January 13, 2026
A variance swap pays the sum of the squares of returns over a period. It provides exposure to volatility no matter the level of the underlying. Unlike a call or put option, it never goes out-of-the-money. The most remarkable fact about variance swaps is that they do not require a mathematical model of the underlying price, only futures on the underlying and puts and calls that expire at the maturity of the variance swap.
Consider an underlying with price x at the beginning of a period and price X at the end of the period. Consider an option with payoff (X - x)^2 at the end of the period having price A and an option with payoff X^2 - x^2 having price B. Futures pay X - x and have price 0. If there is no arbitrage then A = B. This follows from simple algebra: X^2 - x^2 = 2x(X - x) + (X - x)^2. Either contract can be used to replicate the other using the zero price futures contract.
Note this is Taylor’s formula for f(x) = x^2 since cubic and higher order terms are 0.
Variance swap contracts are specified by an underlying instrument and observation times. If the price of the underlying is X_t at time t and the observation times are {t_0 < t_1 < \cdots < t_n} then the payoff on unit notional at time t_n is the average realized variance minus the par variance \frac{c}{t_n - t_0}\sum_{j = 0}^{n-1} R_j^2 - \sigma_0^2, where c is a constant specified in the contract that is not far from the number 1, R_j is the return over [t_j, t_{j+1}], and \sigma_0 is the _par volatility The par volatility is quoted to make the price of the contract at t_0 equal to zero. Variance swap contracts are quoted using the par volatility.
Almost all variance swap contracts specify the return as the log realized return {R_j = \log X_{j+1}/X_j = \Delta \log X_j} where \log is the natural logarithm with base e approximately equal to 2.718281828. I confess to being surprised when I first saw this approximation in an actual contract.
The standard way of specifying return is R_j = (X_{j+1} - X_j)/X_j = \Delta X_j/X_j.1 It does not drag in logarithms and is simpler to work with. By Taylor’s theorem \Delta \log X = (\Delta X/X) - (1/2)(\Delta X/X)^2 + (1/3)(\Delta X/X)^3 + O((\Delta X/X)^4). The cubic term (\Delta X/X)^3 explains the P&L of a variance swap hedge, as we will see later.
Exercise. Show (\Delta \log X)^2 = (\Delta X/X)^2 - (2/3)(\Delta X/X)^3 + O((\Delta X/X)^4).
The exercise shows the cubic term in the standard return contract is twice as sensitive in the opposite direction to the log return contract. When Peter Carr and I were implementing the variance swap model at Banc of America Securities we suggested selling two log return contracts for each standard return contract to minimize the risk of ignoring the cubic term. They (correctly) thought that clients would get confused by having two types of contracts and only wrote log return contracts to halve the effect of ignoring the cubic term.
Continuous time mathematical treatments assume the realized variance is approximated by {(1/t)\int_0^t (d\log X_s)^2\,ds}. If X is an Ito process satisfying {dX/X = \mu\,dt + \sigma\,dB} then d\log X = dX/X - (1/2)(dX/X)^2 = \mu\,dt + \sigma\,dB - (1/2)\sigma^2\,dt so {\int_0^t (d\log X_s)^2\,ds = \int_0^t \sigma^2\,ds}. Under a risk-neutral measure the par variance is {\sigma_0^2 = (1/t)E[\int_0^t \sigma^2\,ds]}.
Black, Scholes, and Merton had to work harder to show the drift of the underlying was irrelevant. The stunning thing about variance swaps is that valuing and hedging them do not require any assumptions on a model for the dynamics of the underlying. They only require futures and options traded at a sufficiently wide range of strikes on the underlying.
The payoff of a variance swap can be approximately replicated by a static hedge and a dynamic hedge. The static hedge is a European log contract and the dynamic hedge uses futures at each observation date except the last. The options, futures, and variance swap expiration must be the same.
For any thrice differentiable function f we use a telescoping sum and Taylor’s theorem with remainder to get \begin{aligned} f(X_n) - f(X_0) &= \sum_{0\le j < n} f(X_{j+1}) - f(X_j) \\ &= \sum_{0\le j < n} f'(X_j)\Delta X_j + \frac{1}{2} f''(X_j) (\Delta X_j)^2 + S_j \end{aligned} where S_j = (1/2)\int_{X_j}^{X_{j+1}} f'''(t) (X_{j+1} - t)^2\,dt. Note f(X_n) - f(X_0) is a European option payoff with expiration t_n and f'(X_j) \Delta X_j is the cash flow at time t_{j+1} from purchasing f'(X_j) futures on X at time t_j.
The quadratic term can be used to replicate a variance swap payoff. If f''(x) = 2/x^2 then (1/2)f''(X_j)(\Delta X_j)^2 = (\Delta X_j/X_j)^2 is the square of the return. We have f'(x) = -2/x + c where c is a constant, and f(x) = -2\log x + cx. It is convenient to choose c = 2/z for some constant z so -2\log X_n + 2 X_n/z + 2\log X_n - 2 X_0/z = \sum_{0\le j < n} (-2/X_j + 2/z)\Delta X_j + (\Delta X_j/X_j)^2 + S_j Rearranging terms and simplifying gives \sum_{0\le j < n} (\Delta X_j/X_j)^2 = -2\log X_n/X_0 + 2(X_n - X_0)/z + \sum_{0\le j < n} (2/X_j - 2/z)\Delta X_j - S_j This shows a variance swap can be replicated using a static hedge and a dynamic hedge with error \sum -S_j. The static hedge is a European option that can be approximated using the Carr-Madan formula. If z = X_0 then the initial dynamic futures hedge is zero.
The static hedge is -2\log X_n/X_0 + (X_n - X_0)/z and can be approximately replicated with a cash position, a forward, and a portfolio of puts and calls. Recall the Carr-Madan formula for a twice differentiable function f\colon [0, \infty)\to\mathbf{R} is f(x) = f(a) + f'(a)(x - a) + \int_0^a f''(k) (k - x)^+ \, dk + \int_a^\infty f''(k) (x - k)^+\,dk If puts with strikes (L_j) and calls with strikes (H_j) are available at t_n then given K we consider the piecewise continuous linear function \tilde{f} determined by by the points (L_j, f(L_j)), L_j < K, and (H_j, f(H_j)), H_j \ge K. We assume linear extrapolation on the left using the two lowest put strikes and on the right by the two highest call strikes. Typically K is chosen to be close to the forward at option expiration.