Forward-Start Option Volatility and Negative Implied Forward Variance
Summary
The document examines an interview question that asks for the implied volatility of a one-year option beginning one year in the future, given two maturity-specific implied volatilities. It explains that combining volatilities requires working with variance over time, but the stated inputs imply negative forward variance under the simple calculation. The answer interprets this as an inconsistent term structure that permits calendar arbitrage: in the example, a longer-dated option is priced below a shorter-dated option on the same strike and underlying.
A second answer highlights a limitation in what can be inferred. A forward-start option whose strike will be set at the future at-the-money level is not the same as a fixed-strike option, and its value cannot be determined from the two quoted implied volatilities alone. The illustration assumes matched at-the-money options and simplifies rates and dividends. Thus, the negative forward variance signals problematic inputs under the setup, but it does not supply a general fair volatility for every forward-start contract.
Key ideas
- Forward variance is inferred from total variance across maturities, not by averaging volatilities directly.
- The example's volatility inputs imply negative forward variance under the stated setup.
- A longer-maturity option priced below a shorter one at the same strike can indicate calendar arbitrage.
- A future at-the-money forward-start option differs from an option with a strike fixed today.
- The quoted maturities alone do not determine the value of every forward-start option.
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# Forward-Start Option Implied Volatility
# Forward-Start Option Implied Volatility
I am preparing for an interview on Monday and I came across a practice question which has me stumped.
"The implied volatility of a 1 year option is 20% and the implied volatility of a 2 year option on the same underlying is trading at 10%. Estimate the implied volatility of a one year option forward starting in 1 year."
Its basically asking, "what is the annualized volatility the market expects for the second year?" correct?
We can't just do $\frac{0.2 + x}{2} = 0.1$ because volatilities are not additive correct? So we have to convert to variances first. So now its $\frac{0.04 + x}{ 2} = 0.01$, which doesn't make any sense.
## Answer by dm63 (score 3)
https://quant.stackexchange.com/a/31099
The answer given above is probably the intended answer to the interview question. However it is not the whole story. Suppose the 1yr and 2yr options are both at the money and struck at usd100 (and that there are no dividends and interest rates are zero, to make things simple). Then if the options are priced at 20% and 10% respectively, the implied price of a usd100 call one year forward, is negative (an arbitrage, as stated). However, the price of a 'then at the money' 1yr option whose strike is determined 1yr from now, cannot be determined from the information we have. One cannot replicate this option using usd100 calls of 1yr and 2yr maturities. I would answer the question by saying that the forward option will be cheap, since we can replicate all the usd100 options we want for nothing, but there is a chance that the stock will be at usd200 by then , and we have no way to lock in the volatility of a 1yr usd 200 call at that time.
## Answer by emot (score 0)
https://quant.stackexchange.com/a/31081
Yup, the issue is that with your implied volatility structure, the options with longer maturity are cheaper than the ones with shorter maturity - try checking this with your Black Scholes formula. This implies calendar arbitrage, you should short the shorter one and go long with the longer maturity option. The implied forward variance is actually negative in your example, your calculations are correct.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.