VIX

A variance swap contract specifies observation times t_0 < t_1 < \ldots < t_n and an underlying X with price/quote X_t at time t. Let {\Delta X_j = X_{j+1} - X_j} where {X_j = X_{t_j}}. At time t_n the variance swap pays the average realized variance minus the par variance {1/(t_n - t_0) \sum_{j=0}^{n-1} (\Delta X_j/X_j)^2 - \sigma^2} where \sigma is agreed on at t_0 so the contract has price 0.

Actual contracts use n - b, where b is a small integer, divided by days per year (often 252) instead of t_n - t_0. As OTC contracts near execution the days per year and b tend to get played with by sell-side firms hoping their buy-side clients won’t notice.

Every variance swap contract I’ve seen specifies realized return as \log_e X_{j+1}/X_j = \Delta \log X_j where e is approximately 2.718281828... instead of \Delta X_j/X_j. It is true that if you make the mistake of assuming X_t is an Ito process then {(d\log X_t)^2 = (dX_t/X_t)^2}. For a discrete difference \Delta \log X_j = \Delta X_j/X_j - (1/2)(\Delta X_j/X_j)^2 + O((\Delta X_j/X_j)^3) so (\Delta \log X_j)^2 = (\Delta X_j/X_j)^2 - (\Delta X_j/X_j)^3 + O((\Delta X_j/X_j)^4).

If the Taylor series for f converges then \begin{aligned} f(X_n) - f(X_0) &= \sum_{j=0}^{n-1} f(X_{j+1}) - f(X_j) \\ &= \sum_{j=0}^{n-1} \sum_{k=1}^\infty f^{(k)}(X_j) (\Delta X_j)^k/k! \\ \end{aligned}

Taking f''(x) = 2/x^2 so f''(X_j)(\Delta X_j)^2/2 = (\Delta X_j/X_j)^2 produces the variance swap payoff. We have f'(x) = -2/x + 2/z for some constant z and f(x) = -2\log x + 2x/z. We have \sum_{j=0}^{n-1} (\Delta X_j/X_j)^2 = -2\log X_n/X_0 + 2(X_n - X_0)/z + \sum_{j=0}^{n-1} (2/X_j - 2/z) \Delta X_j + O(((\Delta X_j/X_j)^3) The first two terms on the right hand side are the payoff of a European option, the static hedge. The terms in the right hand summation are the payoffs at t_{j+1} of 2(1/X_j - 1/z) futures contracts on X entered into at time t_j, the dynamic hedge. This means we should purchase 2/X_{j+1} - 2/z - (2/X_j - 2/z) = 2/X_{j+1} - 2/X_j = 2\Delta X_j/X_{j+1} X_j futures at t_{j + 1} after entering the initial futures position.

Since f'''(x) = -4/x^3 the third order term is (2/3)(\Delta X_j/X_j)^3.

The Carr-Madan formula is f(x) = f(z) + f'(z)(x - z) + \int_0^z f''(k) p(k)\,dk + \int_z^\infty f''(k) c(k)\,dk where p(k) = \max\{k - x,0\} is the put payoff and c(k) = \max\{x - k,0\} is the call payoff at strike k.

Assume the strikes {k_0,\ldots,k_n} are available for trading. We replace f by a piece-wise linear and continuous function \bar{f} determined by the points (k_j, f(k_j))_{j=0}^n. To replicate the static hedge a portfolio of a cash position, a forward at z, puts with strikes less than z, and calls with strikes greater than z is used. Note we do not need a put at strike k_0 or a call at strike k_n to do this. Since f''(x) > 0 this super-replicates the payoff on [k_0,k_n]. We can choose k_0 < k_1 and k_n > k_{n-1} at will since we do not need to trade options at these strikes. Traders tend not to buy far out of the money puts since they are expensive in downwardly sloping implied volatility environments. They often choose a low k_0 which loads up on the k_1 strike put. If the underlying moves toward k_1 then they sell the lowest strike put and buy puts at lower strikes that have become more affordable. The value used for k_n is less critical since f(x) is nearly linear for large x.

For 0 \le j < n we have \bar{f}'(x) = m_j = (\bar{f}_{j+1} - \bar{f}_j)/(k_{j+1} - k_j) on (k_j, k_{j+1}).

For 1\le j < n we have {\bar{f}''(k_j) = \bar{f}''_j = (m_j - m_{j-1})\delta_{k_j}} where \delta_k is a point mass at k. Note \bar{f}''(k_0) = 0 = \bar{f}''(k_n).

The Carr-Madan formula for the piecewise-linear approximation is \bar{f}(x) = \bar{f}(z) + \bar{f}'(z)(x - z) + \sum_{k_j < z} \bar{f}''_j (k_j - x)^+ + \sum_{k_j > z} \bar{f}''_j (x - k_j)^+ Let k^- = \max\{k_j\mid k_j < z\} be the strike of the highest strike put and k^+ = \min\{k_j\mid k_j \ge z\} be the strike of lowest strike call.

The formula for par variance times t = t_n - t_0 uses f(x) = -2\log x + 2x/z so f'(x) = -2/x + 2/z and f''(x) = -2/x^2. \sigma^2t = E[-2\log z + 2(X_n - X_0)/z + \bar{f}'(X_n)(X_n - z) - \bar{f}'(X_0)(X_0 - z)] + \sum_{k_j < z} \bar{f}''_j p(k_j) + \sum_{k_j > z} \bar{f}''_j c(k_j) where p(k) is the put price and c(k) is the call price at k.

Out-of-the-money puts on equity indices tend to be expensive. In practice, traders load up on the lowest strike put they beleive is affordable. If the index stays above this strike then the static hedge replicates the payoff. If the index goes below this strike then traders sell off some of the loaded up put and purchase lower strike puts to replicate the payoff.

We can sub-replicate on [k_0, k_n] by shifting \bar{f} down.

Given k_j and k_{j+1} define m_j = (f(k_{j+1}) - f(k_j))/(k_{j+1} - k_j). We want to find k with f'(k) = m_j. Since -2/k + 2/z = m_j we have $k = z/(1 - mz/2).