As previously discussed, options can be used to hedge portfolios against moves that are contrary to that portfolio’s orientation. For example, buying put options can be used to hedge a portfolio that is on the long side of the trade. Oppositely, buying calls can be used to hedge a portfolio that is primarily focused on the short side.

The difficulty in assessing options is figuring their value at a given point in time since they lose part of their extrinsic value as their witching (expiration) day approaches. Fortunately, there are a couple different models we can use to make quantitative predictions.

Binomial Option Pricing Model

The Cox, Ross, and Rubenstein Binomial Model is probably the simplest of the mainstream models available, and is flexible in that it works both on options that can be exercised over a period of time such as American options, but also on options with specific exercise dates such as Bermudan options. The model is built on the simple assumption that a stock can move in one of two directions in a given time period. The price can move up Su with a probability of p or it can move down Sd with a probability of (1-p). The figure below shows 3 generations of such binomial possibilities at each stage. Once the binomial lattice is constructed, the probabilities at each time point can be calculated at various standard deviations from the mean.

When pricing options, it must be considered that you’re really dealing with several axes; the first being the price of the underlying asset as it compares with the option’s strike price and the second being the time factor. Let’s look at some other models, and then run a few live examples.

Black-Sholes Option Pricing Model

The Black-Sholes Model applies to equities whose price follows Geometric Brownian Motion or a stochastic price process. For simplicity’s sake, the prerequisite for using the model is price action characteristics that are often observed in financial instruments such as equities.

There are several assumptions that the Black-Sholes model makes, some of which are not able to be met in the real-world investment environment, but similarly to the notion of the perfect competition / pure monopoly continuum, such constructs can be valuable despite the fact that they are seldom achievable in their purest forms.

• Borrowing can happen at will at a constant and known risk-free rate.

• The price follows a Geometric Brownian motion with constant drift and volatility. (This is extremely rare in our markets)

• Transaction costs are set to zero.

• There are no dividend payments to navigate. (Dividend harvesting strategies often upset predictable Brownian motion)

• All securities are perfectly divisible. (Fractional shares such as in a DRIP type setting are available)

• There are no restrictions on short selling.

• There is no opportunity for arbitrage.

There are several variations of the Black-Scholes model; one that deals strictly with the price of the equity security, and a partial differential equation (PDE) that deals with derivatives of an underlying equity. While the formula is shown only for reference purposes, the general idea is to understand that one can use this modeling process to estimate the value of the derivative security (in this case an option) at a specific known point in time.

Bjerksund-Stensland Model

This model comes in handy if you happen to be dealing with options whose underlying securities pay dividends. It is a hybrid of the Black Model, the Black-Scholes Model, and the Garman-Kohlhagen Model. However, there are still some assumptions that are not realistic. The model depends on constant dividend yields, discrete dividends, and continuous dividends. While there is a negligible amount of securities (if any) that fit these assumptions entirely, the model has through repeated study shown itself to be more accurate than the Quadratic Approximation Formula in nailing the value of long-dated options.

I mention this model due to its applicability to dividend paying stocks and its affinity to long options for those interested in using it for analysis of LEAPS options.

Exercises

For the purposes of this piece we’re going to run two simulations; one with the Diamonds as our core investment and a single protective put option and the second (also with the Diamonds) using what is known as a Butterfly, which actually involves buying two options and selling a third. We’ll see how the various strategies predict gains at different price levels a year from now according to the three aforementioned models.

Let’s list our assumptions for the first exercise:

-We’re going to buy 1000 shares of DIA

-We’ll buy 25 puts on DIA, which expire on 12/21/2012 with a strike price of $90.

-We will setup our price point analysis as follows:

$140.00 (assumes the Dow makes a new high – bullish case)

+10%, live, -10% of today’s price

$65.00 (assumes a retest of 3/6/2009 lows – bearish case)

-Interest will be set to the “risk-free” (sic) rate of 4.56%, which equals 30-year Treasury Bond yields on 1/27/2010.

-We assume the trades are made today at market prices and we’re looking at P/L one the trades one year from today.

Here are the results:

Black-Scholes Model

Binomial Model

Bjerksund-Stensland Model

What all three of these models are telling us is that the best way to make money with this setup is if the Dow moves big in one direction or the other. If the Dow maintains the status quo and stays within +/- 10% over the next year, this allocation will lose money. The tools are predictive, but the user has to provide the assumptions. If you have a strong belief that the market is going to go down and retest those March 2009 lows, then you can use these models to build yourself a hedging program and then predict the results of such a program. You can tweak the variables, interest rates, etc as well. You can even add additional positions and see how the scenarios play out. This is one of the techniques that we’ve used over the past few years to mitigate periods of market downturn in our newsletter’s portfolio model.

Changing the Strategy

Obviously the first exercise provided a significant amount of risk and depended greatly on us being right in our prediction. If we’re wrong, we could lose a significant amount of money. Clearly there has to be a better middle ground; and there is.

Let’s change up the variables a bit and see if we can’t get something a little more balanced. For this comparison, we’ll use what is known as a Butterfly. Essentially what a Butterfly does in our case is purchases 1000 shares, buys 10 puts at $100 strike price, buys 10 more puts at $80 strike price and sells 10 puts at $90.

Let’s look at the results; again. We have added some price levels between the $65 and the $92 range to give a better illustration, however, all computational variables are the same as before.

Black Scholes – Butterfly

Binomial – Butterfly

Bjerksund-Stensland – Butterfly

Obviously the butterfly strategy worked very well in terms of smoothing the returns across a wide variety of prices. Keep in mind in both cases we were working with around a $100,000 corpus of capital and limiting the action to +/- 1% is especially beneficial if that capital happens to yield more than the Diamonds and you’re collecting significant dividend income.

So, depending on your intentions and purposes, you can either use your option strategy to make a bet on the overall direction of the market and try to make a profit off of it or use an option strategy that allows you to collect dividends without worrying so much about a major market downturn diminishing the value of the underlying assets.

Some Potential Pitfalls

If you decide to investigate any of these types of strategies, you are going to want to keep your eye on a few things:

1) Does the derivative security appropriately represent what it is you’re trying to protect? For example, in our newsletter’s model portfolio, we have 21 dividend producing assets. Obviously, creating a butterfly strategy might be to our advantage. Not all of our components are equities, and many don’t have option chains so we’ll need to find something that does. But what options should we use? Using the Diamonds puts discussed above might work IF the relationship between my portfolio and the Diamonds is somewhat analogous. If it isn’t, then I might not end up where I should be even if the models happen to be spot on. The importance of this issue cannot be emphasized enough.

2) Use options that come from well defined option chains and are not thinly traded whenever possible. If you are working with an option that has just a few hundred contracts of open interest and low volume, authentic price discovery can be a problem because of wide bid/ask spreads. This is not to say it is impossible if the options are thinly traded, but it makes life more complicated.

3) The options pricing models are not always correct; especially in times of acute distress in the markets since the normal Brownian, stochastic oscillations of securities are disturbed and therefore, a portion of the predictive value of the models is lost.

Conclusions

It must first be pointed out that you don’t need fancy trading platforms or expensive financial software to use these models. While it is exceedingly complicated to do the calculations manually, there are websites on the Internet where you can download Excel Macros and add-ins that will allow you to enter the parameters and the custom software will do the rest. Go to your favorite search engine and key in the name of the model you’re interested in and you’ll quickly find a plethora of potential solutions.

We have presented two approaches to hedging, both with options and with drastically different results. For our purposes here at the firm, the Butterfly strategy obviously makes a lot of sense since a growing portion of our work centers around income investing. Being able to mark time with capital while raking in stellar dividend income is obviously something every fixed income investor would like to be able to do. In this era of near-zero interest rates, which just got stretched out by at least another month by the Fed today, it is not always an option to bail out of a sick market, especially when the income is needed to finance someone’s standard of living.

It is relatively amazing how, despite the overwhelming evidence that we have a long way to go at least in terms of economic fallout, so many are fooled. Their lack of understanding comes from a failure to recognize that the Fed cannot print prosperity, savings or real capital. We are now paying the price for decades of failed Keynesian economic policies which center around monetary inflation and the general belief that the economy can be ‘managed’ so that recessions never occur. This is pure nonsense.

So while Starbucks and GM continue call ‘bottom’, ask yourselves these questions: If GM believes the bottom is in, then why does it need billions in a taxpayer bailout? Shouldn’t it be able to weather the remnants of the storm since things are about to get so much better? And if Starbucks thought the bottom was in, why aren’t they canceling earlier plans to close 600 stores and the development of that many more?

As is usually the case, actions speak louder than words.

• 5 major financial firms have gone broke or required a bailout
• At least 3 others narrowly averted bankruptcy (thanks JP Morgan – oh WHERE do you get all that money?)
• More than a dozen banks have failed with more on the way
• Thousands of jobs have been lost just in the financial sector
• Shareholders didn’t just lose their shirts; they lost their pants and socks too
• US taxpayers will bear the cost for at least 10 generations – Yes I did say 10.

And after all that there is still plenty of money for bonuses? Are you kidding me? Bonuses are supposed to reward a job well done. Instead, these folks have been allowed to bankrupt a nation and will get to hit the cookie jar one more time.

The same has been for medium term lows in commodities and precious metals. During an inflationary period, these scarce items take on monetary characteristics, yet it would appear that everything up to and including Elvis, Cleopatra and the 1955 Yankees are being brought in to stop this ship from righting itself. Our convictions are being tested. Will we hold on to them based on a lack of fundamental change or will we bail and provide a starving silver market with some actual metal? I hope that as many as possible go with me and stick to the former rather than opting for the latter. Although I’ve heard if you sell right now, Goldman Sachs is offering a side order of toast to anyone who gives up a one ounce US Gold Eagle. Keep the faith folks; there’s plenty of company on this bumpy ride.

In FOREX markets, the Dollar is down against the Euro as the pair is very close to breaking $1.60; a critical resistance level. The Japanese Yen is attempting a breakout as well at 105 and is right at the 50-day moving average.

Equity markets around the globe are in a rout today as selling pressures have emerged due to the continued troubles at Fannie Mae / Freddie Mac, slow economic growth, and runaway import prices. This last item gets almost no attention even though it is a fairly accurate indicator in terms of consumer health since so much of what we buy is imported. We discussed recent trends in import prices in our 5/16/2008 issue of My Two Cents which may be read here:

Do you believe in Fairytales?

The technical rally that we have been looking for has been a no-show to this point with the major US indexes continuing to probe newer lows. In fact, as I write, the DJIA is threatening to break below 11,000, currently at 11,024.16.

Looking forward, if the major indices are going to muster any kind of a rally, they’d better get started soon. Earnings will start trickling out and the news is not looking good despite a solid attempt to window-dress the results by the mainstream financial press. Lower earnings make current P/E’s high, and warrant further correction (lower prices). The economic stimulus checks have been hitting Main Street for nearly 2 months now with the last scheduled to be sent within a few weeks. Once the brief jump in spending numbers wanes, there is not a lot of good news for US markets looking forward. 

Even more perplexing and alarming is the rout in energy stocks. Oil is currently at an all-time high above $144/barrel and yet the vast majority of energy stocks are getting pummeled today along with the rest of the market. The good news here is that the value of these companies’ assets and products is going up while the share price is going down. We could be seeing some good bargains in this sector shortly if this action keeps up.

The Story no one is talking about

The markets today are reacting to continued deterioration in the financial sector, rebounding Crude oil prices, and a sharply weaker Dollar. Today’s price action is being made even more dicey because of quadruple witching. Quadruple witching is the date each quarter that stock options, stock index options, single stock futures, and stock index futures expire. The next QW will be in September.

We are still feeling for the bottom in stocks. Short-term, we are oversold on most measures, however, it should be noted that significant declines can still occur even from oversold levels. The 12,000 level, then the 11,700 level are the levels to watch.  The DOW is currently in the 11,900 neighborhood and the S&P around 1325. We’ll need to see if these levels hold through the end of day.

In other markets, oil is flat, the Dollar is down mildly, bonds are up with the yield on the 10-year bond sitting at 4.14%. We discussed the bond situation in May’s issue of the Centsible Investor saying the short-term top in bond yields was either in or very close. We have seen yields drop 10 basis points since.

In a notable news development the Royal Bank of Scotland (RBS) has issued a crash warning for stocks and credit instruments, saying that a significant deterioration in market conditions is likely over the next three months. In their opinion, the credit crisis is nowhere close to being over. I couldn’t agree more. See the full story below:

Read Here