The average volatility of a stock in an index will be less than the index itself. This is because the stock returns are not perfectly positively correlated. When the components of an index have correlation coefficients less than one, the average stock will be more volatile than the whole portfolio of stocks. This is related to the idea of diversification in modern portfolio theory.

Unlike portfolio theory, the owner of a stock option has limited downside. Thus, volatility raises the value of options.

As part of the Capital Purchase Program, banks had to give Treasury warrants on common stock equal in value to 15% of the amount of money invested. Treasury invested $100 million in Old National, so it needed warrants on $15 million worth of common stock. So it got warrants to buy 813,008 shares at an exercise price of $18.45; 813,008 * 18.45 = 15 million, or something very close to it. $18.45 represented the value of the common shares at the time of the investment.

[The math is wrong - the author derived the exercise value not the market value.]

(Ilya Podolyako actually drafted a post about this at the time, but I chose not to publish it because I didn’t want to be hammering Treasury for every little thing they did that helped the banks.

[This bit is editorial - Helped yes, by setting a higher strike, the banks' liability to the Treasury was reduced - but a reasonable argument might be made that the terms under which the warrants were originally issued were made under duress.]

Making some additional assumptions, like zero transaction costs and zero dividends, Fischer Black and Myron Scholes worked out a formula to calculate the value of an option from these parameters (and the risk-free interest rate, since you are looking at the future and money loses value over time), which is now known as the Black-Scholes formula, and has been described as the central pillar, for better or worse, of modern finance. (Nassim Taleb strongly disagrees.)

[This is background - Nassim does not want Black or Scholes names associated with option pricing, because they didn't invent the math, they just figured out how to apply the heat transfer equation to option pricing. By the same logic, the Gaussian (or normal) distribution, should be called the de Moivre distribution. Nassim has some issues.]

In any case, the formula incorporates this useful intuition: To calculate the value of an option, you only need to know the expected value of exercise on the maturity date. This is because, theoretically, that is the only day on which you should ever exercise an option.

[I think these options may be exercised by Treasury before the maturity (called American-style), and thus this statement is incorrect - the boundary condition (essentially, a sell-vs.-hold rule) defining the exercise region must be evaluated at all points in time up to expiration.]

Note that the formula says you can price an option without even having an opinion about the fundamental value of the underlying stock – all you need are its current price and its volatility. This is consistent with a general (though not necessarily correct) principle that stock markets always efficiently price assets,

[This is incorrect in a couple of ways. First, Black-Scholes is "arbitrage-free", so it only requires that the dynamics of the underlying follows geometric Brownian motion - it imposes no requirements on the stock price in relation to fundamental falue. Second, I am pretty confident that my memory is correct that Mark Rubinstein and John Cox had a very cute extension to Black-Scholes in their book Option Pricing from 1979 that included the impact of a subjective evaluation of stock price.]

Also note that the key assumption in the formula is that stock prices will move randomly

[Randomly, in the sense of geometric Brownian motion with drift - so e.g., no discontinuous jumps allowed.]

with constant volatility

[actually with deterministic, not constant volatility - the volatility can be non-constant, just needs to be a deterministic function of time (as long as we are sticking with Black-Scholes, that is).]

You need to know the volatility of the stock price between now and the maturity date, but all you can see is its volatility in the past. This makes option pricing especially difficult right now, because stock price volatility has been much higher over the last eight months than over the previous eight years. (The chart is the implied volatility of the S&P 500 since 2000.)

[The author seems a bit confused about what implied volatility is, or he was not careful in his choice of words. Of course, implied vol is usually interpreted as the market's expectation of future volatility. So this chart is actually a movie of the market's forward looking evaluation of volatility, as it changed over this period of time.]

More fundamentally, using any volatility assumption based on past data falls into the trap of assuming that the future will be like the past. This is never a foolproof assumption, and the longer the timeframe you are looking at, the worse the assumption becomes. It usually may not matter a lot for typical short-dated options (30 days, 60 days, etc.) – unless the world changes during those 30 days – but it matters a lot for long-dated warrants, like the 10-year warrants that Treasury got.

[Probably the author's confusion about what is implied vol versus historical vol prevented him from suggesting that we take the vol surface for the Bank's options at the date we are interested in valuation of the warrants and extrapolate from that forward looking information as necessary to get to the ten year point - a caveat is in order, though, as extrapolation is a very bad thing ordinarily, because the errors are going to be huge in all likelihood.]

Finally, the “Reduction” feature of the TARP warrants throws another wrench into the works. To value the warrants, you have to take into account the fact that half of them could vanish if Old National raises $100 million by issuing stock before the end of the year; and as long as the warrants were outstanding, they had an incentive to raise that money. That involves making guesses about the overall funding climate, and the corporate strategy of Old National, neither of which can be statistically estimated.

[The value per unit is not a function of the "Reduction" feature. The total value is a function of the number of units that are to be sold, of course. It is possible the author is referring to an aspect of warrant valuation that he specifically excluded from the discussion above, specifically that warrants, because they create new shares when exercised, dilute the share price. As a result, their value needs to be adjusted downward for this dilution effect - which problem has been solved a long time ago, and the conditions are easier here, because unlike the general problem of warrant exercise, there is no competitive strategic element to deciding the optimal exercise time (no game to be played against other warrant holders, about who gets less diluted shares by exercising slightly earlier).]

q, a regular commenter here, concludes that the price Treasury got is within the range of reasonableness, given his preferred set of assumptions. However, he also says (agreeing with Nemo) that Treasury should not have negotiated a sale to Old National, but should have simply held onto them until maturity (remember, you don’t want to exercise them early);

[Two things wrong here. First, remember, you may want to exercise them early, if they are American-style, and I think they are. Second, a sale is not an exercise - you will always be willing to sell an option American or any other exercise type, for a fair price (since the fair price incorporates the "premium" for not waiting till expiration). Opinion to follow - The only two reasons for the Treasury not to sell are (1) they couldn't agree a fair price or (2) they want to keep their claws in the bank as long as possible.]

Second, assuming Treasury did not sell the warrants, when Old National bought back its preferred shares, it got the right to buy back the warrants at “fair market value” – but there is no market. (You can get a quote on short-dated options, but not long-dated ones – these are typically over-the-counter.

[Not sure if the author is using accounting-ese here, and saying that because these would be Level III assets, there is no fair value. That is just accounting-ese, however. The fair value is what price the parties _freely_ agree to exchange them for. Don't know, maybe that's just legal-ese. (grin)

Auction participants would know all about option pricing, of course, and would apply a range of assumptions; presumably the sale would go to the buyer with the highest volatility assumption, which would probably (but not certainly) yield a higher price than Treasury got.

[Not sure how the author assesses this likelihood - or the extent to which his editorial biases are involved in the assertion. Nevertheless, it is possible that an auction-determined price might be higher, as a result of a "winner's curse" problem. However, in all fairness, the winner's curse says that the winner overpaid, so if the way that the Treasury can extract more is dependent on assuming they can find a schlub to overpay...]

the model price of long dated warrants depends a lot on your assumptions, and i am choosing assumptions that i think are reasonable here.

i assume a volatility of 25%, and i think this is not a low estimate. my rationale for this is that 100-day volatility for ONB averaged 18% and never went above 27%.

i assume a dividend yield of 3.83 which is the historical dividend yield and which is Wilson’s middle number.

i assume a 30% probability of a stock offering, but note that this offering has a higher probability than a fair sale of these warrants on the open market. i think the treasury would assume a higher probability, perhaps 100%.

i assume a borrow rate of 0.50%. this is a proxy for ‘hedging cost’ or ‘risk premium’ as most private investors would hedge these options (which costs money) or expect a higher expected return from this investment than what they can get for treasury bonds. this 0.50% is added to dividend yield.

the number i get is $1.83 per warrant, or $1.26M.

here is a link to my calculation (ht Sandrew on pointing out that wolfram alpha is capable of this)

http://www78.wolframalpha.com/input/?i=option&a=*C.option-_*Formula.dflt-&a=*FP.FinancialOption.OptionName-_VanillaEuropean&a=*FP.FinancialOption.OptionType-_Call&f4=18.45&f=FinancialOption.StrikePrice_18.45&f5=9.65+years&f=FinancialOption.TimeToExpiration_9.65+years&f6=14.7&f=FinancialOption.UnderlyingPrice_14.7&f7=25%25&f=FinancialOption.Volatility_25%25&f8=4.43&f=FinancialOption.DividendYield_4.43&f9=3.19%25&f=FinancialOption.RiskFreeInterestRate_3.19%25&a=*FVarOpt.1-_***FinancialOption.TimeToExpiration–.***FinancialOption.ExpirationDate-.*FinancialOption.CurrentDate—.*–

The problem, of course, is that no one knew what these things were worth (apparently, we still don’t, and almost 8 months later). If the government bought them early on with overly optimistic assumptions and a rosy baseline scenario – the banks would not merely have been saved – they could have made a killing on worthless securities, with the entire loss distributed to the public.

The warrants were supposed to insure that there would be a mechanism whereby any “undue” gain on the part of the financial institutions, which, presumptively, would be reflected in their share prices over the long term, could be easily recouped by the taxpayer.

But the Treasury didn’t buy the assets from the banks when it recapitalized them, and months later we have yet to remove them through the flawed PPIP apparatus. The evaporation clause might have sensibly extinguished 100% of the warrants “in the event we confuse the entire congress as to what they are doing and don’t actually buy your junk.”

So, since the warrants do not serve their original risk-management purpose, it doesn’t seem unreasonable to me to allow healthy institutions that are willing to fend for themselves to opt-out of government control by paying back every dime they borrowed with interest. The taxpayer has lost nothing on the deal.

An excerpt:

Based on this methodology, Citigroup’s warrants were worth about $166 million on May 20, down from $1.67 billion when issued Oct. 28; Bank of America’s warrants were worth about $221 million, versus $921 million; and Wells Fargo’s, about $1.06 billion, down from $1.48 billion.

On the other hand, this method showed that Uncle Sam has made money on the warrants issued to Goldman, Morgan Stanley and JPMorgan. Goldman’s warrants were calculated to be worth $982 million on May 20, up from $680 million; JPMorgan’s, $1.4 billion, up from $1.2 billion; Morgan Stanley’s, $1.25 billion, versus $410 million. The combined value of those three rose from about $2.3 billion on Oct. 28 to about $3.7 billion, producing a gain of $1.4 billion.

I gather from bits and pieces I remember reading that there is some sort of appraisal process where the bank and Treasury first try to agree on a value, and I believe if that fails then there is supposed to be an auction.

It’s funny how much this paper transaction extemporizing (here the warrants, usually the toxic assets), all trying to maintain the charade of a capitalist “market” where none exists on order to prop up these unviable banks, is redolent of childern pretending to play a game but lacking some of the requisite equipment and just improvising (and improvising the rules as well).

Of course, the banks have their opinion about all this (from the same Fortune article):

Treasury must attempt to liquidate the warrant, the stimulus legislation says. But the ABA decries this as well, saying in its letter that selling the warrant to a third party could unfairly dilute a bank’s shareholders.
In other words, Treasury should just rip up the warrants – even though the warrants were one reason why the banks got investments on such generous terms in the first place. How times have changed since last fall.

Of course no one should have expected the underlying attitudes of the bankers to change. They have always believed and will always believe for as long as they’re allowed to do so that society exists only as a resource for them to mine.

And they absolutely believe society should be grateful for the privilege of serving as this resource.

So it’s not surprising that, since the nominal change in administrations (and what should have been a change in Congress starting in 2007) brought no Change at all where it came to government ideology vis the FIRE sector, “times have changed” in how emboldened the bankers feel in more openly displaying their lack of remorse, their own sense of entitlement and aggrievement, and their ruthless will to continue and intensify their feudal practices.

As Rehoboam said to the supplicants who had hoped he’d spare his father’s whip: “I will chastise you not with whips but with scorpions”.

So none of that’s surprising, given the lack of political Change. What is somewhat surprising, despicably so, is this absolute lack of change, coming from an someone whose fundamental campaign promise was “Change”.