Forward Variance Versus Forward-Start Implied Volatility
Summary
The discussion distinguishes two quantities often called forward implied volatility. One is calculated from the change in implied variance between two maturities for vanilla options at a fixed strike. It represents a forward measure inferred from today’s implied-volatility term structure, rather than a guarantee of the volatility that will prevail over the future interval.
The other quantity is obtained by matching a model price for a forward-start option to a Black–Scholes price over the interval from the first maturity to the second. The answers identify this as a forward-start option implied volatility and distinguish payoffs that scale the future asset price by its earlier level from payoffs based on the future-to-initial price ratio. These definitions are not generally equivalent: the first concerns vanilla option term structure, while the second depends on the forward-start payoff and pricing model. The replies offer limited conditions and caveats, including that vanilla-based interpretation depends on assumptions about underlying dynamics.
Key ideas
- Vanilla forward variance is inferred from implied variances at two maturities and a fixed strike.
- Forward-start implied volatility is defined by matching a forward-start option price to a model price.
- Forward-start contracts can use distinct payoffs based on the future asset price or its ratio to the earlier price.
- The two volatility definitions describe different quantities and are not generally interchangeable.
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Full text
# Forward implied volatility definition
# Forward implied volatility definition
I am struggling to understand the link between two definitions of forward implied volatility. The well kwown model free forward implied volatility from $T_1$ to $T_2$ is defined as:
$$ \sigma_F(T_1, T_2, K)=\sqrt{\frac{\sigma(T_2,K)^2T_2-\sigma(T_1,K)^2T_1}{T_2-T_1}}. $$
In this paper, the forward implied volatility is implicitely defined as (p15):
$$ C_\mathrm{Model}(T_1,T_2, K) = e^{-rT_1}C_{\mathrm{BS}}(K, T_2-T_1, \sigma_F(T_1, T_2, K)). $$
Are both of them equivalent formulas ?
Thank you very much!
## Answer by Frido (score 3, accepted)
https://quant.stackexchange.com/a/81089
They're not equivalent. The first is a measure of the term structure of the implied variance / implied vol of vanilla options.
The second should probably be more accurate called Type I forward start implied volatility, as it is the implied volatility of options with the following payout (called Type I forward start options): $$ E_0 \left[ \left( \frac{S_T}{S_t} - \alpha \right)_+ \right] $$ There is also the Type II forward start implied volatility, which is the IV of options with the following payout (called Type II forward start options): $$ E_0 \left[ \left( S_T - \alpha S_t \right)_+ \right] $$
(Btw: the link does not work, at least not for me.)
## Answer by Arshdeep (score 1)
https://quant.stackexchange.com/a/79674
No, the second one depends on hitting an arbitrary price. If that price is intrinsic, the the implied vol is 0.
First one depends on market information of today, which may imply something different in the future, there is no guarantee that the market is interested in getting the forward vol right when trading vanillas. It is only usable if underlying dynamics are lognormal for sure.
If in the second one, the model is BS, then I think it is the same as first.
If you can tell us how you want to use this, we can help you with more.
Thanks!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.