Dynamic Hedge Portfolio Theory And Management

Writing Options - Writing Calls - Option Theta

Or how it's possible to make a profit without calling the market




Foundation of an Option Hedge Portfolio

Derivative Position Creation

Option Volatility Theory

Black & Scholes Option Model

Putting the Models Together

Stock Price Movement

Hedge Creation

Hedge Management

A Market Neutral Option Portfolio

Stock options offer the investor the chance to earn a profit in many types of situations. There are many different types of strategies available to the option investor. The problem with each and every strategy is that they all require one common element; the ability to predict the movement of a stock.

There are several problems and questions that arise from such a basic tenant. First is the most obvious. Very few people have ever been able to predict, with consistency, the price movement of a stock. The second problem with the many exotic option strategies is if you could predict the movement of a stock why use the strategy at all? Returns would be greatest if one simply purchased a call or a put in line with the investor's expectation of the movement of the stock.

The final problem with exotic strategies is the need to time the market. The market and not the stock drive almost a third of a price change of a stock. This is an extremely critical point of understanding. Just watch the market re-caps every day. When several large Tier 1 companies have positive news the sector is generally affected. By way of example, Dell announces better than anticipated earnings. The only stock that should be affected is Dell. The reality is others such as Gateway and Micron will also show gains for the day.

A truly effective way to invest in stock options would be to develop a way to earn a profit without calling the market. That is, we need to invest in such a way that we don't have to rely on the market moving one way or another for us to earn a profit.

Is there a tool that will accomplish this? Yes, and it is called Dynamic Option Hedge Portfolio Investing.

What is a hedge?


A hedge is the purchase and sale of securities simultaneously that seek to capture a price discrepancy between positions.


An example will make this easy to see.


Stock ABC had a call that expired in 120 days with a strike price of X with a seller willing to take $1.00 for it.


We found a buyer willing to pay $3.00 for the same option.


Our action was to sell a call for $3.00 and purchase a call for $1.00.


The result of our hedge is we have a guaranteed profit of $2.00 no matter what happens to the price of the stock.


Does such an opportunity actually exist? The answer is no. Very rarely will an investor be able to locate and trade an arbitrage situation like this.


We can, however, simulate this situation through the creation of a derivative position.



Foundation of an Option Hedge Portfolio


As previously mentioned, if we could purchase under priced options and sell overpriced options on the same security we would have a guaranteed profit.This would be a perfect hedge, but it rarely exists. 





Stock ABC with a call 120 days out was fair priced at $2.00 and it struck at $30.

It was traded for $1.00, It was under priced so we purchased it.


Stock XYZ with a call 120 days out was fair priced at $2.00 as well also struck at $30.

It was traded at $3.00, It was overpriced, so we sold it


We have a net credit of $2.00 to open the position.




If these two stocks moved with the market we would have a locked in profit of $2.00,

No matter what the market did!


For every move one option had the other would have an equal move. The profit of $2.00 is constant.




To earn consistent profits we need to find two different stocks that have the same relative risk. To earn money we need to create a derivative position.


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Derivative Position Creation

If we could create a derivative and give this derivative the properties of ABC stock then we could create a hedged position without having to have discovered both and over priced and under priced option in company ABC.  This would simulate the perfect hedge.


In a nutshell here is what we are searching for:





Both companies would have to have similar risks associated with them.


 By selling the overpriced option and purchasing the under priced we would open the position at a net credit.


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Option Volatility Theory

There are two kinds of volatility, Historical Volatility and Implied Volatility. Historic volatility is a statistical measurement of past price movements. Implied volatility measures whether option premiums are relatively expensive or inexpensive. Implied volatility is calculated based on the currently traded option premiums. The option equation is essentially solved in reverse.

The challenge it always the same; calculate how a stock with react in the future. If we knew what the future volatility would be, we could make a fortune quite easily. Because we don't know what future volatility is going to be, we try to guess what it will be.

The beginning point for this guess is historical volatility. We first look at the volatility of a stock or other security, over a given period of time. Generally, when evaluating volatility, we look at several different periods. We may look at what the volatility has been for the past week, for the past month, for the past three months, for the past six months, and so forth. The longer time period will yield more of an average volatility. However, don't expect large changes in volatility over time. Stocks or other securities which are volatile on a daily or weekly basis usually remain that way over time. THIS IS A VERY IMPORTANT PART OF THE STRATEGY. WE ASSUME THAT VOLATILITY WILL ALWAYS REGRESS TO THE MEAN VOLALITILITY OF A SECURITY GIVEN ENOUGH TIME. When evaluating the purchase of an option, it is the historical volatility of the underlying security we are looking at. For instance, when deciding whether or not to purchase an option on XYZ, we would look at the historical volatility of XYZ. Basically, what historical volatility boils down to is, what is the probability that this particular underlying security will move a particular distance measured in price on a given day, week, month, etc.

However, there is a different interpretation of volatility not associated with the underlying security. This is implied volatility. There are many different models for pricing options. However, most will yield a price relatively close to each other.

However, what if we use our historical volatility in the formula and come up with a price far away from where the option is trading? What if we do this using several different option pricing formulas and we still come up with a price which is not close to where the option is trading? Why would we come up with a price, which can't be accounted for? We're all using the same inputs. We all use the price of the underlying security, the time until expiration, and the strike price, dividends to be paid by the stock, the current risk free interest rate, and volatility. All of these inputs are known. Or are they?

The one input which is not known and for which we have to take a guess at is volatility. What has happened is that the marketplace is assuming a different volatility other than historical volatility. The way to solve for this implied volatility is to use our option-pricing model in reverse. We know the price of the option and all the other variables except the volatility the marketplace is using. Therefore, instead of using the equation to solve for the option's price, we use the model to solve for the option's volatility. We insert the price into the model, leave out the volatility (which we are looking for), and keep the other variables the same. It is then that we will find out what volatility will yield the current market price.

Most traders refer to implied volatility as premium. To be precise, the word premium refers to the option price relative to the underlying security. Nevertheless, traders will say things like, "Premium levels are high." or "Premium levels are low." What the trader is really referring to is the implied volatility. The implied volatility is high or the implied volatility is low.

The first thing that one thinks about when trying to evaluate historical volatility is that the standard deviation should be used. And if a person is looking for a simple way to measure volatility, the simple standard deviation will work well enough. However, use of the standard deviation assumes that there is a normal distribution. If stock prices were normally distributed, the implication would be that there could be negative prices. This we know is impossible. The furthest a stock's price can fall is to zero. However, the stock price can rise infinitely. Therefore, we take the standard deviation of the logarithmic price changes measured at regular intervals of time.

Formula For Calculating A Stock's Volatility Using a Lognormal Distribution

Xi = ln[Pi divided by ((1+r)/52))Pi-1]


Xi = each price change

Pi = price of the underlying security at the end of the i-th period

r = risk free interest rate (We are using weekly data in this formula. For daily data use 253)

Step two is to calculate the standard deviations lognormal for the data series. This would be the price change for the period under consideration. For example, using weekly data, we would calculate the above calculation for each week for at least 14 weeks.

Step 3 is to sum the answer for each calculation in step 2 and divide by 14. This gives us the mean.

In Step 4 we subtract each calculation from the mean.

In Step 5 we square each number from step 4 and add them all together.

Step 6 The annualized historic volatility is the answer from step 4 X the square root of (365/7)

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Black & Scholes Option Model

Trading options without the Black & Scholes option model is like wandering in the desert without a compass. The first step any option trader should make is to fully know his way around the formula, so as to understand exactly what the inputs are and under what conditions the results can be trusted. Traders that know how to use the B&S option model have a definite advantage over those that do not. The markets are a process by which the best investors make money and can hence stay on in the game, and the worst investors lose money and are forced to quit. Money flows from the pockets of the amateurs into the pockets of the professionals. Nowhere is this truer than in the options markets.

The Black & Scholes option model is an option valuation model. It tells us what the theoretical price of an option should be. Without B&S, evaluating an option price would be an exercise in guesswork. With B&S, it becomes a scientific and hence precise process.

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Putting the Models Together



Over the years there have been many copycat option models. They all handle things slightly different. The simple truth is that they are all flawed without knowing the future volatility of a security.

As in all pricing models we do not have a variable; future volatility. I have several ways to estimate it. It general terms I test the 5 year volatility number, 1 year, 50 days and 10 days. I blend these values and weigh them.

I treat the output of the formula as the central point for determining fair value. It is from this calculation that option quantities are dictated.


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Stock Price Movement


Stocks move in price because of news. The market moves because of overall feelings about the news and about the outlook for sectors and industries. It is like the market is the ocean and individual stocks are boats.

When the swell are up, stocks will tend to move upward. When down, stocks in that sector will tend to move down.


Through statistical analysis it has been determined that 30% of the price of a stock is by sector news and 70% is by company news. That is why there can be no new information for Dell but if a positive story about Computer Manufactures develops, Dell will likely move in an upward direction.

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Hedge Creation


To set up a Hedge Portfolio we need to follow several "rules". We must keep in mind that for the hedge to work there must be equal exposure on both sides of the hedge.



Rule 1: The fair value of the options should be approximately equal. Fair value, is based in Black- Scholes option model.

Rule 2: Because we are purchasing options in quantities to approximate fair value and we are purchasing options that are underpriced and selling options that are overpriced the dollar value the dollar value will not be equal.

Rule 3: The capital required will be much greater on the short side of the hedge versus the long side of the hedge.

Rule 4: For the hedge to be balanced the relative volatility should be reasonable equal.

Rule 5: Diversification. To reduce risk by 65% we need to hold 100 issues.

Rule 7 We need to match long positions to short position by industry and or sector. The more matching we can do the more predictable the results. No more than 10% of the hedge should be invested in any one sector.

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Hedge Management of the Portfolio


Rule 1: If one side of the hedge becomes to large we will need to reduce it to bring it back into balance.


Rule 2: If a position grows in relation to another we need to trim it back. A position that has grown a great deal will exert to much influence on the balance of the hedge. A position that doubles will have twice the influence.


Rule 3: Cashing Out. This is what it is all about. If premiums change significantly we will take profits. When a call that we bought at bargain basement prices under priced is now fairly priced we will sell it. An option that was sold at boutique prices has fallen to fair value we will buy it back.