Variance Swaps

One Period

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 $A$. If an option with payoff $X^2 - x^2$ has price $B$ and the forward paying $X - x$ has price 0 then $A = B$. This follows from the elementary formula $$X^2 - x^2 = 2x(X - x) + (X - x)^2.$$ Note this is Taylor’s formula for $f(x) = x^2$ since cubic and higher order terms are 0. Either contract can be used to replicate the other using the 0 price forward contract.

Exercise. If the price of the forward is $C$ then $B = 2xC + A$.

Note this provides a perfect hedge no matter the value of $X$.

A more realistic one period model has payoff $(\log X/x)^2$, the square of the realized return. Recall Taylor’s formula with remainder for sufficiently differentiable $f$ is $$f(x) = \sum_{k=0}^n f^{(k)}(a) (x-a)^k/k! + \int_a^x f^{(n+1)}(t) (x - t)^n/n!\,dt$$

Exercise. If $f(x) = \log x$ show $$\log X/x = \log X - \log x = (1/x)(X - x) - (1/x^2)(X - x)^2/2 + \int_x^X 2/t^3(X - t)^2/2\,dt$$

Contract

Contracts are specified by an underlying instrument and observation times. If the level of the underlying is $X_t$ at time $t$ and the observation times are $(t_j)_{0\le j\le n}$ then the payoff on unit notional at time $t_n$ is $$(\frac{c}{t_n - t_0}\sum_{0\le j < n} R_j^2) - \sigma_0^2$$ where $R_j$ is the realized return over $[t_j, t_{j+1}]$, $\sigma_0$ is the par volatility, and $c$ is a constant specified in the contract that is not far from the number 1. The par volatility makes the price of the contract at $t_0$ equal to zero.

The realized return is $R_j = \log X_{j+1}/X_j = \log X_{j+1} - \log X_j = \Delta \log X_j$ where $\log$ is the natuaral logarithm with base $e \approx 2.718281828$. Real-world variance swap contracts actually specify this approximation.

Another common way of specifying realized return is $R_j = (X_{j+1} - X_j)/X_j = \Delta X_j/X_j$. It does not drag in logarithms and is simpler to work with.

Exercise. Show $\Delta \log X = (\Delta X/X) - (1/2)(\Delta X/X)^2 + O((\Delta X/X)^3)$.

The cubic term $(\Delta X/X)^3$ explains the P&L of a variance swap hedge, as we will see later.

Exercise. Show $(\Delta \log X)^2 = (\Delta X/X)^2 - (\Delta X/X)^3 + O((\Delta X/X)^4)$.

Continuous time mathematical treatments assume the realized variance is $(1/t)\int_0^t (d\log X_s)^2\,ds$. 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 $(d\log X)^2 = \sigma^2\,dt$ and $\int_0^t (d\log X_s)^2\,ds = \int_0^t \sigma^2\,ds$. Under a risk-neutral measure the par variance is $\sigma_0^2 = (1/t)E[\int_0^t \sigma^2\,ds]$.

The astounding 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 dynamic hedge in futures and a static hedge in a calls and puts.

For any thrice differentiable function $f$ we use a telescoping sum and Taylor’s theorem with remainder to get $$\begin{aligned} f(X_n) - f(X_0) &= \sum_{0\le j < n} f(X_{j+1}) - f(X_j) \\ &= \sum_{0\le j < n} f'(X_j)\Delta X_j + \frac{1}{2} f''(X_j) (\Delta X_j)^2 + R_j \end{aligned}$$ where $R_j = (1/2)\int_{X_j}^{X_{j+1}} f'''(t) (X_{j+1} - t)^2\, dt$. Note $f(X_n) - f(X_0)$ is a European option payoff 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 realized 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 $c = 1/z$ for some constant $z$ so $$-2\log X_n + X_n/z + 2\log X_0 - X_0/z = \sum_{0\le j < n} (-2/X_j + 2/z)\Delta X_j + (\Delta X_j/X_j)^2 + R_j$$ Rearranging terms and simplifying gives $$\sum_{0\le j < n} (\Delta X_j/X_j)^2 = -2\log X_n/X_0 + (X_n - X_0)/z + \sum_{0\le j < n} (2/X_j - 2/z)\Delta X_j - R_j$$ This shows a variance swap can be replicated using a static hedge and a dynamic hedge using futures with error $R_j$. The static hedge can be approximated with puts and calls using the Carr-Madan formula. If $z = X_0$ then the initial furtures hedge is zero.

Static Hedge

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 at-the-money forward at option expiration.

Cubic Term

The cubic term typically describes 95% of variance swap P&L over each period.

Since $f'''(x) = -4/x^3$ the error term $R_j$ over the period $[t_j, t_{j+1}]$ is $R_j = (1/2)\int_{X_j}^{X_{j+1}} -4/t^3 (X_{j+1} - t)^2\, dt$. Using $t$ is between $X_j$ and $X_{j+1}$ we have $$\begin{aligned} |R_j| &\le \frac{2}{\min\{X_j, X_{j+1}\}^3} \left|\int_{X_j}^{X_{j+1}} (X_{j+1} - t)^2\, dt\right| \\ &= \frac{2}{\min\{X_j, X_{j+1}\}^3} \frac{1}{3}|X_{j+1} - X_j|^3 \\ &= \frac{2}{3} \frac{|\Delta X_j|^3}{\min\{X_j, X_{j+1}\}^3} \end{aligned}$$