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Negative Forward Variance and Calendar Arbitrage in Options

Article Quant Q&A · Author: fwd_T
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Summary

The document examines whether negative forward variance inferred from at-the-money-forward implied volatility across two maturities signals an arbitrage. It defines forward variance from the difference in total implied variance over the two time intervals. Because an option’s value increases with implied variance, negative forward variance implies lower modeled value for the longer-maturity at-the-money-forward call than for the shorter one.

The answer connects this price ordering to a calendar arbitrage argument. Under the stated rate assumptions, conditioning the later call payoff on information at the earlier maturity and applying convexity gives a lower bound tied to the earlier call’s value, adjusted for the foreign interest rate. The argument therefore identifies a potential no-arbitrage violation in the quoted volatility curve. Its conclusion depends on the setup, including comparable forward moneyness and the assumptions about rates and option pricing; the post does not discuss market frictions or broader surface consistency.

Key ideas

  • Forward variance compares total implied variance across maturities, scaled by the interval between them.
  • A negative value means the shorter maturity has greater total implied variance than the longer maturity in this construction.
  • Since call value rises with implied volatility, that ordering can produce a longer-dated call priced below the shorter-dated call.
  • A conditional expectation and convexity argument links this price ordering to a calendar arbitrage under the stated assumptions.

Tags

OptionsVolatilityArbitrageDerivatives pricing
Full text
# Is negative forward variance an arbitrage?


# Is negative forward variance an arbitrage?












I believe that having a negative forward variance on a ATMF implied volatility curve of a volatility surface could imply the existence of a static arbitrage (for example, a calendar arbitrage). Although I tried to look at this from multiple perspectives, I could not come up with an answer. Therefore, I thought it might be appropriate to ask the community, in case someone has more insight into this than I do.

## Answer by fwd_T (score 4)

https://quant.stackexchange.com/a/75827

Let $$ V_t^{T_1,T_2}=\frac{(T_2-t)V_t^{T_2}-(T_1-t)V_t^{T_1}}{T_2-T_1} $$ be our forward variance where $t<T_1<T_2$, $V_t^{T_1}$ is the ATMF implied vol as seen at time $t$ for slice at maturity $T_1$ and $V_t^{T_2}$ is the AMTF implied vol seen at time $t$ for tenor $T_2$. Let $P_1$ be the PV at time $t$ of an ATMF long call position of maturity $T_1$, and $P_2$ be the PV at time $t$ of an ATMF long call position of maturity $T_2$. We know that $$ \frac{\partial P_1}{\partial V_t^{T_1}}>0; \frac{\partial P_1}{\partial V_t^{T_1}}>0 $$ Let $\widetilde{V}_t^{T_1}=(T_1-t)V_t^{T_1}$ and $\widetilde{V}_t^{T_2}=(T_2-t)V_t^{T_2}$. Then, $$ \frac{\partial P_1}{\partial \widetilde{V}_t^{T_1}}=\frac{\partial P_1}{\partial V_t^{T_1}}\frac{\partial V_t^{T_1}}{\partial \widetilde{V}_t^{T_1}}=\frac{1}{T_1-t}\frac{\partial P_1}{\partial V_t^{T_1}}>0 $$ and similarly for $T_2$ superscript. But since the forward variance is negative, we have $\widetilde{V}_t^{T_1}>\widetilde{V}_t^{T_2}$ and since the PV of the option is an increasing function of this variable, this means that the PV of the call option of maturity $T_2$ is smaller than the PV of the call option of maturity $T_1$. So $P_1>P_2$. This can be shown to be a calendar arbitrage as explained here: Check for arbitrage - European calls with same strike price, different duration and price . More specifically, under no-arbitrage we would have: $$ P_2=e^{-(T_2-t)r_d}\mathbb{E}_{\mathbb{Q}}\left[\left(S_{T_2}-F_{t,T_2}\right)^{+}|\mathcal{F}_t\right]=e^{-(T_2-t)r_d}\mathbb{E}_{\mathbb{Q}}\left[\mathbb{E}_{\mathbb{Q}}\left[\left(S_{T_2}-F_{t,T_2}\right)^{+}|\mathcal{F}_{T_1}\right]|\mathcal{F}_t\right]\geq\\ \geq e^{-(T_2-t)r_d}\mathbb{E}_{\mathbb{Q}}\left[\left(\mathbb{E}_{\mathbb{Q}}\left[S_{T_2}|\mathcal{F}_{T_1}\right]-F_{t,T_2}\right)^{+}|\mathcal{F}_t\right]=\\ =e^{-(T_2-t)r_d}\mathbb{E}_{\mathbb{Q}}\left[\left(\underbrace{S_{T_1}e^{(r_d-r_f)(T_2-T_1)}}_{F_{T_1,T_2}}-F_{t,T_2}\right)^{+}|\mathcal{F}_t\right]=\\ =e^{-(T_1-t)r_d}e^{(T_1-T_2)r_d}\mathbb{E}_{\mathbb{Q}}\left[\left(\underbrace{S_{T_1}e^{(r_d-r_f)(T_2-T_1)}}_{F_{T_1,T_2}}-F_{t,T_2}\right)^{+}|\mathcal{F}_t\right]=e^{-(T_2-T_1)r_f}P_1 $$

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