In addition, the Black-Scholes calculations incorporate volatility inputs which are generally not observable. At March 31, 2008, the estimated fair value of these contracts was $6,171 million and the weighted average volatility was 22.8%. The impact on fair value from changes to volatility is summarized below. The values of contracts in an actual exchange are affected by market conditions and perceptions of the buyers and sellers. Actual values in an exchange may differ significantly from the values produced by any mathematical model. Dollars are in millions.

Hypothetical change in volatility (percentage points) Hypothetical fair value

Increase 2 percentage points $ 6,777

Increase 4 percentage points $ 7,379

Decrease 2 percentage points $ 5,563

Decrease 4 percentage points $ 4,956

Let's assume, as I think is the case, that the four indices which are underlying the puts are: SP500, Nikkei, Ftse, and Eurostoxx50. I'm pretty sure that the puts at initiation were 100% (or less) of the spot on the strike date, i.e. the inception date. And for the sake of simplicity, suppose the duration remaining on the options is 15 years. If these assumptions are correct, then it looks like Berkshire was undervaluing the puts by using a too-low volatility on March 31.

Why do I say this? There are two possible kinds of volatility to use when marking options: historical and implicit volatility. To make things simple (for me anyway), use the formula p = f(v) to represent the relationship between the option price p and the volatility v. f is an increasing function; when v1 > v2, then f(v1) >= f(v2), and equality holds only in trivial cases. If p is the market price of an option, then there will exist a unique v such that p = f(v). v is called the implicit volatility. If you trade an option and mark it implicitly, then your profit and loss on the day of trade should be close to 0.

Historic volatility is calculated by looking at the day-to-day variation on the index, over some pre-determined time period. Only institutions who want to die early deaths mark with historical volatility. The old UBS tried it before being taken over by SBC, and one of the reasons they had to be taken over by SBC was because they were marking some or all of their books with historical volatility. They exploded once historical volatility started trending sharply higher, but they couldn't buy back their position, except at a huge loss, because implicit vol was even higher.

I can imagine Berkshire actually using historic volatility; it's what people who are external to option trading would think you're supposed to do. It's what an insurer would think you're supposed to do. If it does, then I understand how they got into this extraordinary position of selling all those puts. Historical vol is at 15, but these idiot traders are pricing it at 24! What a great opportunity to make 9 points! Meanwhile, the option traders are selling the volatility they bought from Berkshire at 25 (or higher...), and they return to Berkshire to get some more. Rinse and repeat. Eventually Berkshire has to stop selling. When it does, implicit volatility explodes - the greatest fool is done, and there's no one left to sell. Berkshire can't buy back its position, except at a huge loss, so it has to sit on it, and pray the market goes its way. Praying, though, is a lousy trading strategy.

I've been sidetracked. The essential point is: if you want to get the market price of the option - what it's really worth - you need to use implicit vol. For the rest of the post, "volatility" will mean "implicit volatility".

Volatility depends on strike; the higher the strike, the lower the volatility. To have got a 22.8 volatility on the 15-year SP500 on March 31st you'd have to go almost to a 200% strike. On the Nikkei the lowest volatility on any reasonable strike was more than 3 points higher than 22.8. On the FTSE you'd have needed a 150% strike. And on the Eurostoxx you'd have needed a 130% strike. It looks best on the Eurostoxx, but the all-time high of the index was 4558, which is only 125% of the index value on 31 March.

So, even if Berkshire sold *all* its puts only on the Eurostoxx and only at the all-time high of the index, you can't justify a 22.8% volatility. It's at least a point off. And if you assume any kind of reasonable weighting on the indices and strike prices that are sometimes below the all-time highs on the index, you end up concluding that the volatility is undervalued by 2 to 4 points.

In any case, that was then. Volatility has exploded since March 31, up say 6 to 7 points on the same strike. So my best guess is that, volatility has gone up by 8 to 11 points compared to Berkshire's markings. Using its handy table, that turns out to be a loss of around 3 billion Usd since 31st March *only on the volatility marking*. It took a bath as well on the lower spot.

So Berkshire is under serious water here. Its position is too big; it won't be able to buy it back, at any price. If I were a better man, I'd feel sorry for it.

ADDITIONAL THOUGHT: A Bloomberg article tells us that Berkshire's maximum exposure has decreased recently to 35.5 billion Usd. This must be because of the rise of the dollar. So, inversely, if the dollar were to fall, Berkshire's exposure in terms of dollars would increase - and is theoretically unlimited.

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