Variance Swap Fair Strike via Log-Contract Replication
Consider a frictionless Black-Scholes world where a stock follows geometric Brownian motion with constant volatility $\sigma$.
- Show that the fair variance swap strike $K_{\text{var}}$ -- the fixed leg that makes the swap have zero value at inception -- equals the expected annualized quadratic variation of the log price, i.e., $K_{\text{var}} = \sigma^2$.
- Now consider the more general (and more useful) result: using the log-contract replication approach, express the fair variance swap strike as an integral over out-of-the-money (OTM) European calls and puts across all strikes. Specifically, show that
$K_{\text{var}} = \frac{2}{T} \left[ \int_0^{F} \frac{P(K)}{K^2} \, dK + \int_{F}^{\infty} \frac{C(K)}{K^2} \, dK \right]$
where $F$ is the forward price, $P(K)$ and $C(K)$ are OTM put and call prices at strike $K$, and $T$ is time to expiry.
- Explain why the /K^2$ weighting appears and what it means practically for variance swap exposure across strikes.
Hints
- Start with Ito's lemma on $\log S_t$ to connect realized variance to the log contract payoff.
- Use the Carr-Madan spanning formula: any payoff $f(S_T)$ with $f'' $ well-defined can be decomposed into a continuum of calls and puts weighted by $f''(K)$. What is $f''(K)$ when $f(S) = -\log(S)$?
- Split the replication at the forward price $F$: puts below $F$ and calls above $F$, each weighted by /K^2$. The risk-neutral expectation of the log contract then gives you the fair strike integral directly.
Worked Solution
How to Think About It: A variance swap pays the difference between realized variance and a fixed strike. The question is: what fixed strike makes the swap fair (zero cost) at inception? In the constant-vol Black-Scholes world, realized variance is deterministic, so the answer is trivially $\sigma^2$. The deeper result -- and the one interviewers really care about -- is how to replicate this payoff using vanilla options in a world where vol is NOT constant. The key object is the log contract, $\log(S_T/S_0)$, because Ito's lemma connects it directly to realized variance. And a log contract can be replicated by a continuum of OTM options weighted by
/K^2$.Quick Sanity Checks: - In a constant-vol world, all the option prices are determined by $\sigma$, so the integral must collapse to $\sigma^2$. Good consistency check. - The
/K^2$ weighting means low-strike puts contribute more per unit strike width than high-strike calls. This matches the observation that variance swaps have significant downside (crash) exposure. - As $T \to 0$, the fair strike should approach the instantaneous variance, regardless of the model.Derivation:
*Part 1: Constant-vol case.*
Under Black-Scholes, the stock follows $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$. By Ito's lemma:
$d \log S_t = \left(\mu - \frac{\sigma^2}{2}\right) dt + \sigma \, dW_t$
The quadratic variation of $\log S_t$ over $[0, T]$ is:
$\langle \log S \rangle_T = \int_0^T \sigma^2 \, dt = \sigma^2 T$
The annualized realized variance is $\sigma^2 T / T = \sigma^2$. Since this is deterministic (constant vol), the risk-neutral expectation is also $\sigma^2$. The fair strike is:
$K_{\text{var}} = \frac{1}{T} E^{\mathbb{Q}}[\langle \log S \rangle_T] = \sigma^2$
*Part 2: Log-contract replication.*
The key identity comes from Ito's lemma applied to $-\log(S_T/S_0)$:
$-\log \frac{S_T}{S_0} = -\int_0^T \frac{dS_t}{S_t} + \frac{1}{2} \int_0^T \sigma_t^2 \, dt$
Rearranging for the realized variance:
$\frac{1}{T} \int_0^T \sigma_t^2 \, dt = \frac{2}{T} \left[ \int_0^T \frac{dS_t}{S_t} - \log \frac{S_T}{S_0} \right]$
The first term, $\int_0^T dS_t / S_t$, is the return from a continuously-rebalanced portfolio holding
/S_t$ shares -- under the risk-neutral measure, its expectation is $(r T)$ (or zero if rates are zero). The second term is the log contract payoff.Now the crucial step: replicate $-\log(S_T / F)$ (where $F = S_0 e^{rT}$ is the forward) using a static portfolio of vanillas. The Carr-Madan spanning formula gives:
$-\log \frac{S_T}{F} = \int_0^{F} \frac{(K - S_T)^{+}}{K^2} \, dK + \int_{F}^{\infty} \frac{(S_T - K)^{+}}{K^2} \, dK$
Taking risk-neutral expectations and discounting:
$E^{\mathbb{Q}}\left[-\log \frac{S_T}{F}\right] = \int_0^{F} \frac{P(K)}{K^2} \, dK + \int_{F}^{\infty} \frac{C(K)}{K^2} \, dK$
where $P(K)$ and $C(K)$ are undiscounted OTM put and call prices. Combining with the Ito identity above:
$K_{\text{var}} = \frac{2}{T} \left[ \int_0^{F} \frac{P(K)}{K^2} \, dK + \int_{F}^{\infty} \frac{C(K)}{K^2} \, dK \right]$
*Part 3: Why
/K^2$?*The
/K^2$ weighting arises from the second derivative of $-\log(S/F)$ with respect to $S$: $\frac{d^2}{dS^2}(-\log S) = 1/S^2$. In the Carr-Madan spanning formula, any twice-differentiable payoff $f(S_T)$ can be written as a portfolio of calls and puts weighted by $f''(K)$. For the log payoff, $f''(K) = 1/K^2$.Practically, this means low-strike puts get much heavier weight than high-strike calls. A put at strike $K = 50$ gets four times the weight of a call at $K = 100$. This is why variance swaps have significant crash exposure -- a 20% drop in the stock contributes far more to realized variance than a 20% rally, and the replicating portfolio reflects this through heavier put weighting.
Answer: The fair variance swap strike is $K_{\text{var}} = \sigma^2$ under constant vol, and more generally:
$K_{\text{var}} = \frac{2}{T} \left[ \int_0^{F} \frac{P(K)}{K^2} \, dK + \int_{F}^{\infty} \frac{C(K)}{K^2} \, dK \right]$
The
/K^2$ weighting comes from the log payoff's convexity and gives variance swaps their characteristic skew/crash exposure.Intuition
The variance swap is one of the cleanest examples of the deep connection between options and realized volatility. The reason you can replicate a variance swap with vanillas at all is that Ito's lemma turns the log of the stock price into a running integral of instantaneous variance plus a stochastic integral (which has zero expectation under the risk-neutral measure). So pricing realized variance reduces to pricing the log contract, and any smooth payoff -- including the log -- can be decomposed into a strip of calls and puts via the Carr-Madan formula.
The practical takeaway is the
/K^2$ weighting. It tells you that variance swaps are not neutral bets on volatility -- they are heavily skewed toward downside moves. This is why, even before 2008, variance swaps traded at a premium to ATM implied vol: the market prices crash risk into the put wing, and the/K^2$ weighting amplifies it. When you see the VIX index (which uses this same integral formula), you are really looking at a put-heavy average of implied vols across the entire strike spectrum, not just ATM vol.