Why Only a Giant Can Mark the Close: Bending a a VWAP settlement

How you would bend a VWAP settlement to your own options book, and why the maths says only a giant can do it

The second of a three-part series on how Indian index options settle. The first post built the settlement as a volume-weighted average and the legitimate edge that lived inside it; this one turns the same object into a weapon; the third walks through the Jane Street case, where SEBI alleges someone did exactly this.

In the first part of this series I argued that an Indian expiry option is really an Asian option on a volume-weighted average. The settlement is not the closing price but

S^{*} = \int_{0}^{1} S(\tau)\, d\tau

the price averaged over the last half hour in the clock of the market's own volume. From that one functional fell the convergence trade, the legitimate edge I traded for years: forecasting an average the screen had not caught up to. I was a price-taker, computing a functional of a path I did not control.

This post is about what happens when you stop forecasting that average and start steering it. It is, as far as I can tell, exactly the mathematics underneath the ₹4,843 crore SEBI impounded from Jane Street in July 2025, and the next post walks through that case in full. Here I just want the equation.

The same functional, manipulated

Flip the role. Suppose you hold an options book with settlement-delta $D = \partial\Pi/\partial S^{*}$ (say you are net short calls and long puts, so $D < 0$ and you profit if $S^{*}$ comes in low), and suppose you have the balance sheet to trade the constituent stocks and futures in size. Can you move the average your way, and when does it pay? When I first sat down to work this out properly, I expected the answer to be messy. It wasn't.

What follows is a stylized model, a deliberately simple caricature meant to expose the shape of the incentive, not a reconstruction of anyone's trading. Its value is that two of its predictions match the public record exactly.

Model your cash-and-futures activity as a position trajectory $x(\tau)$ in the volume clock, with $x(0) = 0$ and permanent price impact $\lambda$ : your accumulated position pushes the constituent prices, and so the index, by $\lambda \cdot x(\tau)$ above the honest path. The effect on the settlement (the volume-clock average) is

\Delta S^{*} = \lambda \int_{0}^{1} x(\tau)\, d\tau

To depress the settlement you carry net short pressure, $x(\tau) < 0$ , across the window. Your book gains $D \cdot \Delta S^{*}$ , linear in how far you drag the average. But the trajectory costs you. With a standard quadratic (Almgren-Chriss) trading cost, and the requirement that you end the day flat in cash ( $x(1) = 0$ , because you do not actually want the inventory), the problem is

\begin{gathered} \min_{x} \; \int_{0}^{1} \eta\, \dot{x}(\tau)^{2}\, d\tau \\ \text{subject to} \quad \lambda \int_{0}^{1} x(\tau)\, d\tau = -A, \qquad x(0) = x(1) = 0 \end{gathered}

where $A > 0$ is the depression you are targeting. The Euler-Lagrange condition forces the inventory's acceleration to be constant, so the optimal path is a parabola:

\boxed{\; x^{*}(\tau) = 6G\,\tau(\tau - 1), \qquad G = A/\lambda \;}

negative throughout, building smoothly to a trough in the middle of the window and unwinding into the close. Read that off the page: the cost-minimising way to mark a volume-weighted average down is not a slam at 3:29. It is sustained, smoothly distributed selling pressure spread across the whole window. Prediction one: spread, don't slam.

Put the optimal path back in and the execution cost is $12\eta A^{2}/\lambda^{2}$ , quadratic in the depression achieved. So net profit, as a function of how hard you push, is

P(A) = \underbrace{D\,A}_{\text{linear gain}} - \underbrace{\frac{12\eta}{\lambda^{2}}\, A^{2}}_{\text{convex cost}}

A^{*} = \frac{D\lambda^{2}}{24\eta}, \qquad \boxed{\; P^{*} = \frac{D^{2}\lambda^{2}}{48\eta} \;}

The punchline is the $D^{2}$ . Manipulating an average pays in proportion to the square of your options book. The gain is linear in how far you push; the cost is convex. For a small book the convex cost swamps the linear gain and the whole thing is negative: you would just be burning impact to move a number you barely benefit from. It only turns positive, and then explosively, once $D$ is enormous. Prediction two: only a giant can run it.

And notice where the leverage lives. Profit scales as $\lambda^{2}/\eta$ , the square of how easily you move the index over how expensive your trading is. A thin cash market (large $\lambda$ , easy to push) beneath a deep options market (huge $D$ ) is the ideal hunting ground, and the Indian expiry supplies it almost to caricature: a shallow underlying carrying an enormous, concentrated options market on top.

There is a bitter irony here. The exchange chose a volume-weighted average over a last-traded price precisely because an average resists single-print manipulation: you cannot move a half-hour VWAP with one trade at 3:29:59. The averaging meant to defeat the cheap manipulator is exactly what rewards the expensive, sustained, well-capitalised one. The very feature designed to protect the close is what makes the parabola pay.

The same idea, without the equations

If the calculus of variations isn't your first language, here is the whole thing without a single symbol.

Picture a wholesale vegetable market. The "official price" of tomatoes for the day is not the last price anyone pays; it is the average price across every sale in the final half hour of trading. And all sorts of contracts around town settle on that official average, so it matters far beyond the stalls.

Now suppose two things are true about you. First, you have placed an enormous side bet with a bookmaker, one that pays out for every rupee the official tomato price comes in below a certain level. Second, you are a big enough player that when you sell tomatoes hard, the price actually moves.

Put those together and a temptation appears. In that final half hour you start selling aggressively, more than you otherwise would, at prices a little worse than you'd like. Each sale loses you a little. But every rupee you drag the average down makes your giant side bet pay, and the bet is so much larger than your tomato business that the winnings swamp the losses. You are, quite literally, paying to move an average and being paid many times over for it.

Two things fall straight out of that picture, and they are the two the mathematics pins down exactly.

You spread the selling out; you don't dump it at the bell. Slam every tomato in the last thirty seconds and only those thirty seconds count toward a half-hour average: you would barely move it, and you would crater the price against yourself doing it. Far better to lean on the market steadily across the whole window.

Only a whale can play. It only makes money if the side bet is vast next to what it costs to push the price. A small punter dumping vegetables to swing the average would lose more on the tomatoes than the bet could ever return. The strategy belongs to someone whose bet dwarfs the market they lean on.

The mirror image works too: if the bet paid out when the price came in high, you would spend the half hour buying instead, propping the average up. And the play can run in two acts: buy heavily in the morning to push the index up and draw others in, then sell into the close so the official average lands low, paying a book that was bearish all along. Same logic, staged over a day rather than a half hour.

That, stripped of every equation, is the strategy. In the next post it acquires a name, a market (Bank Nifty rather than tomatoes), an options book in place of the bookmaker's slip, and a regulator alleging that a firm ran it at a scale measured in tens of thousands of crores.

Next: the Jane Street case, where SEBI alleges exactly this pattern, the ₹4,843 crore it impounded, and the appeal that is still unresolved.

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