8.12 Case studies

Case A. Black, Scholes, Merton and LTCM: the model and its limits#

Context Fischer Black and Myron Scholes derived the option formula by a hedging argument; Robert Merton supplied the continuous-time replication proof. The paper was rejected by more than one journal before the Journal of Political Economy published it in its May–June 1973 issue (Black and Scholes 1973; Merton 1973). The Chicago Board Options Exchange had opened a month earlier, on 26 April 1973 (Cboe history; MarketsWiki).

Timeline

• 1973: CBOE opens (26 Apr); Black–Scholes and Merton papers appear.
• 1976: Black publishes the futures version (8.8).
• 1979: Cox, Ross and Rubinstein publish the binomial model (8.4).
• Feb 1994: Long-Term Capital Management (LTCM) is founded, with Merton and Scholes among its principals (Federal Reserve History, "Near Failure of Long-Term Capital Management").
• 1997: Merton and Scholes receive the Nobel memorial prize in economics "for a new method to determine the value of derivatives"; Black had died in 1995.
• End-1997: LTCM's leverage is about $30 of debt for each $1 of capital (Federal Reserve History).
• August 1998: Russia defaults; LTCM loses 44% of its value in the month (Federal Reserve History).
• 23 Sep 1998: under the auspices of the Federal Reserve Bank of New York, 14 banks and brokerage firms agree to invest $3.625 billion for 90% of the fund; no public money is committed (Federal Reserve History; US GAO letter GGD-00-67R).

Mechanism LTCM's trades were mostly relative-value bets (spreads between related bonds, and sales of equity-index volatility), not mispriced options in the BSM sense. The link to this module is structural. Every replication argument here assumes (A3): you can always trade, at the model price, to rebalance the hedge. In August–September 1998 liquidity vanished and correlations between supposedly unrelated spreads jumped, so positions that were hedged "on paper" could not be adjusted or financed. Leverage turned mark-to-market losses into a funding crisis, as with Metallgesellschaft in M07 · 7.8. The Black–Scholes PDE says a hedged option earns the risk-free rate; it says nothing about the margin you must post while the hedge converges.

Case B. SEBI's 2025 interim order against Jane Street: index options and the expiry-day index#

Status first What follows is an interim, ex parte order with prima facie findings. It is not a final determination, Jane Street disputes it, and proceedings were continuing as of Nov 2026. The figures to 21 Jul 2025 are taken from the primary documents: SEBI's interim order WTM/AN/MRD/MRD-SEC-3/31516/2025-26 of 3 Jul 2025 (105 pages, https://www.sebi.gov.in/enforcement/orders/jul-2025/interim-order-in-the-matter-of-index-manipulation-by-jane-street-group_95040.html) and SEBI press releases 40/2025 and 45/2025, all read on 6 Nov 2026. Later steps rest on press reports and are marked as such.

Context Jane Street is a global proprietary trading firm. SEBI's FY26 study found that prop traders and FPIs made money in Indian index options while individuals lost, with about 99% of their gross profit coming from algorithmic entities (SEBI FY26 study; M01 · 1.11 Case 1).

Timeline

• Feb 2025: NSE, as first-line regulator, sends Jane Street Singapore Pte Ltd and JSI Investments Pvt Ltd a caution letter (6 Feb 2025) advising them to refrain from large cash-equivalent positions and from certain trading patterns; the group replies on 6 and 21 Feb 2025 (order, background table and para 17.1).
• 15 May 2025: on a Nifty weekly expiry, the order finds a further instance of the "extended marking the close" pattern after the caution letter (order, para 17).
• 3 Jul 2025: SEBI passes the interim order under Sections 11(1), 11(4), 11B(1) and 11D of the SEBI Act, 1992, against JSI Investments Pvt Ltd, JSI2 Investments Pvt Ltd, Jane Street Singapore Pte Ltd and Jane Street Asia Trading Ltd. It bars them from the securities market and impounds ₹4,843,57,70,168 (about ₹4,843.6 crore) of alleged unlawful gains (order, paras 62.1 ff.).
• The order describes two alleged patterns, "intra-day index manipulation" and "extended marking the close". On 17 Jan 2024, the most profitable day examined (₹734.93 crore of profit in Bank Nifty options), the group net bought ₹4,370.03 crore of Bank Nifty constituents in cash and stock futures between 09:15:00 and 11:46:59, the largest net buyer by far, while its Bank Nifty option book added a short delta worth ₹32,114.96 crore in cash-equivalent terms, 7.3 times the long cash position; it then sold the constituents for the rest of the session (order, paras 14.7 and 15.3–15.27).
• Over the examination period, 1 Jan 2023 to 31 Mar 2025, the group's total profit across NSE segments was ₹36,502.12 crore: ₹43,289.33 crore in index options and ₹899.99 crore in stock options, against losses of ₹7,208.23 crore in stock futures, ₹190.81 crore in index futures and ₹288.17 crore in cash, that is ₹7,687.21 crore of losses outside options (order, Table 4 and para 14.2). SEBI analysed 18 Bank Nifty trading days in that period in depth (paras 14.4–14.6); the impounded amount sums its computed unlawful gains over 21 days (Table 44): 15 days of the intraday pattern and 3 days of extended marking the close within the period, plus 3 Nifty expiry days of the latter pattern in May 2025.
• 14 Jul 2025: SEBI confirms that Jane Street credited ₹4,843,57,70,168 to an escrow account with a lien in SEBI's favour, "without prejudice" to its rights, and asked for the conditional restrictions to be lifted (SEBI PR 40/2025).
• 21 Jul 2025: with the escrow in place, the directions in paras 62.2–62.5, 62.7, 62.8 and 62.10 of the order cease to apply; the entities remain under a cease-and-desist direction (para 62.12), and the exchanges are directed to monitor their positions until the investigation and any proceedings end (para 62.13) (SEBI PR 45/2025).
• Sep 2025: Jane Street appeals to the Securities Appellate Tribunal (SAT), alleging bias in the probe and seeking investigation documents; SAT asks SEBI to reply within three weeks (Business Standard, 3 Sep 2025; Moneylife; secondary).
• 25 Feb 2026: SAT adjourns the hearing (Business Today; secondary).
• 21 Oct 2026: SAT reserves its order on Jane Street's plea for disclosure of SEBI's documents (unredacted investigation report, NSE correspondence). This concerns access to documents, not the merits; no SAT ruling and no final SEBI order had been reported as of 6 Nov 2026 (NewsBytes; Business Standard topic page; secondary).

Where secondary sources conflict, and what the primary documents settle This is the course's reference timeline; M05 · 5.11 Case 3 and M09 · 9.12.2 refer here. (i) The interim order is dated 3 Jul 2025 (its signature line); some outlets give 4 Jul, the date of the press coverage. (ii) The impounded amount is ₹4,843,57,70,168; press figures of ₹4,843.57 crore truncate it, and dollar conversions range from US$560 to 567 million. (iii) Press reports of a net profit of "about ₹36,671 crore" do not match the order's Table 4 total of ₹36,502.12 crore. (iv) The "18 versus 21 days" conflict is two different counts: 18 Bank Nifty days examined in depth, 21 days in the unlawful-gains computation. (v) Whether, and from exactly when, the group resumed trading after 21 Jul 2025 is not stated in PR 45/2025; reports differ. (vi) A request by Jane Street for further documents reported as "denied" in Nov 2025 appears to be SEBI's refusal to supply them, since SAT was still hearing the disclosure plea in 2026.

Mechanism, analysed with this module's tools Consider a trader who holds both an underlying position (constituent stocks and futures, total delta $D_u$ index units) and an option position on the index with delta $D_o$. For a move $\delta S$ in the index, P&L to first order is $(D_u+D_o)\,\delta S$, plus gamma terms. Two facts drive the alleged strategy.

1. Option prices follow the index through no-arbitrage. Put–call parity (8.5) and the delta of 8.9 tie option prices to the futures and the index. If the index rises, calls reprice up and puts down within seconds, because market makers' models are anchored to the underlying.
2. Price impact is not symmetric in size. Buying ₹4,370 crore of constituents in two and a half hours pushes their prices up. That is a cost to the buyer: the average purchase price is above the starting price, and when the same shares are later sold, the average sale price is below the peak. Viewed alone, the cash leg loses money, which matches the reported losses in cash and futures.

Now combine them. In the morning, with the index pushed up, the trader sells calls and buys puts at prices reflecting the elevated index, building $D_o<0$ with $|D_o|$ much larger than the cash-leg delta. In the afternoon, selling the cash leg pushes the index back down. The option book gains about $|D_o|\times$(index fall), while the cash leg loses its impact cost, which scales with $D_u$. If $|D_o|\gg D_u$, the option gain exceeds the cash loss. Near expiry this is amplified: ATM gamma scales like $1/\sqrt\tau$ (8.9), so a small index move changes ATM option values a lot, and settlement against the closing index value (the "marking the close" allegation) makes the last minutes decisive.

A hypothetical illustration, not the reported positions: on an expiry morning with Bank Nifty at 46,600 and $\tau=0.5/365$ (half a calendar day, the calendar-time convention used throughout this module), an ATM option at 15% volatility and $r=6.5\%$ has delta 0.51 and gamma 0.0015 per point. (On a trading-day clock, $\tau=0.5/252$, the gamma would be 0.0013; 8.7 and 8.10 explain why the time convention matters most for short-dated options.) A book short 10,000 lots of calls and long 10,000 lots of puts (lot 30) has delta $-30\times10{,}000\times(0.51+0.49)\approx-300{,}000$ index units. A 300-point fall earns about ₹9 crore on that book. A cash leg with delta $+30{,}000$ index units loses its round-trip impact cost; if that is, say, 60 index points on average, the loss is about $30{,}000\times60=\text{₹}18$ lakh (₹0.18 crore). The ratio of option delta to cash delta (10 to 1 here) is what makes a loss-making cash trade part of a profitable combined position.

The economics shows why regulators treat this as manipulation rather than arbitrage, if proved. A no-arbitrage price is one no trader can move by trading; this strategy, as alleged, profits precisely by moving the reference price of a derivative market much larger than the cash market that sets it. The 1 Oct 2024 SEBI measures (one weekly expiry per exchange, extra expiry-day ELM, intraday position monitoring) and the 2025 FutEq position limits act on the same mechanism.