Case analysis: moving the reference price

A stylised version of the mechanism alleged in SEBI's 2025 interim order (notes 8.12, Case B). In the morning a trader buys $Q$ index units' worth of constituents. Each unit bought moves the index up by $\lambda$ points permanently (linear permanent impact) and costs an extra $\eta$ points in temporary impact (spread and slippage). At the top the trader takes an option position with delta $-D$ (index units) through a short call and long put at the same strike, at prices consistent with parity at the prevailing index. In the afternoon the trader sells the $Q$ units with the same impact parameters, returning the index to its starting level, and closes the options. (a) Show that the cash leg's P&L is $-2\eta Q$ (in index-point rupees per unit) to first order, i.e. permanent impact nets out over the round trip. (b) Show that the option leg earns $D\lambda Q$ if option prices follow the index one for one, and $\beta D\lambda Q$ if they follow with pass-through $\beta\in[0,1]$. (c) Derive the condition for the combined strategy to be profitable. Interpret it as a lower bound on the ratio of option delta to cash-leg size. (d) Why does the strategy fail in a market where option prices are set by a deep, independent options market rather than mechanically by the index? Why does expiry day help it? Which of SEBI's measures in Oct 2024–2025 act on which parameter? (e) This is an interim order with prima facie findings. List the facts you would need to see before concluding that a particular trading pattern was manipulation rather than hedging or arbitrage.