Forward Variance Swap Theta, Vega, and Volatility Carry
Summary
The document examines the Greeks of a forward variance swap built from two spot-starting variance swaps. It questions how the position can have approximately zero dollar gamma and declining vega while still carrying theta. The response distinguishes model theta from volatility carry and argues that the result depends on how market data are assumed to evolve through time.
A deterministic-volatility example makes the point: the forward variance value is an integral over its fixed future interval, so it has no time dependence in that model. By contrast, holding the two spot-starting fair strikes fixed as time passes effectively assumes a particular daily evolution of the volatility surface. The response attributes the apparent theta to this carry and to recalibration, and notes that derivatives of the component strikes with respect to time were omitted. The example is deliberately simple and does not address skew effects or establish a universal Greek convention; the interpretation depends on the chosen market-data dynamics.
Key ideas
- A theta calculation requires specifying how market data change over time or what is held constant.
- The apparent theta may represent volatility carry rather than stochastic-calculus theta.
- In a deterministic volatility model, forward variance over a fixed future interval has no time dependence.
- Holding spot-starting variance swap strikes constant as time passes imposes a particular surface evolution.
- The calculation should account for how the component fair strikes change with time.
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# Greeks of Forward Variance Swap
# Greeks of Forward Variance Swap
If I'm not mistaken, ignoring skew delta, the Greeks of forward variance swaps read
\begin{align} \mbox{Dollar Gamma} &\approx 0\\ \mbox{Vega} &\approx 2K_t - \frac{t}{T_2 - T_1} (2K_{t,2} - 2K_{t,1})\\ \mbox{Theta} &\approx -\frac{1}{T_2 - T_1} \cdot (K_{t,2}^2 - K_{t,1}^2) \end{align} where $$K_t^2 = \frac{T_2}{T2 - T_1} \cdot K_{t,2}^2 - \frac{T_1}{T2 - T_1} \cdot K_{t,1}^2$$ is the fair strike of the forward variance swap, and $K_{t,1}$ and $K_{t,2}$ are the fair strikes of the two spot-starting variance swaps, expiring at $T_1$ and $T_2$, respectively.
My question is: My gamma is zero, and vega is decaying, for what am I paying theta? Volga and Vanna? I thought theta is used to price-in all $(dW_t)^2$ terms.
## Answer by Andrea (score 1)
https://quant.stackexchange.com/a/82003
A precise definition of theta requires assumptions on how the market data evolves in time, or (which is the same) what stays constant.
I think that
- what you call theta is actually carry (vol carry, a bit like yield curve carry)
- what you pay for is the fact that the model is constantly rebased forward
- (and that) you have forgotten the derivative of $K_{t,1}$ (and 2) wrt to $t$
To see this, let's take a (simple) model that shows a term structure of fair strikes (and no delta). Moreover, I set $r=0$ since it is irrelevant here.
The model is
$dS_t=\sigma(t) S_t dW_t$
for deterministic $\sigma(t)$.
$K_t^2(T)=\frac{1}{T-t}\int_t^T\sigma(s)^2 \, ds$
the fair value of the forward variance swap is
$V(t, T_1, T_2) = (T_2 - T_1) \, K_t^2(T_1, T_2)=\int_{T_1}^{T_2}\sigma(s)^2 \, ds$
Which has no dependency on $t$ and so no theta (at least no stochastic calculus theta).
In your case, you have kept constant $K_{t,1}$ (and 2), assuming the fair strike for dates 1 and 2 (fixed) are the same every day.
This is not easy (or even possible) to achieve with a consistent model and the daily "recalibration" generates the vol-carry you have seen.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.