Are There Flaws In Options Pricing?

[Martinghoul]

Quote from VGSSD:
You are right that put call parity doesn't follow from the validity of the risk free rate, it follows from no arbitrage.

However, the argument does hinge pretty heavily on the existence of some form of safe investment of capital. If the economy were in a truly horrible state at which there was some sort of nontrivial probability of default on even the so called safest bonds, one would need to re-think the put call parity argument. The replicating portfolio would no longer stay positive with probability one.

This is incorrect... The probability of default is fully priced into the funding rate used to build the replicating portfolio. This rate going bonkers (the way LIBOR did in 2007-08) doesn't mean that put-call parity needs re-thinking.

The argument that you could make that would have implications for put-call parity is a different one. Specifically, you could claim that the potential event of default causes mkts to become incomplete, in which case put-call parity might not hold. However, the completeness of mkt assumption is such a funny beast anyways, it doesn't really matter.

 

[rew]

Quote from dmo:

You can make of that what you wish. Warren Buffett a few years ago looked at the fact that the DJIA had risen from about 40 in 1932 to about 14,000 in 2000 and decided the "random walk" assumption for stocks is absurd and that OTM puts priced under a random walk assumption are way overpriced. He put his money where his mouth was, and sold some several billion dollars worth of FOTM SPX long-term puts. Time will tell if he was right or wrong.

The assumption in option pricing is that the log price of the underlying is a random walk with a constant drift. In terms if the price, that means that you can have random motion around an exponential increase in price. In fact, put-call parity forces the underlying rate of growth to be the same as the risk free rate.

So the fact that the DJIA has grown enormously in nominal dollars over many decades is not in itself inconsistent with the assumptions used in pricing options.

 

[VGSSD]

Quote from Martinghoul:

This is incorrect... The probability of default is fully priced into the funding rate used to build the replicating portfolio. This rate going bonkers (the way LIBOR did in 2007-08) doesn't mean that put-call parity needs re-thinking.

The argument that you could make that would have implications for put-call parity is a different one. Specifically, you could claim that the potential event of default causes mkts to become incomplete, in which case put-call parity might not hold. However, the completeness of mkt assumption is such a funny beast anyways, it doesn't really matter.

Perhaps we are thinking of different ways to derive the put call parity relation, but I was referring to the most elementary MBA level argument where one literally assumes that you can put 10 dollars at the bank and have it come out at 10(1+r) a year later. How is the possibility of default account for here? It seems to me that one is assuming there is no possibility of default in this sort of argument. Perhaps what you mean is that one can start with a different model of the short rate and still get put call parity, in which case you are probably correct.

 

[dmo]

Quote from rew:

The assumption in option pricing is that the log price of the underlying is a random walk with a constant drift. In terms if the price, that means that you can have random motion around an exponential increase in price. In fact, put-call parity forces the underlying rate of growth to be the same as the risk free rate.

So the fact that the DJIA has grown enormously in nominal dollars over many decades is not in itself inconsistent with the assumptions used in pricing options.

In his letter to shareholders, Buffett explains that due to the effect of inflation and retained earnings over time, the probability of stocks dropping over the long term is far less than the probability of stocks rising. Therefore, according to him, the Black/Scholes model produces "absurd results" when applied to long-term options (15-20 years in his case).

Buffett doesn't specifically mention the term "lognormal distribution" in his letter, but it seems clear to me that's the assumption he finds absurd in the case of long-term S&P 500 options.

 

[Martinghoul]

Quote from VGSSD:
Perhaps we are thinking of different ways to derive the put call parity relation, but I was referring to the most elementary MBA level argument where one literally assumes that you can put 10 dollars at the bank and have it come out at 10(1+r) a year later. How is the possibility of default account for here? It seems to me that one is assuming there is no possibility of default in this sort of argument. Perhaps what you mean is that one can start with a different model of the short rate and still get put call parity, in which case you are probably correct.

The replicating portfolio argument and all those other "MBA-level arguments" are all sort of "locally linear" concepts, i.e. all the conclusions are established as the time intervals and price movements tend to 0. So as long as you assume mkt completeness over these infinitesimally small timeframes (not for a year) for all your assets, including the funding, you're OK, regardless of whether the rate implies some probability of default. However, all no arbitrage arguments, including put-call parity, do fall apart if the whole completeness assumption is broken. At any rate, that's how I understand it.

 

[atticus]

Quote from dmo:

You can make of that what you wish. Warren Buffett a few years ago looked at the fact that the DJIA had risen from about 40 in 1932 to about 14,000 in 2000 and decided the "random walk" assumption for stocks is absurd and that OTM puts priced under a random walk assumption are way overpriced. He put his money where his mouth was, and sold some several billion dollars worth of FOTM SPX long-term puts. Time will tell if he was right or wrong.

Actually he sold ATM based upon the prevailing mark on the day the deals were struck. He failed to report a >$5B marked-loss to shareholders. He used a similar ploy when failing to mark equity option losses to his book because, as he stated (paraphrasing), the fundamental outlook of these companies remains strong".

He eventually reported the MTM and rolled into 800-900 strike puts on SPX, but only after the SPX had dropped 600 points.

These were Euro-style vanilla puts and were not subject to any clearing or settlements, so the counter-parties could only net the contract with another party or at Buffett's request for an offset. The banks involved did in fact net the Buffett puts... the banks had been hedged from inception (from 12% vol) in long straddles, which was the natural hedge in the Buffett ATM short puts (bank longs). I have heard of one bank/dealer that covered the long synthetic straddle at better than a double in 18 months.

 

[dmo]

Quote from atticus:

Actually he sold ATM based upon the prevailing mark on the day the deals were struck. He failed to report a >$5B marked-loss to shareholders. He used a similar ploy when failing to mark equity option losses to his book because, as he stated (paraphrasing), the fundamental outlook of these companies remains strong".

He eventually reported the MTM and rolled into 800-900 strike puts on SPX, but only after the SPX had dropped 600 points.

These were Euro-style vanilla puts and were not subject to any clearing or settlements, so the counter-parties could only net the contract with another party or at Buffett's request for an offset. The banks involved did in fact net the Buffett puts... the banks had been hedged from inception (from 12% vol) in long straddles, which was the natural hedge in the Buffett ATM short puts (bank longs). I have heard of one bank/dealer that covered the long synthetic straddle at better than a double in 18 months.

Right, they were ATM puts. My bad.

That's funny about his failing to report the loss, especially since he made a big point in his letter to shareholders about how he would value the options using B/S pricing even though he disagreed with it. Add that to his shameless and very self-interested defense of Moody's, and I wonder how he maintains his paragon-of-virtue status.

 

[rew]

Quote from dmo:

In his letter to shareholders, Buffett explains that due to the effect of inflation and retained earnings over time, the probability of stocks dropping over the long term is far less than the probability of stocks rising. Therefore, according to him, the Black/Scholes model produces "absurd results" when applied to long-term options (15-20 years in his case).

Buffett doesn't specifically mention the term "lognormal distribution" in his letter, but it seems clear to me that's the assumption he finds absurd in the case of long-term S&P 500 options.

One legitimate issue is that BS necessarily assumes that the underlying asset grows on average at the risk free rate. (This is enforced by put-call parity.) Of course futures pricing does the same thing -- the no arbitrage future price of a non-dividend paying stock is its current price times exp(rT). But people expect that the average long term growth of stocks is greater than the risk free rate, otherwise nobody would ever invest in them (they'd buy Treasury bonds instead). So, on one hand we have the principle that higher risk demands higher average returns, and on the other hand the no arbitrage price for derivatives requires that stock prices be modeled as if they were growing at the risk free rate. Maybe that is what Buffet was complaining about. Any comments from the local quants as to how this particular circle is squared?

 

[VGSSD]

Quote from rew:

One legitimate issue is that BS necessarily assumes that the underlying asset grows on average at the risk free rate. (This is enforced by put-call parity.) Of course futures pricing does the same thing -- the no arbitrage future price of a non-dividend paying stock is its current price times exp(rT). But people expect that the average long term growth of stocks is greater than the risk free rate, otherwise nobody would ever invest in them (they'd buy Treasury bonds instead). So, on one hand we have the principle that higher risk demands higher average returns, and on the other hand the no arbitrage price for derivatives requires that stock prices be modeled as if they were growing at the risk free rate. Maybe that is what Buffet was complaining about. Any comments from the local quants as to how this particular circle is squared?

There are no issues here, BS does not assume stocks grow at the risk free rate. Stocks are free to grow as they please, the point is that true growth rates are actually irrelevant from a no arbitrage pricing perspective.

This is a fairly common misconception I think.

 

[Emilio_Lizardo]

Quote from rew:

BS necessarily assumes that the underlying asset grows on average at the risk free rate.

Not true. Usual hypothesis is that mu = r + (sigma^2)/2.

The condition mu = r corresponds to the process S(t)B(t)^-1 being a martingale -- a fairly implausible assumption.