If you are like me, you will enjoy the heck out of Dan McCrum's FT Alphaville series (part 1, part 2, part 3) on Berkshire Hathaway's index put options, but I cannot guarantee that you are like me. I was mildly titillated by his use of the term "negative cross gamma," if that gives you an idea.

I also cannot guarantee that it contains the full truth about Buffett's derivatives. It is based on some papers written by Pablo Triana, a professor at Esade Business School, which speculate a bit about what might be going on with the put options that Berkshire wrote a few years ago. Berkshire's disclosure of those trades is notoriously wispy,1 so the speculation is not conclusive. But it is interesting! Here (and here), for instance, is an effort to reverse-engineer exactly what the trades look like and to figure out their risk sensitivities, which, again, is not something that Berkshire Hathaway itself makes easy.2

Yesterday's post is even more interesting, and even more speculative. As far as I can tell from the related Triana paper, an old Lehman Brothers internal presentation washed up on a beach somewhere and Triana stumbled over it.3 Here is the presentation. The thing that interests Triana and McCrum -- and me, and you, if you're like me -- is a tantalizing slide that seems to say that Buffett sold Lehman what is somewhat romantically referred to as a "rainbow put," or more prosaically as a "Worst of Basket Put Option." Like so:

Triana says "Aided by the boxes and arrows so familiar to derivatives people," and it is a fair point, the boxes and arrows are not doing a lot of work here.
Triana says "Aided by the boxes and arrows so familiar to derivatives people," and it is a fair point, the boxes and arrows are not doing a lot of work here.

This is some pretty racy stuff for a guy who famously went around saying that derivatives are "time bombs" whose model-based valuation "can bring on large-scale mischief." A 20-year vanilla put on the S&P 500 Index is somewhat dicey to value, but more or less has a market-like price.4 A 20-year worst-of put on three different indices in three different currencies is a purely model-driven beast, and it seems unlikely that even Warren Buffett would have a good intuition for what it's worth.

Lehman, on the other hand, had an intuition. Or a model anyway. From the next slide:

Day 1 P/L of this trade was approximately $16.5 million after liquidity valuation adjustments of 0.75 volatility point for equity volatility and 5% correlation for equity / equity correlation. As such, Day 1 MTM was $221.5 million, equal to upfront premium of $205 million plus Day 1 P/L of $16.5 million.

Got that? Lehman cooked up a worst-of put option that was worth $221.5 million to it. Then it convinced Berkshire to sell it that put option for $205 million, making it a (mark-to-market) profit of $16.5 million on the first day.5

Here's what I like -- and, again, you may not be me, which is my loss I'm sure -- about this slide: Berkshire paid Lehman for this trade. I mean, not literally: In this trade, Berkshire sold Lehman a put, and Lehman wrote Berkshire a check for $205 million. (For all of the index put trades that Berkshire did with Lehman and Goldman Sachs, Berkshire was paid a total of $4.5 billion of premium.) But as far as the derivatives structurers at Lehman were concerned, they'd gotten paid $16.5 million to do the deal. After all, they'd been handed something worth $221.5 million, and only paid $205 million for it.

Generally speaking, there are two ways that a bank could find itself doing a trade: By paying for it, or by being paid for it.6 In the first type of trade, the bank is looking to buy something. Here, that something would be protection. The bank actively wants to hedge a risk or whatever, and goes to a deep-pocketed highly rated counterparty and asks it to write the bank some insurance.

It feels intuitive that that would be what Goldman and Lehman were doing with Berkshire. The banks are broadly exposed to equity markets in their business; Berkshire is a giant pile of money that can insure that risk for them; it seems natural that there should be a trade for them to do. Here, for instance, is a guy:

“When people need protection against the downside, Berkshire is the place to get it,” said Tom Russo, a partner at Gardner Russo & Gardner in Lancaster, Pennsylvania, which holds Berkshire stock. “Very few organizations have the capacity.”

But when that happens, the bank will pay for protection. It is buying a service, and the service costs it money. And that's not what happened here. Berkshire's services did not cost Lehman money. Lehman made a $16.5 million profit the day it did this trade.

So it would appear to be the second type of trade: one where the bank is selling a product to a customer. The bank is fulfilling a client demand, and so of course it's getting paid for fulfilling that demand. You buy stock, you pay a commission. You buy call options, you pay the bid/offer spread. You sell put options, you pay the bid/offer spread, in the form of the bank writing you a check for a smaller number than the "fair" price.

The fact that Lehman charged Buffett $16.5 million for this trade means that it's a customer-facilitation trade. (Also the fact that Lehman hedged the trade; the slides say, "The desk has hedged with 10-year OTC options." If Lehman was buying these puts for its own purposes, it presumably wouldn't hedge -- those purposes would be, in effect, to hedge something else.)

This doesn't mean that Buffett literally came to Lehman saying, "here is a trade I want to do, will you do it for me? I will pay you." "Customer facilitation" often means something more like, Lehman salespeople came out to Omaha, proposed this trade to Buffett, talked him into it, and then charged him a nice profit for it.7

By the way, how nice is this profit? It's $16.5 million on a $1 billion notional, or 1.65 percent of notional. Compare Goldman's profit of 2.5 to 6.7 percent of notional on the options it sold to Libya, which Libya is mad about. And those options were shorter-dated (3 years) and pretty vanilla; the Buffett ones run for decades and seem to have been a bit more exotic. If you amortize the $16.5 million over 20 years of (uncollateralized!) derivative exposure, it's only like 8 basis points a year, which is very skinny. Which is not a huge surprise: You're really not supposed to rip off Warren Buffett.8

Anyway, I don't know that any of this is particularly news, but it's interesting to get a little peek at Berkshire's dealings with its banks on these trades. I had a vague sense that Warren Buffett got into them -- as he seems to get into a lot of his trades with banks -- as sort of a favor to the banks, or that his special Sage-of-Omaha status meant he didn't have to do anything as pedestrian as paying banks for his derivatives exposure. Apparently, though, he was just a customer like everyone else. And, like a lot of other customers, he seems to have been unable to resist the occasional exotic derivative.

1 Here's how they were originally disclosed, in Warren Buffett's 2007 letter to shareholders:

The second category of contracts involves various put options we have sold on four stock indices (the S&P 500 plus three foreign indices). These puts had original terms of either 15 or 20 years and were struck at the market. We have received premiums of $4.5 billion, and we recorded a liability at yearend of $4.6 billion. The puts in these contracts are exercisable only at their expiration dates, which occur between 2019 and 2027, and Berkshire will then need to make a payment only if the index in question is quoted at a level below that existing on the day that the put was written. Again, I believe these contracts, in aggregate, will be profitable and that we will, in addition, receive substantial income from our investment of the premiums we hold during the 15- or 20-year period.

Here's the latest 10Q:

The equity index put option contracts are European style options written on four major equity indexes. Future payments, if any, under any given contract will be required if the underlying index value is below the strike price at the contract expiration date. We received the premiums on these contracts in full at the contract inception dates and therefore have no counterparty credit risk. We have written no new contracts since February 2008.

The aggregate intrinsic value (which is the undiscounted liability assuming the contracts are settled based on the index values and foreign currency exchange rates as of the balance sheet date) of our equity index put option contracts was approximately $2.3 billion at September 30, 2013 and $3.9 billion at December 31, 2012. However, these contracts may not be unilaterally terminated or fully settled before the expiration dates which occur between June 2018 and January 2026. Therefore, the ultimate amount of cash basis gains or losses on these contracts will not be determined for many years. The remaining weighted average life of all contracts was approximately 7.25 years at September 30, 2013.

2 And it is not an entirely successful effort, in that Berkshire's mark-to-market losses are less than Professor Triana would have expected, which means ... something. Most straightforwardly, that the trades are not what Professor Triana thought they were. Or that Berkshire's valuation is idiosyncratic. Or ... something.

Also, lotta rho in these trades. Surprising (to me) amounts of rho. "Lotta rho" is not something I say all that often, so I'm savoring it down here in this footnote.

3 Its provenance is not really discussed but presumably it's from the voluminous Lehman bankruptcy evidence. (Here's one stash, and here is the mother lode; if you can find this deck in the Jenner repository you win a cookie.) From the face of it, it is from a January 16, 2008, "Capital Markets and IBD Finance -- Offsite," and by the Valuation and Control Group. Man, the Lehman valuation group in January 2008. Wouldn't it be fun if, like, page 17 said "yeah, we've stopped having any controls, let's just all go nuts!"? It doesn't, though, it's pretty boring.

4 There's interesting discussion in the FT Alphaville comments about how Berkshire values the index puts as Level 3 assets. The idea is "vanilla S&P puts have a market price so it looks dicey to treat them as Level 3." Of course if they're three-currency three-index worst-of options then that argument goes away. But even vanilla index puts strike me as pretty Level 3 if they have a 20 year tenor; it's not like 20-year put options regularly trade.

5 Plus some adjustments. The idea is basically that the model tells Lehman that the thing is theoretically worth $30 million, or whatever, but then it haircuts that model for liquidity risks that prevent it from realizing that full theoretical value. I don't know exactly what those haircuts are worth. If I price up a vanilla $1 billion notional 20-year at-the-money S&P 500 index put in Bloomberg OV, I get $7.9 million of vega, so 0.75 volatility point is worth about $6 million. Assuming the same thing was true in 2007 (meh?), and that the actual rainbow put had similar vega to a vanilla S&P put (meh??), and that the correlation adjustment was of the same order of magnitude as the volatility adjustment (meh???????), the theoretical day-one profit to Lehman, before liquidity adjustments, may have been around $25-30 million.

6 Where "paying" means "paying the bid/offer spread," more or less, not literally writing a check. A whole lot of the business is about finding ways to be paid without the client writing you a check. Clients enjoy not writing checks, and banks enjoy being paid.

Arguably there is a third flavor, too, which is a pure arms'-length transaction where each side thinks the other is paying the spread. So if Lehman trades with Goldman, say, or maybe Citadel. This is sort of a degenerate case of the other two though: Even smart hedge funds who think that they're buying cheap from their bank also understand that their bank is charging them a spread.

7 There are things to say about the Volcker Rule here I guess? We talked a while back about Greg Smith's interpretation of the Volcker Rule:

Market making should be when a customer approaches a bank to facilitate a trade. Today, the bank decides what trade it wants to do -- then it lines up clients to take the other side of the trade. This is no longer market making in my view -- it is akin to proprietary trading.

This is not the approach taken by, you know, the Volcker Rule, but what do you think about it? Does it matter whether Lehman came to Buffett to try to snooker him into this trade, or whether Buffett came to Lehman begging them to put him into a 20-year rainbow index put? Does this trade feel prop? Does anything regulatory turn on whether it feels prop? Should anything?

8 On the other hand, if you've got a model for what this is worth, and he doesn't, why price so tight? Presumably because Lehman and Goldman and maybe others are bidding against each other for these trades? And you'd hate to be the guy who lost the Warren Buffett trade by pricing too wide.

To contact the writer of this article: Matt Levine at mlevine51@bloomberg.net.

To contact the editor responsible for this article: Tobin Harshaw at tharshaw@bloomberg.net.