Buffett alternative to Black Scholes model

[nickenumbers]

I have seen where Buffett is not a fan of using the Black Scholes model for valuing options that are long dated.  I believe that his issue is with the volatility input as the long dated nature of some options make the volatility a little less relevant.

 

Has anyone seen a model or a calculator for long dated options that Buffett prefers?

 

 

I am a little fuzzy on this next comment, but I think Buffett said that the valuation of PUT options over the long run was more wrong than the value of CALLs but, I am not 100% on that.

 

Thoughts, input, opinions?

 

:-* 8) :P

[LC]

1 - there are ways to value long dated derivatives taking this into account

 

2 - this may be an interesting starter read: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.613.1657&rep=rep1&type=pdf

 

 

[wabuffo]

Has anyone seen a model or a calculator for long dated options that Buffett prefers?

 

I vaguely remember a quote from Buffett from the late 90s when he was railing about forcing companies to expense the cost of their employee options on the income statement.  He used an example that Berkshire/Buffett would just replace the 10-year employee options issued annually for cash at about a third of their strike price when BRK made an acquisition of a company with employee options.

 

At the time, I worked out similar numbers (using a 7% long-term borrowing cost and a 35% tax rate).  Buffett thinks of it as being economically similar to the company borrowing the full cost of the shares at the strike price at the time of issuance.  If you then bring the total cost of the interest paid on the loan over the ten years by the company back to present-value and deduct taxes, you get a good estimate for the value given up by the Company in issuing the options every year.

 

Today, at long-term borrowing costs of 4% and a 21% tax rate, the cost would be lower -- probably around a quarter of the strike price over 10 years.

 

Of course, he was talking generically and not specifically about overvalued or undervalued situations.

 

wabuffo

[SHDL]

Buffett was basically saying that he doesn’t think a log-normal diffusion is a good model for the S&P 500.  As far as I know he’s never explicitly stated which stochastic process he prefers, but I imagine it’s something closer to an exponential trend plus a mean reverting process.  (At least that is what most value investors are essentially using, even though they may not be able to articulate their beliefs as such.)

[Hielko]

I am a little fuzzy on this next comment, but I think Buffett said that the valuation of PUT options over the long run was more wrong than the value of CALLs but, I am not 100% on that.

 

Thoughts, input, opinions?

 

:-* 8) :P

If Buffett would have said that, which I have no clue about, he would have been wrong since there is a thing called Put-Call parity.

[rb]

I am a little fuzzy on this next comment, but I think Buffett said that the valuation of PUT options over the long run was more wrong than the value of CALLs but, I am not 100% on that.

 

Thoughts, input, opinions?

 

:-* 8) :P

If Buffett would have said that, which I have no clue about, he would have been wrong since there is a thing called Put-Call parity.

Yea the prices of calls and puts are linked through put-call parity. So if one is correct so is the other, and vice-versa.

 

In fact there is no Black-Scholes formula for puts. Black-Scholes only calculates for the price of calls. The price of puts is determined through put-call parity.

[rb]

I wouldn't be surprised if Buffett doesn't like Black-Scholes-Merton model (BSM). Buffett is a value guy and BSM does not calculate the value of options.

 

What BSM does is calculate the price of an option through an arbitrage-free framework. What you get is actually the cost of the option, and therefore it's price. To put this another way, if there was no active market for the underlying shares then the price of options would very likely be different from BSM.

[Cigarbutt]

nickenumbers,

 

The issue you describe was debated when Mr. Buffett discussed his investment (as a seller) in long-term European-type put options on various stock indices.

 

The BSM model works well for approximations (especially short term) and its use is commonplace so it can be used as a language (which Mr. Buffett supported for interim reporting purposes). I also have used "adjusted" approximations like wabuffo describes.

 

Mr Buffett's point, I think, was that the implied volatility used as an input (which is derived from the market itself with the obvious risk of circularity) gave a put price which deviated clearly from expected historical results and common sense (inflation, retained earnings etc) and that this probable deviation was more likely to occur with long term puts because the inherent approximation gets larger with a long term outlook. Mr. Buffett IMO simply decided to take advantage of that opportunity which was not guaranteed to result in a profit but was associated with a significant possibility of reaching an excellent return.

 

The space-based navigation systems usually work very well but sometimes (if you keep your eyes on the road), it can be beneficial to override the system.

 

I'm not sure what you mean with your post and call question but put options on equities are usually more expensive than call options on equities because, in the main stocks tend to go up but, when they go down, they tend to do so faster, with obvious repercussions on "volatility".

[nickenumbers]

Great info everyone.  Thank you.

 

Cigarbutt-  Thanks for that detailed answer.  It shows that I have a lot more to learn..  I am going back to Options school!  ;)

[LC]

FWIW for vanilla products a standard is to use trinomial tree with persistant skew. Plenty of documentation online. A better place for these type of questions is probably here: https://quant.stackexchange.com/

 

[sleepydragon]

BSM relies on no arbitrage profit theories. The price of the options equal to cost of continually hedging the derivatives by chasing gamma( buy high sell low). It’s from the perspective of the market maker who sell the option. So it needs to know the volatility how frequently they need to adjust the hedge,  and will only works for short term American options, but not suitable to long term European style options

[Cigarbutt]

Great info everyone.  Thank you.

If you're interested:

https://www.cfainstitute.org/research/foundation/2013/fundamentals-of-futures-and-options-corrected-april-2014

I guess section 5 (67-97) in the downloadable pdf book contains the relevant pricing info.

On page 91, you'll find what rb referred to ie using call-parity to derive the put value from the call value.

The section may also explain well how to use BSM even for American options (with the underlying paying dividend) because early exercise is rare (early exercise of puts is more frequent).

[LC]

I wonder if we have another other quants here on the forum?

[SHDL]

nickenumbers,

 

I thought I’d give you two pieces of advice that you might want to keep in mind as you learn about this stuff:

 

1. Pay special attention to how the price sequence of the underlying security is modeled.

 

As I pointed out in my (rather terse) message above, this is really the key to understanding the disagreement between Buffett and Black-Scholes.

 

A short summary (which is probably still too dense but will hopefully make sense to you some day) is the following. 

 

First, B-S is built on the assumption that the underlying stock follows what’s called a geometric random walk. To understand what that is, imagine a world where stock prices either gain or lose a fixed percentage each day according to a coin toss.  So for example you start today at 100, you flip a coin, it’s heads, so you gain 2% for the day and you go to 102.  Then tomorrow you flip a coin again, this time it’s tails, so you lose 1% and you end at (around) 101.  The day after tomorrow yay it’s heads again so you gain 2% and you’re up to (around) 103.  And so on. 

 

One consequence of this assumption is that any drop in stock prices is permanent — it has a lasting impact that does not go away with the passage of time.  As a result, people who believe in this model are always worried about the possibility of having something like a 10% drop one day, followed by a 5% drop the next day, followed by an 8% drop the day after, …. with absolutely no end in sight.  This compounds over time and creates a great deal of pessimism with regard to long term returns. 

 

Note that this is very different from how value investors tend to view price drops. As you know, their view is that there is this thing called intrinsic value and that stock prices tend to gravitate toward it over time (even though they may deviate from it in the short run).  So if you own a stock that you think is worth 20 and its price drops from 15 to 10 you expect a rebound sooner or later.  Could the stock go to 7 before that?  Sure, but that only makes a rebound even more likely.  In this sense price drops are temporary (as long as IV is intact!) unlike in the B-S model. Believers of this model therefore tend to be more optimistic with regard to long term returns.

 

And you can probably see how this difference can explain Buffett’s disagreement with B-S.  Being a value investor, Buffett was a lot more optimistic about the future long term performance of the S&P in 2008 than what a blind application of B-S would have suggested.  He was therefore the natural seller of those long dated puts while the B-S users were the natural buyers.

 

2. Do ask yourself if you really need a mathematical option pricing model (be it B-S or not) if your goal is to do well as an investor.

 

I’d say you do not.  What you need instead, really, is (a) the ability to describe your views about the underlying security in terms of a probability distribution over its future returns, (b) knowledge on how you can use various options (strategies) to “modify” that distribution, and © the ability to make reasonable portfolio allocation decisions accordingly.  And as you will probably find out soon enough, these skills actually do not have much in common with the skills needed for serious derivative pricing.

[Lupo Lupus]

nickenumbers,

 

I thought I’d give you two pieces of advice that you might want to keep in mind as you learn about this stuff:

 

1. Pay special attention to how the price sequence of the underlying security is modeled.

 

As I pointed out in my (rather terse) message above, this is really the key to understanding the disagreement between Buffett and Black-Scholes.

 

A short summary (which is probably still too dense but will hopefully make sense to you some day) is the following. 

 

First, B-S is built on the assumption that the underlying stock follows what’s called a geometric random walk. To understand what that is, imagine a world where stock prices either gain or lose a fixed percentage each day according to a coin toss.  So for example you start today at 100, you flip a coin, it’s heads, so you gain 2% for the day and you go to 102.  Then tomorrow you flip a coin again, this time it’s tails, so you lose 1% and you end at (around) 101.  The day after tomorrow yay it’s heads again so you gain 2% and you’re up to (around) 103.  And so on. 

 

One consequence of this assumption is that any drop in stock prices is permanent — it has a lasting impact that does not go away with the passage of time.  As a result, people who believe in this model are always worried about the possibility of having something like a 10% drop one day, followed by a 5% drop the next day, followed by an 8% drop the day after, …. with absolutely no end in sight.  This compounds over time and creates a great deal of pessimism with regard to long term returns. 

 

Note that this is very different from how value investors tend to view price drops. As you know, their view is that there is this thing called intrinsic value and that stock prices tend to gravitate toward it over time (even though they may deviate from it in the short run).  So if you own a stock that you think is worth 20 and its price drops from 15 to 10 you expect a rebound sooner or later.  Could the stock go to 7 before that?  Sure, but that only makes a rebound even more likely.  In this sense price drops are temporary (as long as IV is intact!) unlike in the B-S model. Believers of this model therefore tend to be more optimistic with regard to long term returns.

 

 

I thought this was a very nice point, going much beyond option pricing. Value investors view price drops as changes in market perceptions without (commensurate) changes in true values. Efficient markets however suggest that prices drop because fundamentals have changed. The truth of course lies somewhere in between. Yes prices drop when fundamentals change, but sometimes they do not correctly reflect changes in fundamentals, which opens the door for value investing strategies.

[Cigarbutt]

^Interesting value-based insight.

 

From the 2003 annual meeting:

 

Buffett: The Black-Scholes model is an attempt to measure market value of options. It cranks in various variables, mainly past volatility of the asset involved, which are not the best judge of value.  ... Berkshire had a very low beta – experts like to give complex Greek names to simple things – but that doesn’t mean the option value to anyone who understood it was lower than another stock with higher volatility.

 

As Charlie said, Black-Scholes can give silly results over longer terms. Last year, we made one large commitment in which somebody on the other side was using Black-Scholes and we made $120 million. We love the idea of someone else using mechanistic formulas. They may be right 99% of the time, but we can pass 99 times and only invest the one time they’re wrong.

 

Munger: Black-Scholes is a know-nothing system. If you know nothing about value – only price – then Black-Scholes is a pretty good guess at what a 90-day option might be worth. But the minute you get into longer periods of time, it’s crazy to get into Black-Scholes. For example, at Costco we issued stock options with strike prices of $30 and $60, and Black-Scholes valued the $60 ones higher.  This is insane.

 

Buffett: We like this kind of insanity. We will pay you real money if you deliver someone to our office who is willing to offer us three-year options that we can pick and choose from.

-----

Whether it's options or else, the market is there to make offers and it's up to you to figure out if it's a good deal.

 

[nickenumbers]

All,

 

Excellent information, I need to read and review it and work thru it all over the next couple of days.  Thanks!

[Dynamic]

I really loved this thread, particularly SHDL's response.

 

I've not really gone deep into trying to understand Black-Scholes, so its possible my understanding isn't quite right.

 

Black-Scholes and the variations of it for American style options and dividend-payers and those incorporating expected growth trends, were designed based on averages over large numbers of underlying stocks, and fit pretty well on average (averaging over time and over different stocks) and over shorter time periods. On average over time and over a wide range of stocks, the market behaves mostly pretty efficiently, so they bake the assumption of perfect efficiency and of volatility meaning risk into Black-Scholes and back test it and forward test it and it seems to work remarkably well on average. It's also easier to test it on frequently traded options with small spreads, which mostly means relatively short-dated contracts that actually produce a much larger data set to test the formula. If you look at LEAPS contracts, some of the last trade dates can be days or weeks ago and the published Bid Ask spreads can be as high as 5-10%, so the data set is pretty sparse for testing Black Scholes' validity in such conditions.

 

Value Investors know that there are both times and securities for which the market is not efficient (and it can be a large discrepancy occasionally, which is the moment we achieve our Margin of Safety and place our infrequent trades). In those situations Black-Scholes will not provide an appropriate price, perhaps skewing the probability-weighted risk-return distribution significantly towards or against the buyer or seller of the option, just as mispricing of stocks presents opportunities to the buyer or seller of the stock.

 

One key point is that I think Black-Scholes assumes a relatively symmetrical probability distribution for future prices. This was explained earlier in the thread, with the comparison between a random walk that incorporates memory of recent prices set against the idea of a temporary mispricing based on widespread optimism, pessimism or poor analysis before the price reverts back towards Intrinsic Value.

 

To Value Investors, the larger the margin of safety gets, the lower the likelihood and magnitude of potential loss becomes (even in the short-to-medium term, but especially in the longer term) and the higher both the likelihood and magnitude of potential gains become. While I've had plenty of stocks decline further soon after my purchase, perhaps by 10% over a month or a couple of quarters, I've had rather more positions rise by more than 10% surprisingly quickly after I finally pulled the trigger. This effect is sometimes very valuable if another higher conviction idea happens to occur a few months later and I sell some of the first position to fund the second. If it doesn't happen the second position would have to be even cheaper to be worth the switch.

 

Lower risk (even in the short-to-medium term, but more certainly in the long term) can occur simultaneously with higher reward at those rare times when patient Value Investors finally pull the trigger.

 

Efficient Market proponents completely miss the idea of this highly skewed and asymmetric probability distribution that accompanies an unusually high Margin of Safety, and Black Scholes type models do not incorporate such skewness because these are rare events that tend to average out to near zero over a long period of time and a wide range of stocks.

 

Equally the companies that are predictable and anti-fragile enough for Quality oriented Value Investors to invest heavily in for the long term and for them to assign an Intrinsic Value to, tend to be more likely to rebound from bad news on the political or economic front than the remainder of traded companies out there.

 

So it's possible that IV is so uncertain and thus is not a particularly useful concept for the majority of companies whose prospects get tossed around by the winds of external events. If that were the case, it might explain why on average, Black Scholes seems to work fine and obtain a faithful following while ignoring the outliers that Value Investors aim to pick off and profit from.

 

To be clear there are rare times when markets in aggregate are significantly undervalued and have a positively skewed return distribution, as well as rare stocks among all the stocks out there that are undervalued at any one time and also have a positively skewed return distribution.

 

Buffett's equity index puts were of the former type when markets were statistically cheap and the spread of expiry dates could still be profitable if a market crash fell on one or two of those expiry dates.

 

A Value Investor's purchases at times other than juicy bear markets tend to be of the latter type.

 

On rare occasions, perhaps 2009, Value Investors get a bite of both cherries at once as out-of-favour companies get beaten down further in a fearful overall market.