9.6 Variance swaps and the VIX

9.6.1Log-contract replication#

Proof

Step 1 (Itô). $d\ln S_t=dS_t/S_t-\tfrac12\sigma_t^2dt$, so

$$\int_0^T\sigma_t^2\,dt=2\int_0^T\frac{dS_t}{S_t}-2\ln\frac{S_T}{S_0}. \tag{9.23}$$

The realised variance equals a dynamic position (hold $2/S_t$ units of the underlying, rebalanced continuously) minus two log contracts. Under $\mathbb Q$, $E[\int dS/S]=rT$.

Step 2 (static spanning). For any twice-differentiable $f$ and any $S^*>0$, Taylor's theorem with integral remainder gives

$$\begin{aligned} f(S_T)&=f(S^*)+f'(S^*)(S_T-S^*) \\ &\quad+\int_0^{S^*}f''(K)(K-S_T)^+dK+\int_{S^*}^\infty f''(K)(S_T-K)^+dK. \end{aligned} \tag{9.24}$$

(Check: if $S_T>S^*$, only the second integral is non-zero and equals $\int_{S^*}^{S_T}f''(K)(S_T-K)dK=f(S_T)-f(S^*)-f'(S^*)(S_T-S^*)$ by integration by parts; symmetric if $S_T<S^*$.) With $f=\ln$, $f''(K)=-1/K^2$:

$$\ln\frac{S_T}{S^*}=\frac{S_T-S^*}{S^*}-\int_0^{S^*}\frac{(K-S_T)^+}{K^2}dK-\int_{S^*}^\infty\frac{(S_T-K)^+}{K^2}dK.$$

Step 3 (price). Take $E^{\mathbb Q}$, using $E^{\mathbb Q}[S_T]=F$, $E^{\mathbb Q}[(K-S_T)^+]=e^{rT}P(K)$, $E^{\mathbb Q}[(S_T-K)^+]=e^{rT}C(K)$:

$$E^{\mathbb Q}\ln\frac{S_T}{S_0}=\ln\frac{S^*}{S_0}+\frac{F}{S^*}-1-e^{rT}\left[\int_0^{S^*}\frac{P}{K^2}dK+\int_{S^*}^\infty\frac{C}{K^2}dK\right].$$

Substitute into (9.23): $TK_{var}=2rT-2E^{\mathbb Q}\ln(S_T/S_0)$. Choosing $S^*=F$ makes $\ln(F/S_0)=rT$ and $F/S^*-1=0$, which leaves (9.22).∎

The hedge is static in options: hold $2/(TK^2)\,dK$ of each OTM option, plus a dynamic futures position. The $1/K^2$ weight makes the strip's dollar gamma $\tfrac12\Gamma S^2$ independent of $S$, which is exactly what removes the path dependence of 9.3. Low strikes get the largest weights, so a steep put skew raises $K_{var}$ above the ATM variance.

9.6.2The VIX and India VIX formula#

The VIX discretisation With listed strikes $K_i$, $K_0$ the first strike at or below $F$, $Q(K_i)$ the OTM option mid-price (put below $K_0$, call above, the average of the two at $K_0$), and $\Delta K_i$ the half-distance between neighbouring strikes,

The last term corrects for $K_0\ne F$: with $S^*=K_0$ in Step 3, the extra terms are $2[\ln(F/K_0)-(F/K_0-1)]/T\approx-(F/K_0-1)^2/T$. The VIX is $100\sqrt{\sigma^2_{30}}$, where $\sigma^2_{30}$ interpolates (9.25) from two expiries to a 30-day horizon. (9.25) is the function variance_swap_strike in m09_strategies.py.

India VIX NSE's India VIX applies this methodology (adapted from CBOE's VIX) to Nifty options: best bid–ask quotes of near- and next-month Nifty options are used to bracket a 30-calendar-day horizon, OTM strikes only, with the at-the-money strike set just below the forward, and the relevant-tenor NSE MIBOR as the rate. The methodology rolls to the next and far expiries when the near month has 3 trading days left, and fills missing quotes by interpolation (natural cubic spline), per the NSE white paper and fact sheet linked in the readings. Confirm the current version on NSE's India VIX page before relying on the details. India VIX is an annualised 30-day risk-neutral volatility, in percent.

9.6.3Vega notional and convexity#

Variance vs volatility Variance swaps are traded in vega notional $N_{vega}=2K_{vol}N$, so that for small moves the P&L is $\approx N_{vega}(\sigma_R-K_{vol})$. The P&L is convex in $\sigma_R$: a seller loses more when realised vol rises by 5 points than she gains when it falls by 5 (Problem 9). A volatility swap (paying $\sigma_R-K$) has a lower fair strike than $\sqrt{K_{var}}$ by Jensen's inequality, $E\sqrt{X}\le\sqrt{EX}$; the gap is the convexity adjustment and grows with vol-of-vol.