Introductory Context
"Variance and volatility swaps are introduced conceptually in Topic 18.11 (from the volatility asset class perspective). This topic develops the pricing theory and hedging mechanics at the expert level -- showing how a variance swap is replicated by a portfolio of options (the 'log contract'), how the fair strike is determined, and how the instruments relate to the VRP that the retail option seller captures approximately through simple credit spreads. "
Variance Swap Mechanics
A variance swap has a payoff at maturity: N × (σ²_realised - K²_var). Where N is the variance notional (e.g., Rs 1 lakh per variance unit), σ²_realised is the realised variance over the observation period (computed from daily log-returns: σ²_realised = (252/n) × Σᵢ (ln(Sᵢ/Sᵢ₋₁))²), and K_var is the fair strike (the forward variance) agreed at contract initiation. The variance buyer profits if realised variance exceeds the strike (the actual market was more volatile than expected). The variance seller profits if realised variance is below the strike (the market was calmer than expected -- the VRP was positive). The fair strike K_var = E[σ²_realised] under the risk-neutral measure.
The Log Contract and Replication
The most important theoretical result in variance swap pricing: a variance swap can be replicated (approximately) by a 'log contract' -- a position in a continuum of options of all strikes, weighted by 1/K² (the inverse of the square of the strike). The replication: buy a portfolio of European call options at all strikes above the current forward price F₀, weighted by 1/K², plus a portfolio of European put options at all strikes below F₀, also weighted by 1/K². This portfolio's payoff replicates the variance swap's payoff, establishing that the fair variance swap strike equals the weighted sum of all options prices -- the entire implied volatility surface's information is contained in the fair strike.
Practical implication: the S&P 500 VIX index (and by analogy India VIX) is computed using exactly this replication formula -- it is the square root of the implied fair variance swap strike, computed from the observed prices of all available near-term options. This connects India VIX (the headline volatility indicator) directly to the variance swap's fair strike: India VIX = √(K_var × 12) where K_var is the monthly implied fair variance and the ×12 converts to annualised units. The entire discussion of India VIX as a measure of market fear (Modules 14-20) is ultimately grounded in the variance swap replication theory.
The Convexity Adjustment - Variance vs Volatility Swaps
The volatility swap (which pays N × (σ_realised - K_vol)) is harder to price than the variance swap because of convexity. By Jensen's inequality: E[√X] < √E[X] for any random variable X. Therefore: the fair volatility swap strike K_vol < √(K_var), where K_var is the fair variance strike. The difference √(K_var) - K_vol is the convexity adjustment -- it represents the discount from the square-root of the fair variance strike to the fair volatility strike. The convexity adjustment is proportional to the 'vol-of-vol' (how much volatility itself varies): K_vol ≈ √(K_var) × (1 - Var[σ]/(8 × K²_var)). The higher the vol-of-vol, the larger the convexity adjustment, and the cheaper the volatility swap relative to the square-root of the variance swap price. This is why the Heston model's ξ parameter (vol-of-vol) directly affects the relative pricing of variance and volatility swaps.
Variance vs Volatility Swap Comparison
Variance swap: payoff = N × (σ²_realised - K_var). Payoff is convex in volatility. Larger gain per unit of realised vol above strike. Easier to hedge (log contract replication is cleaner). Volatility swap: payoff = N × (σ_realised - K_vol). Payoff is linear in volatility. More intuitive (directly compares realised to implied vol). Harder to hedge (no perfect replication). Fair relationship: K_vol < √K_var (the convexity adjustment). Institutional use: variance swaps for hedging and trading, volatility swaps for more intuitive PnL expression.
Variance Swaps and the VRP in Indian Markets
The positive VRP in Indian markets (implied variance systematically exceeds realised variance -- from Topic 18.1) creates a structural advantage for variance swap sellers, just as it creates an advantage for simple option sellers. A variance swap seller who locks in K_var at the fair strike and subsequently observes lower realised variance than K_var earns the VRP directly and cleanly: the seller's entire P&L is determined by (K_var - σ²_realised) × N, with no path dependency, no delta management, no gamma effects, and no expiry-day surprises. This directness is why variance swaps are the preferred VRP capture vehicle for institutional volatility desks -- they eliminate all the complications of options-based VRP capture (gamma risk, theta management, strike selection) in exchange for a simple, clean exposure to the variance-realised differential.
Variance swaps represent the endpoint of the journey that began in Module 1: from buying and selling simple calls and puts, through the complexities of multi-leg strategies and position management, through the mathematical depths of advanced pricing models, arriving finally at the instrument that most directly and purely expresses the volatility trader's fundamental trade -- whether realised volatility will be above or below implied volatility. The variance swap is the institutional options trader's version of the retail options seller's iron condor: the same structural bet (implied volatility will exceed realised), in the most efficient and analytically clear possible form. Understanding this connection -- from the complex to the simple, from the advanced to the fundamental -- completes the advanced options practitioner's intellectual journey.
NSE Has Proposed VIX Futures That Would Function Like Variance Swap Approximations
NSE has periodically discussed the introduction of India VIX futures contracts that would allow direct VIX trading by retail participants -- analogous to the VIX futures on CBOE in the US. If introduced, NSE VIX futures would provide a retail-accessible vehicle for pure implied volatility trading, without the complications of options' delta, gamma, and expiry management. The pricing of VIX futures is directly derived from variance swap theory: the VIX future price is the risk-neutral expectation of future VIX levels, which is connected to forward variance through the convexity adjustment discussed in this topic. Monitor NSE's product announcement page for updates on any VIX derivative product introduction.