An Analytic Second-Order Dividend Correction for Variance Swap Contracts on Stock Indices

We present an analytic dividend-payout modification to the valuation of a typical equity index variance swap contract. Including these dividends in the valuation reveals the risk that the contracts have to changes in the expected future dividends or, more precisely, to market dividend swap rates. This adjustment is also necessary from a calibration point of view because the volatility curve “basis” in lognormal stock price change models is usually the dividend-free volatility curve, σ (t), that is used in the lognormal process dS = Sµ (t) dt + Sσ (t) dz . (1) Known future dividends are typically introduced as instantaneous downward jumps in the paths of future stock prices, independently of the above process. Assuming piecewise lognormal index-level evolution, we display the effect of future dividends on the the index variance swap contract to second order in dividends as an analytic expression. Moreover, we demonstrate that this term must be at least second order, ie one cannot generally go to only first order in dividends when assuming piecewise lognormality. The resultant dividend risk is found to be typically quite small compared to the cash value at risk in common contracts. For short-tenored contracts, however, say one month, this dividend adjustment is arguably not negligible, particularly in view of current bid-offer spreads on these contracts