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 days per year divided by n - b, where b is a small integer, instead of {1/(t_n - t_0)}. As OTC contracts near closing the days per year and b tend to get played with by sell-side firms hoping their buy-side client won’t notice.

Every variance swap contract I’ve seen specifies realized return as \log_e X_{j+1}/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}. Since {\Delta \log X_j = \Delta X_j/X_j - (1/2)(\Delta X_j/X_j)^2 + O((\Delta X_j/X_j)^3)} we have (\Delta \log X_j)^2 = (\Delta X_j/X_j)^2 - (\Delta X_j/X_j)^3 + O((\Delta X_j/X_j)^4).

Let’s used the idealized payoff and deal with details later.

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}

Let f''(x) = 2/x^2 so f'(x) = -2/x + 2/z for some constant z and f(x) = -2\log x + 2x/z. Usually z is taken to be X_0 so the initial futures hedge is 0. Solving for the quadradic term gives

\sum_{j=0}^{n-1} (\Delta X_j/X_j)^2 = f(X_n) - f(X_0) - \sum_{j=0}^{n-1} f'(X_j)\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 summation are the payoffs of 2(1/X_j - 1/z) futures contracts on X entered into at time t_j, the dynamic hedge.

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\} and c(k) = \max\{x - k,0\}.

Given strikes {k_0,\ldots,k_n} 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.

A possible improvement might be to find \bar{f}_j\approx f(k_j) that minimizes the L^\infty norm of f - \bar{f} over [k0, k_n]. Attention must also be paid to to the extrapolated slopes {(\bar{f}_1 - \bar{f}_0)/(k_1 - k0)} and {(\bar{f}_n - \bar{f}_{n-1})/(k_n - k_{n-1})}.

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}}. 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)^+

The formula for par variance is \sigma^2 = \bar{f}(z) + \bar{f}'(E[X] - 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 at k 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.