Valuing Large Options in the Absence of Collateralisation

Update 16.15pm: BAH - edited plainly incorrect figures resulting from Moody's-like "model error". Incorrect figures left struck out so you can see how daft I was.

Seemingly everyone in (my) blogoshpere has some sort of view on these Berkshire put options so, feeling somewhat left out, here is my potshot.

What is the value of a very large put option, purchased on an equity index, where the owner of the put option does not benefit from collateralisation, that is, the owner is exposed to the credit risk of the option writer? As appears to be the case for Berkshire Hathaway's options sold to (or through) Goldman Sachs, there is a nasty correlation effect in play: as the put option develops more intrinsic value, whoever owns it is increasingly worried about getting paid and BRK's credit spread has blown out.

In his 2007 letter to shareholders, Warren Buffet wrote "the puts in these contracts are exercisable only at their maturity date" and "in all cases we hold the money, which means we have no counterparty risk". Neither of these statements makes it clear BRK has no collateralisation obligations. If Warren were forced to disclose that there are potential cash-calls, perhaps as as a result of credit rating triggers, mark to market losses or some other mechanism, all bets are off - they would be in horrible shape. What follows is based on the hypothesis that there are no such triggers.

Let us assume that BRK sold $40bn notional 20 year puts (over 4 indices) in 2006-2007 at an average equivalent S&P 500 level of 1400. At the prevailing swap rate and dividend yields, and implied volatility of around 24%, this would have realised premia of approximately $4.5bn, close enough to the premia actually received not to worry too much about the exact details of the transactions.

The undiscounted future value of this liability, ie the fair value expectation of payment in 2027, is presently around $33 $19bn. (At the money long dated volatility has expanded to 38%; this option now is well in the money and the skewed volatility for 1400 strike is more like 33%). The present value of this liability, before the impact of credit spreads, is around $19 $10bn using the current swap curve.

So far, so simple. But this valuation does not take account of the credit spread of the writer of the put. If you're not into the dirty technicalities, skip the next bit and move on to "in other words". Otherwise, wrap a wet towel round your head and stay with me.

From put-call parity, we have

P(rf) = C(rf) + X * Z(rf) - S

where Z is a zero coupon bond, S is the spot price, X is the strike and the subscript "rf" denotes an instrument in the absence of credit risk. Let us make a heroic assumption that the issuer is quite positively correlated with equity markets, so a call option written by it has the same value as a risk free call (ie it is highly unlikely that a rising market correlates with BRK going bust), so C(rf)= C(brk), where the subscript "brk" denotes an instrument in the presence of credit risk.

We then have:

P(rf) = C(brk) +X * Z(rf) - S, and using put call parity again, we can replace C(brk)

P(rf) = [P(brk) + S - X * Z(brk)] + X * Z(rf) - S

= P(brk) + X*(Z(rf)-Z(brk))


In other words (are you back?), a put written by BRK without collateralisation is worth less than a fully collateralised put by an amount equal to the notional of the put times the difference between a risk free zero and and a zero valued at BRKs credit spread. With a BRK credit spread of 50bps our credit-risked put is worth 9.1bn, but with current CDS levels of, say, 400bps the credit-risked put is valued considerably considerably less than $10 $5bn! (1)

It will be interesting therefore to see if BRK choose to mark the present value of the puts discounted at their own credit spread in much the same way as banks have realised mark to market gains on their fixed income liabilities as their credit spreads have blown out. (I doubt they will).

Whoever actually still owns these puts/BRK credit risk (GS is still my favourite) likely hedged some credit risk on BRK at inception. Not enough! As equity markets fell and implied volatilities have risen, their credit exposure to BRK has increased enormously. The put owner has been forced into purchasing a lot more credit cover in a nasty cross gamma effect. No wonder BRK's credit spreads have gone bananas; they will likely remain volatile as there is a short cross gamma hedger out there for the next 20 years.

Moreover, as a result of the credit hedger's scramble for CDS protection, his mark to market on the original option is potentially worth only half the value had he been fully collateralised. Time to boot this into the level 3 asset pool I suspect, even if most of the pricing inputs are observable in the interdealer market. Note however the converse is also true - if the market rallies and volatility subsides, the put owner will be long way too much CDS protection on BRK.

Meanwhile, Warren sits there like a contented actuary. He has the cash, he doesn't have to post collateral (we hope), he's comfortable that in 18 years the market will be 65% higher, and he claims he doesn't care about the mark to market risk. I'm not sure he thought it through completely though: even if he has no collateral issues, the action of the person covering themselves against his risk certainly does, and that person is screwing up Warren's ability to finance himself elsewhere.
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(1) This is quite an exaggeration of the haircut in reality - of course if BRK went bust with nil recovery today the put would have zero value and this simplistic model would give it a negative value which is of course impossible. If you wanted to develop this idea further, and I guarantee you the capital structure arbitrage mob do, you would be considerably more rigorous regarding correlations between the credit and equity risk instruments here, but simplicity will suffice for this post.


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