Instability and the cover rule

In the old days of equity derivatives, one of the main instruments was the covered warrant. This was a call option (usually - put warrants are uncommon) issued by a bank and traded as a security: it gave the holder the right but not the obligation to buy some number of underlying shares at a fixed price. This market still exists, and is reasonably active in some countries.

The term 'covered' come from the early regulatory framework. In order to prove to the exchange that the issuer of the warrant could meet their obligations, they had to keep some or all of the underlying. This position in the underlying was known as the cover: it ensured that if the warrant ended up in the money, the issuer could deliver shares to the warrant holders. (Obviously if a corporate issues warrants on itself, then there is no problem: an entity can always print more of its own shares. The issue arises when a bank issues warrants referencing shares in another corporation, often without that corporation's permission or support.)

The cover requirements were supposed to ensure that squeezes did not happen whereby the issuer was forced to pay higher and higher prices to buy shares against the warrants that they hold. This is an issue with less liquid underlyings, especially ones where most of the liquidity is controlled by a small number of parties. By forcing banks to buy the underlying before issueing the warrant, exchanges made market disruption much less likely.

There is definitely something that we can learn from this piece of market history. When derivatives traders are forced by regulation to have a matching position in the underlying, then:

• There is a natural limit on the size of the market;
  
• Both derivatives and underlying markets are more orderly; and
  
• The issuer's risk is automatically limited.

When that is not the case, things can get a little crazy. Let's look at two examples.

The first is the CDS market. I am a supporter of this market, and I view many of those who wish to limit CDS trading as uninformed, hysterical or both. (People called Gillian who have a book to plug may well fall into this category.) However, there is one reasonable objection to CDS, and that is that it sometimes allows the tail (the derivatives market) to wag the dog (the underlying bond or loan market). I have no objection to letting people short credits, but doing so by CDS can provide more protection sellers than there are bonds, creating exactly the sort of squeeze post default that the cover requirements eliminated for warrants. The lack of a borrow market for corporate bonds is the real culprit here. Perhaps one solution would be to keep the CDS market as is, but to require that naked shorts pay a credit borrow fee to a holder of a deliverable instrument. This fee would be in exchange for the bond or loan holder agreeing not to buy protection on it or lend it to anyone else: the fee would automatically ensure that no more CDS protection was sold than there were bonds (or loans) extant which would at least make it more likely that the CDS settlement was orderly.

Second, the commodities market, specifically oil. This post was inspired by a fascinating article on the oil market from the Oil Drum (via FT alphaville). One part of the author's arguments is that the existence of an enormous market in financial contracts on oil has resulted in considerable price volatility - perhaps even price manipulation - which is in the interests neither of producers nor consumers of physical oil, but which benefits intermediaries such as the investment banks considerably. Certainly if one believed that this is true - and the evidence is impressive - then again the solution is obvious: require all derivatives positions to have a physical hedge. If you are short, then you have to own the underlying. If you are long, then you have to borrow the underlying. A given barrel of oil can act as the hedge for just one contract. And you can only use deliverable oil - stuff in tanks - not oil that is still in the ground.

Cash up front please

The Telegraph reports an idea of Paul Tucker's: making the banks pay to clean up their own mess:

The banking industry could be told in advance that, if ever there was another crisis, the ultimate cost would come from banks themselves. In the midst of a crisis, that would not be possible. A government would have to pump in new equity. But when the dust had settled and the government had sold its shares, the loss (if any) could be calculated - and then collected from the industry via a levy.

This isn't a bad idea. But there is a better one. Make them pay before the crisis.

There are various ways to do this. One is to take cash from the banks, via a beefed up version of the way the FDIC works. In order to be a financial institution, you need an annually renewable license, and the license should be expensive.

A more intriguing one, though, is to make the banks hand over each year not cash, but one year call options on their stock. The regulator would then hedge these options. The bank's shareholders would only be diluted if the stock went up, sugaring the pill for them, while the hedging process would ensure the regulator made money whether the stock went up or down. Indeed, as the position is long gamma, a big fall would be particularly profitable to the hedging strategy.

On Black Scholes Hedging

Further to yesterday's post about CB arb, a short note on hedging options.

If you sell an option at an implied vol of v, and the Black-Scholes assumptions hold (in particular the underlying is a diffusion with constant delivered volatility w, and you can trade instantaneously at zero spread) then the expected P/L of a delta hedged position depends only on v and w. In fact, as Dupire amongst others pointed out, it depends on v versus the gamma weighted delivered volatility. Thus, on average, if you sell an option at v, and v > w, you will make a profit. If you buy an option at v and v < w, you similarly expect to profit.

Two things can screw you up. Firstly this result only holds if the underlying is a diffusion. Therefore in the real world, with jumps, you can buy a 'cheap' option (i.e. one whose implied is less than realised) and still lose money on hedging. Secondly all the other imperfections (bid/offer spreads, variable interest rates, stock borrow costs etc.) hurt, so in practice you need at least a 2 vol point difference between realised and implied to have a good chance of a profit.

Thus, to return to yesterday, you will judge a CB to be cheap if the implied vol needed to recover the price of the CB is significantly less than your expectation of future delivered vol of the underlying during its life. If you are right, you can make money delta hedging the embedded option. Before 2004 or so CBs were often cheap - the issuers discounted them a bit to be sure of getting the issue off - and so CB arb could be profitable. The problem is that buying a CB to get the option is inherently deleveraged, since the option is only a fraction of the total (the rest being the bond floor). Hence callable convertible asset swaps. But that is a story for another day.

Risk and climate change policy

Just before I went away Paul Krugman posted on the economics of catastrophe, and I have been meaning to follow up for a while. Krugman picks up on some research by Marty Weitzman looking at the distribution of outcomes of climate change. The basic idea is to look at the uncertainties and hence come up not just with a single prediction of the path of global average temperatures, but a path of distributions. There is a lot of model risk in this analysis - it is bad enough predicting financial distributions were we do at least have a lot of high frequency data - however the results are interesting. Krugman says:

Marty surveys the existing climate models, and suggests that they give about a 1% probability to truly catastrophic change, say a 20-degree centigrade rise in average temperature.

Twenty degrees would be game over. Even if it is only 0.01% chance, this is an outcome worth hedging. Clearly then it is not just the expected temperature change that we should be concerned with, it is the variance of that change, or more accurately the upside tail of the distribution. As Krugman says, mobilizing people to protect against low probability but catastrophic outcomes is crucial. Hedging far from the money is cheap, but you do actually have to buy those options.

What does delta hedging a tranche mean?

Some old research from, of all people, Bear Stearns, makes fascinating reading. It is about delta hedging CDX and iTraxx tranches, just about the simplest possible hedging problem in structured credit (in that index itself is the hedge, and both that and the tranches are liquid).

Suppose we have sold a tranche of the CDX. What it the delta with respect to the index? The standard definition would say something like

delta = (price of tranche at index spread plus 1bp - price of tranche at index spread) / 1bp

But there is a hidden correlation assumption: we calculate this delta at constant base correlation. Thus delta hedging will only be P/L minimising if

• spread movements are small;
• rehedging is possible after a small spread movement; and
• base correlation remains constant.

The first two assumptions have not held recently with even the hitherto liquid CDX and iTraxx displaying jumps and illiquidity. But interestingly even back in 2004 the last one was known not to hold either. Here is the tracking error of delta hedging each of the CDX tranches from the Bear's research:
And here are the realised deltas (i.e. I think the best deltas ex post) vs. the calculated ones (ex ante from the model):
And remember, that is the easiest hedge in structured credit. If the simplest position to hedge when the market was not particularly troubled gives you 3% tracking errors, what is it like trying to delta hedge a bespoke hybrid CDO at the moment?