Impact of ranking-induced discontinuities on index futures volatility

Determine how the ranking-based construction of the market index I_t, which induces discontinuities in the index’s volatility at random points, affects the volatility dynamics of the corresponding index futures F_{t,T} = E[I_T | 𝔽_t]. Specifically, characterize the volatility process of F_{t,T} when stock prices S^j_t follow dS^j_t = S^j_t ∑_{k=1}^d σ^{jk}_t (ρ^{jk} dB^k_t + √(1−(ρ^{jk})^2) dW^k_t) and the index aggregates the top-ranked stocks via I_t = ∑_{j=1}^{ar{n}} w_j S^{(j)}_t.

Background

The paper models market indexes by ranking constituent stocks and summing the top-ranked names, which introduces discontinuities and local time terms into the index dynamics. Options on equity indexes are typically written on index futures rather than directly on the index level.

Because the ranking mechanism makes the index’s volatility discontinuous at random times, understanding how this feature propagates to the tradable futures F_{t,T} is nontrivial and directly relevant for pricing and hedging index options. The authors explicitly state that this effect is presently unclear.