conditional expectations weighted by the Black–Scholes gamma, while the average implied volatility over the remaining life of the option is itself defined through that forward variance. The author asks how this pair of definitions is interpreted and used together.
It also examines the stated link between the time evolution of a Black–Scholes option value and the forward implied variance, and how that relates to expected profit on a delta-hedged option. The central issue is whether the deterministic forward variance can vary over time and still appear in the pricing equation when the average implied volatility is used. The document presents these as unresolved conceptual questions rather than supplying an answer or empirical evidence; it does not establish a trading rule or assess hedge performance.
Key ideas
- Forward implied variance is defined through gamma-weighted conditional expectations of realized variance.
- The remaining-life average implied volatility is linked back to the forward variance function.
- The discussion asks how a time-varying forward variance enters the Black–Scholes pricing relation.
- Expected delta-hedged option profit depends on the difference between realized variance and the hedge’s implied variance assumptions.
- The document poses questions but does not resolve them or provide empirical tests.
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Full text
# Expected vs Implied Volatility discussion - Gatheral chapter 3
# Expected vs Implied Volatility discussion - Gatheral chapter 3
On page 27-28 of Gatheral's book The Volatility Surface, the Black-Scholes forward implied variance is defined as
> $$ \nu_{K,T} (t) = \dfrac{\mathbb{E} \left[\sigma_t^2 S_t^2 \Gamma_{BS}(S_t, \bar{\sigma}(t)) \vert \mathcal{F_0}\right]}{\mathbb{E} \left[S_t^2 \Gamma_{BS}(S_t, \bar{\sigma}(t)) \vert \mathcal{F_0}\right]} $$ where $$ \bar{\sigma}^2 (t) := \dfrac{1}{T − t} \int^T_t \nu_{K,T}(u) \, du$$
At first glance, what I see here is that $\nu_{K, T}$ is defined in terms of $\bar{\sigma}(t)$. But $\nu_{K, T}$ is given in terms of $\bar{\sigma}$ and right after $\bar{\sigma}$ is now defined in terms of $\nu_{K, T}$. So the first question is, how is this possible?
Then, it is discussed that the expected realized profit of a delta hedged position should be when selling the option assuming an implied volatility of $\bar{\sigma}$, delta-hedging it using the deterministic forward implied variance function $\nu_{K,T}$ and the realized volatility being $\sigma_t$.
The book states
> Now, of course, $C_{BS} (S_t, K, \bar{\sigma}(t), T − t)$ must satisfy the Black-Scholes equation (assuming zero interest rates and dividends) and from the definition of $\sigma(t)$, we obtain: $$ \dfrac{\partial C_{BS}}{\partial t} = -\dfrac{1}{2} \nu_{K,T} (t) S_t^2 \dfrac{ \partial^2 C_{BS}}{\partial S_t^2}$$
The thing is, in the BS equation for $C_{BS}$ a constant volatility may be assumed, as we could claim that in a BS world $\bar{\sigma}(t)$ is defined not to change for all $t$ up until $T$. However, I do not see a reason for $\nu_{K,T}(t)$ to be constant, and I do not see why $\nu_{K,T}(t)$ can be traded for $\bar{\sigma}^2 (t)$ in that equation. In fact, from the definition of $\nu_{K, T}(t)$ I agree that stochasticity is removed, but I would expect it not to be constant in $t$ (as well as $K$ and $T$). Thanks for helping me out here!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.