There were not enough options in the world. That is the single most arresting finding in Carole Bernard and Phelim P. Boyle's 2009 paper in the Journal of Derivatives, and it is the place a technically inclined reader should begin. By the mid-2000s, the notional size of Bernard L. Madoff Investment Securities LLC ("BLMIS") advisory book was so large that, had Madoff actually executed the options leg of his stated strategy on the listed market, the contracts required would have exceeded the entire open interest on the S&P 100 index option (ticker OEX). The strategy he described could not have been run, by him or anyone else, at the size he claimed to be running it. Markopolos reached the same conclusion by inspection in 2000; Bernard and Boyle formalize it in 2009 (see [markopolos-no-one-would-listen]).
Carole Bernard & Phelim P. Boyle, Mr. Madoff's Amazing Returns: An Analysis of the Split-Strike Conversion Strategy, 17 J. Derivatives 62 (2009), is the closest thing the Madoff literature has to a forensic-mathematical autopsy. It is indispensable reading for a law student because it demonstrates that the fraud was visible not only to the qualitative skeptic but to anyone who would do the arithmetic — and, crucially, that the arithmetic was undergraduate-level.
The Strategy, Glossed
Madoff claimed to run a split-strike conversion (buying a basket of large-cap equities, selling out-of-the-money index calls, and buying out-of-the-money index puts). The premium from the sold calls funded the purchased puts; the puts capped the downside; the calls capped the upside. A collared portfolio is, by construction, a deterministic payoff function of the underlying index over the holding period, up to the basis between the stock basket and the index itself. That determinism is what makes the strategy unusually easy to audit from the outside and unusually difficult to fake from the inside.
A Sidebar for the Non-Quant Reader
- No-arbitrage bound. A pricing constraint derived from the principle that two portfolios with identical payoffs must trade at identical prices. If they did not, a trader could buy the cheaper and sell the dearer for a riskless profit — an arbitrage — and the market would close the gap. Applied to a collar, no-arbitrage logic yields a closed-form upper and lower limit on what the strategy can earn in any month, given the realized path of the index.
- Put-call parity. The identity that, for European options on a non-dividend-paying asset, a long call plus a short put at the same strike replicates a forward position in the underlying. Parity is the structural backbone the bounds rest on: deviations from parity in the listed-options market are vanishingly small, so any reported return inconsistent with parity is, in the strict sense, impossible.
- Implied volatility. The volatility figure that, plugged into the Black-Scholes formula, reproduces the observed market price of an option. Implied volatilities for OEX options were directly observable throughout the period Bernard and Boyle study, removing any excuse for guessing at option premia.
- Split-strike conversion. See the canonical gloss in [markopolos-no-one-would-listen]; the term refers to the long-stock, long-put, short-call structure Madoff described to investors and to the SEC.
The Closed-Form Bounds
The heart of the paper is a derivation of upper and lower bounds on the monthly return a split-strike conversion could have generated in each month from December 1990 through May 2008, using actual S&P 100 closing prices and observed OEX option prices. Bernard and Boyle treat the strategy charitably. They let the hypothetical manager pick the best feasible strikes after the fact — a generous concession that biases the bounds toward Madoff — and they allow the stock basket to track the index with realistic slippage. Even with the thumb on the scale, the result is stark: in a meaningful fraction of months, Madoff's reported return lies above the upper bound. The reported returns are not merely lucky; they are inconsistent with the no-arbitrage structure of the claimed payoff. In derivatives language, Madoff was reporting a free lunch that the market did not serve.
Two further constraints sharpen the impossibility. First, the open-interest ceiling already noted: the listed OEX market simply did not contain enough contracts. Second, an over-the-counter explanation — that Madoff had bilateral options with unnamed European counterparties — collapses on inspection, because no dealer's books showed exposures of the relevant size, and the implied counterparty risk would have been a disclosable concentration. The paper's complementary returns-based style analysis is taken up at greater length in the next-but-one chapter (see [clauss-roncalli-weisang-risk-lessons]), which shows that even loosened to a generic equity-with-options style, Madoff's returns fit no factor model the public data can construct.
The Smoothness Problem
Bernard and Boyle's second contribution concerns the shape of the return distribution rather than its level. A collar reduces variance relative to the underlying index but does not eliminate it; the collared portfolio still moves with the market within the strike corridor, and so its monthly returns must inherit a substantial portion of the index's monthly volatility. The authors compute what that residual volatility ought to look like and compare it to what BLMIS actually reported through its flagship feeder, Fairfield Sentry Limited ("Sentry").
The reported standard deviation is implausibly low. The correlation with the S&P 100 is too weak for a manager allegedly holding the index basket, and the frequency of losing months is too small for a strategy whose downside protection is, by construction, partial. The smoothness is the tell. A real options-collar return stream is bumpy in characteristic ways. Madoff's was sandpapered. For a reader interested in the law of materiality, this is the empirical core: the misrepresentations were not buried in footnotes but visible in the moments of the return distribution itself.
The Doctrinal Hand-Off
Bernard and Boyle do not write as lawyers, but their bounds reverberate through several corners of the case. Section 206 of the Investment Advisers Act of 1940, 15 U.S.C. § 80b-6, imposes a fiduciary antifraud duty on registered investment advisers; Rule 10b-5, 17 C.F.R. § 240.10b-5, forbids material misstatements in connection with the purchase or sale of any security. The quantitative work supplies the evidentiary spine for both theories: the returns BLMIS reported were not exaggerated but structurally impossible given the strategy the firm described. The auditors who blessed those returns — taken up in the next chapter (see [fuerman-solo-auditor-red-flag]) — had access to the same closing prices and the same option chains. The SEC's missed opportunities, catalogued in Investigation of Failure of the SEC to Uncover Bernard Madoff's Ponzi Scheme, Report No. OIG-509 (Aug. 31, 2009), look worse, not better, once one accepts the Bernard-Boyle premise that the analytical tools required were neither novel nor proprietary.
A note on the pull quote above: the language is a close paraphrase of Bernard and Boyle's discussion of the open-interest constraint (Bernard & Boyle, supra, at 70-71). Where this chapter quotes the paper, it does so verbatim; where it summarizes, it brackets or rewords.
Doctrinal Takeaway
For the JD or LLM reader, the chapter's payoff is the doctrine of inquiry notice under the federal securities laws — the rule that a reasonable investor, presented with storm warnings sufficient to suggest the probability of fraud, is charged with the diligence to investigate (Merck & Co. v. Reynolds, 559 U.S. 633 (2010)). Bernard and Boyle establish that mathematical impossibility, derivable from public closing prices by a single afternoon's work, was among the storm warnings available to every feeder fund, every auditor, and every examiner from 1992 forward. If quantitative replication is now understood as basic due diligence, its omission ought to satisfy inquiry notice as a matter of law, tolling no statute and excusing no fiduciary. The operational-due-diligence chapters that follow take up what that standard demands in practice.