A stylised model of expiry-day index pushing

A trader holds an option book whose P&L is $-D(S_T - S_r)$, i.e. it gains $D$ rupees per index point that the settlement level $S_T$ falls below a reference $S_r$ set earlier in the day. The trader can buy $Q$ units of the index basket in the morning, which raises the index by $\lambda Q$ (and lets her set up the option book at the higher reference level), and sell them before settlement, which lowers it by $\lambda Q$. Each leg costs $\tfrac12\lambda Q^2$ in impact. (a) Show that if settlement is the instantaneous index level, the trader's profit is $D\lambda Q - \lambda Q^2$; find the optimal $Q$ and profit. (b) If settlement is the average index level over a window during which the selling is spread evenly, show that the option gain halves and compute the new optimum. (c) What does this say about VWAP versus last-price settlement? (This is a toy model of the mechanism alleged in SEBI's 2025 order, not a description of any firm's trades.)