Author Topic: Black Scholes and long dated options  (Read 4840 times)

Arden

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Black Scholes and long dated options
« on: July 03, 2012, 12:38:30 AM »
I've been making some calculations with the BS formula, and I reach some results that seem strange to me. For example- WFC warrant with a maturity in 6.5 years trade at 8.5, while 1.5 year options trade at almost 4. This seems weird to me, because it assumes that the  1.5 years starting now are worth a big chunk of the warrant price, while in my opinion the years after that are the more valuable ones, because if WFC compounds at 10, the year from year 5 to year 6 is worth a lot more than the year from now. This got me reading about the formula.

Black Scholes' formula assumes stocks move at a random brownian motion with a drift, which is the risk free rate- meaning, the forumula assumes every risk is hedged perfectly, so everything should move at that rate.

Putting aside the formula ignoring the company and its own compounding of its equity, is it logical to make an assumption that the risk free rate is the return for any stock? Maybe it's true for a commodity, but in stocks This seems to me not only a little off, but off by orders of magnitude in the long term.

It seems the WFC warrant pricing closely tracks the Black scholes value for it, so if you assume WFC compounds at 15% or even at 5%, you're seeing something completely different of what a guy pricing with the risk free drift assumption sees. If I could rewrite the formula and assume a specific drift for WFC(the drift doesn't appear in the final formula) I would assume a drift of about 5-10% (to be conservative), which I suspect would change the warrant price by a lot.

Am I understanding BS correcly?  Do you think perhaps the risk free drift assumption does make sense?
Sol Arden


twacowfca

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Re: Black Scholes and long dated options
« Reply #1 on: July 03, 2012, 06:13:37 PM »
Arden, your understanding of Black Scholes is correct.  It's as good as it gets for pricing short term options, but it can be way off when pricing LEAPS.  LEAPS are usually priced more accurately using scenario analysis by assigning probabilities and expected values to future states.  :)

Your idea of substituting a different rate for the risk free rate makes sense for LEAPS.   Using the industry or sector specific cost of capital for very long dated warrants would be appropriate.   You might be able to lever that idea into a Nobel Prize if you are an academic.  :)

The WFC warrant you describe does seem to be a better buy compared to the option assuming the variables for each are about the same except for the duration.  Referring to the risk free rate as being the assumed drift rate is one way of looking at it, but the risk free rate is also related to the cost (real or implied) to finance the purchase of the stock in order to write a covered call.  As that rate goes up the cost to write the call increases and the call gets more expensive to write.  I'm always sceptical about theories that assume Mr. Market is omniscient.
« Last Edit: July 05, 2012, 07:06:51 AM by twacowfca »

Arden

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Re: Black Scholes and long dated options
« Reply #2 on: July 04, 2012, 01:34:18 AM »
Thanks for your reply mate, glad to know I haven't gone too far with trying to make sense of BnS.

A nobel prize may be nice, But if this method is more correct, then I would rather make truckloads of money instead :D

As far as I can see, the warrants are priced using black and scholes, and the difference between cost of capital and risk free rate, especially now, is just huge, this could cause an unbelievable mispricing in some warrants.

I think, and tell me if you agree, that as a value investor your biggest edge would be in long warrants , in stocks with a good return(or cost, I'm not sure) on capital with low volatility, because in volatile stocks the volatility that goes into the formula will compensate for its mistake in using risk free rate, thus making you lose your edge. This may be the reason BAC warrants don't have same leverage as the WFC warrant- WFC is a pretty calm stock (not that this means anything).

Sol Arden

twacowfca

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Re: Black Scholes and long dated options
« Reply #3 on: July 04, 2012, 07:58:16 AM »
Thanks for your reply mate, glad to know I haven't gone too far with trying to make sense of BnS.

A nobel prize may be nice, But if this method is more correct, then I would rather make truckloads of money instead :D

As far as I can see, the warrants are priced using black and scholes, and the difference between cost of capital and risk free rate, especially now, is just huge, this could cause an unbelievable mispricing in some warrants.

I think, and tell me if you agree, that as a value investor your biggest edge would be in long warrants , in stocks with a good return(or cost, I'm not sure) on capital with low volatility, because in volatile stocks the volatility that goes into the formula will compensate for its mistake in using risk free rate, thus making you lose your edge. This may be the reason BAC warrants don't have same leverage as the WFC warrant- WFC is a pretty calm stock (not that this means anything).

I think you are right again.  The current risk free rate is very low compared to the potential for appreciation of certain stocks.  To some extent, the market compensates for this through a high implied volatility in the valuation of certain leaps, especially those that have experienced high historical volatility recently.  BAC is a salient example of this phenomenon.  There may be some sleepers like WFC with relatively modest implied vol that could be better buys.

 Be aware that rising interest rates at some time in the not to distant future such as with increased taxes on the investing class in the US in 2013, could pinch net interest margins for banks if they are unable to raise the rates they charge on loans in a time of weak demand as much as they may have to increase the rate they offer to attract deposits.

I don't think banks are a better business when considering their long term prospects through the entire credit cycle.  Every decade or two the government takes the worst of the lot out and shoots them.  Then the survivors are put under many restrictions.  They no longer have the monopoly they enjoyed decades ago on attracting deposits.  Even the big money center banks that have a lock on profitable business that small banks can't handle have a big downside with loss of control and lack of liquidity on the derivatives they write.
« Last Edit: July 04, 2012, 08:31:48 AM by twacowfca »

Arden

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Re: Black Scholes and long dated options
« Reply #4 on: July 04, 2012, 08:17:47 AM »
Thanks again :)

A rising interest also benefits banks, doesn't it? Theoretically, they could still give customers a very low interest and raise it less than the FED does in the future. Why do you think it'll hurt them?

About taxation- I agree that rising taxes cause some risk, but ,listening to some of the statements both Obama and Romney made, It seems probable they will lower corporate tax rate after the elections (With Romney it's pretty certain, Obama less so) . Since WFC pays pretty much the full rate, it will be really significant, raising its rate of return several percentage points. I really hope this happens.
Sol Arden

twacowfca

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Re: Black Scholes and long dated options
« Reply #5 on: July 04, 2012, 08:45:55 AM »
Thanks again :)

A rising interest also benefits banks, doesn't it? Theoretically, they could still give customers a very low interest and raise it less than the FED does in the future. Why do you think it'll hurt them?

About taxation- I agree that rising taxes cause some risk, but ,listening to some of the statements both Obama and Romney made, It seems probable they will lower corporate tax rate after the elections (With Romney it's pretty certain, Obama less so) . Since WFC pays pretty much the full rate, it will be really significant, raising its rate of return several percentage points. I really hope this happens.

A rise in short term rates would be reflected immediately with a rise in money market rates.  Banks that depend on deposits for their funding would have to match those rates and pay more, probably before they could raise the rates on loans that roll over.  However, the effect on each bank would be different depending on their long/ short exposures to changes in interest rates.  The recent stress tests that banks published tells a lot about what the effect of a substantial rise in rates would be for each bank.  In general, I don't think that a rise in rates would be favorable for most banks. 

Taxes on the investing class to pay for increased Medicaid enrollment are definitely going into effect Jan. 1, 2013.  Politics involving who's in and who's out after the election could make it difficult to gain consensus to extend the Bush tax cuts or repeal the spending cuts scheduled to take effect in January.  I think stocks will decline in price as the effective owners earnings take home pay is cut by tax increases.


Most insurance companies, especially those with long tailed liabilities, may, however, benefit from a rise in interest rates.  :)
« Last Edit: July 06, 2012, 02:49:22 AM by twacowfca »

Arden

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Re: Black Scholes and long dated options
« Reply #6 on: July 05, 2012, 05:55:02 AM »
Do you know if there's any connection between beta and the volatiliy we enter into the BS formula? Are they the same?
Sol Arden

twacowfca

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Re: Black Scholes and long dated options
« Reply #7 on: July 05, 2012, 07:56:10 AM »
Do you know if there's any connection between beta and the volatiliy we enter into the BS formula? Are they the same?

Beta refers to the propensity of the price of a stock to rise or fall more or Less than the market rises or falls.  It's a useful concept in some circumstances. 

For example, in March, 2009, we rotated out of some of our low beta stocks that were great businesses and had fallen less than the market or had even risen during the market meltdown.  We used cash from those sales to buy high beta stocks like BAC that had tanked, on the supposition that  the enormous liquidity the government was pumping into the system would stabilize the market before it fell all the way to the sub basement level reached in The Great Depression.  We sold those high beta stocks a few months later when the market rebounded after they rose about four times as much as the market went up.   Then, we put the proceeds back into the low beta stocks of the great businesses that should outperform the market enormously over time.

Beta is useful when trading in and out of stocks, but it is not very relevant  for valuing the long term  worth of a business.  Beta is however an extremely important concept to understand for fund managers who could lose their jobs if they underperform in the short term, but who will hold on to their jobs if their performance fairly closely tracks the mediocre performance of other fund managers.

Volatility means how much percentage a stock (or the stock market) goes up and down, not compared to the market like beta, but compared to itself, in other words compared to the mean of its own price fluctuations.  The width of the standard deviation is the usual measure of volatility.  Fortunately for those with a long term perspective, volatility is often computed based on the average daily fluctuation over a relatively short interval of a few months.  Volatility has a strong tendency to regress to the average percentage or mean.   

Therefore, those who have the advantage of being able to invest (or sometimes trade) with a long term perspective can take advantage of volatility.  As Warren has pointed out, volatility is the friend of the value investor who isn't overexposed to Mr. Market's moods and who has dry powder available to shoot game when the hunting is good.   :)
« Last Edit: July 06, 2012, 02:40:48 AM by twacowfca »

Arden

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Re: Black Scholes and long dated options
« Reply #8 on: July 05, 2012, 09:05:08 AM »
Great answer, thanks!
Sol Arden

twacowfca

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Re: Black Scholes and long dated options
« Reply #9 on: July 05, 2012, 11:34:17 AM »
Great answer, thanks!

You're very welcome.  :)