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** Theoretical framework**

**History of the warrant market**

Trading of covered warrants on the Stockholm stock exchange started in year 1995. In the first five years the turn over and closed deals per year steadily increased. The yearly turn-over peaked in year 2000 when it was over 32 billion SEK, which is more than year 1999 and 2001 together. Last year, 2004, the turnover closed at approximately 9 billion after a really strong first six months which was 6 billion. Meaning that during the time July to De-cember 2004 it was a decrease in the turnover for the first time since the dramatically drop in year 2001 and 2002 (Hedensjö, 2004).

In February 2005 a new type of covered warrant was introduced to the market, called turbo warrants. The first issuer was the French bank Société Général and the warrants are listed on NGM’s derivatives exchange market called NDX. The turbo warrants directly hit the top four positions of the chart of the most traded warrants in Sweden. The turnover on NDX increased by 31 million SEK in Marsh after the introduction of the turbo warrants. On the Stockholm stock exchange the turnover for warrants took an opposite direction, it hit a history low 385 Million SEK in March, almost a 50 percent decrease compared to February same year (Huldschiner, 2005).

According to Optionsanalys there are some crucial differences between the historical trad-ing volumes in options, forwards and warrants. They are asking the question whether this is how it is supposed to be or just a lack in efficiency on the Stockholm stock exchange. The figures they are referring to is from the first six months during year 2002. Throughout this period the OMX index decreased with devastating 28 percent. The Swedish stock market was now longer in this bubble with hyper active trading and overvaluation which it had been in since year 1999. It was no adjusting to new levels. Despite this tremendous drop in stock prices, the number of closed deals and turnover for warrants the first six months in 2002 was higher than the same period for year 1999 (Optionsanalys, 2002).

The drop in trading with warrants is almost exactly in line with the drop in general stock market trading. The drop from year 2001 to 2002 for trades with stock and interest rate re-lated derivatives is about 7 percent. However, the total premium value from derivatives trading has dropped with over 45 percent for the same period, from 181 million to 100 mil-lion SEK. The thing that makes this interesting is that the drop in the warrant market was 58 percent from the previous year, which can be compared to the total derivatives market drop of 7 percent. The drop in total amount of option contracts and forward contracts was about 6.5 percent while the drop of warrant contract is greater than 50 percent. It is inter-esting why the warrant market more than halved while the options and forwards market just insignificant decreased (Optionsanalys, 2002).

Optionsanalys is giving some explanations to this. First of all they argue that the trade with warrants has been somewhat like a fashion thing for investors. It is something that can generate great outcome fast and easy. The investors do not need any extra knowledge or authorization to be able to trade with warrants, in addition to stocks. A warrant investor is mostly speculating in an increase of the stock market and when the market is in a regres-sion the investor easily loses too much money and abandons the warrant market after some terrible deals (Optionsanalys, 2002).

Investors using options has in general a higher level of knowledge and experience about the market than investors using warrants. To be able to trade with options, investors have to sign a special contract and that makes it more difficult to be an option investor. Option investors are aware of what they are doing and they are alert to the risks but can also see market opportunities by reducing risk for other investments. Option investors are said to use options like the tool it is supposed to be, while warrant investor rather see the warrant derivatives like a lottery ticket (Optionsanalys, 2002).

During the time period 1999 to 2002 about 90 percent of the existing warrants on the Stockholm stock exchange where call warrant. There are several reasons for this, the strongest one is that the warrant market started in the mid nineties and from that point of time to year 2000 the stock market it self had a tremendous increase in value which led to that call warrants being more profitable. The fascinating thing is that after the IT-boom in March 2000 when the stock market started to decrease heavily, warrants investors still in-vested predominantly in call warrants rather that put. This is explained by the fact that war-rant investors rather buy call than put warrants. The results was that instead of purchasing put warrants in a declining market, investors rather waited to find a low position of the market to buy calls (Optionsanalys, 2002).

**Covered Warrants**

A warrant is a kind of option that is traded on the stock exchange. When a warrant is bought the right to either buy or sell an underlying asset is actually purchased. This asset is mostly a stock but can also be an index, a basket of stocks, a currency or a commodity. When a warrant is bought a future price as well as exercise price and expiration date is agreed upon. The party that sells the warrant is called an issuer. The issuer is a traditional bank or an investment bank operating on the stock exchange. At the expiration date the is-suer of the warrant is obligated to buy or sell the underlying asset if the buyer wants to ex-ercise the warrant. Because the buyer of a warrant has the right not to participate at the ex-piration date the investor has to pay compensation to the issuer. This compensation is sim-ply the price of the warrant (SG, 2002). In chapter 2.2 and 2.3 SG (2002) will be used al-most exclusively. The authors have decided to use this reference heavily because there is no other publicly known information available about covered warrants characteristics.

**Fundamentals of warrants**

Parity demonstrates the total number of warrants needed to buy or sell one underlying as-set. The parity must be regarded when several key ratios are calculated; this will be obvious further down in the paper. Therefore it is crucial to understand the parity and what it stands for. Different warrants have different parity. The issuer is using the parity to make the trading more practical for investors. If the parity for a call warrant is 20, the investor needs 20 contracts to be able to buy one share at expiration.

An investor with the fundamental believes that a certain asset will increase in value will buy a call warrant. Vice versa, an investor that speculates in a decrease of an asset will buy a put warrant. These two types of warrants can be compared to call options and put options. If the exercise price of a call warrant is greater than the spot price in the market of the under-lying asset there is no point of exercising the warrant. This is the case because it is then cheaper to buy the underlying asset on the market. Let’s take an Ericsson call warrant as an example to illustrate this better. The exercise price of this call warrant, held by an investor is 25 SEK. If the Ericsson share is traded at 22 SEK on the market at the expiration date the call warrant will logically becomes worthless. On the other hand the price of the under-lying asset might move in a favorable way for the investor and finally exceed the exercise price. This results in that the investor will receive the difference between the exercise price and the spot price.

If the spot price of the underlying asset is below the exercise price of the call warrant, the warrant is out of-the-money. If the spot price is greater than the exercise price the warrant is in-the-money. Vice versa applies for put warrants. For both put and call warrants the fol-lowing is true; if the spot and exercise price is exactly equal to each other the warrant is at-the-money.

**Break even**

Break-even is a frequently used tool to analyze if a warrant is interesting to invest in. It is calculated when the investor wants to find out the price of the underlying asset which must be achieved if an investment in the warrant will be profitable on the expiration date or not. This example will illustrate this; lets imagine that an investor buys a call warrant in Nokia for 5 SEK with the exercise price 100. In this case one call warrants gives the investor the right to buy one share in Nokia. If the spot price of Nokia rises above 100 SEK he starts to make money but because he paid 5 SEK for the warrant, the underlying spot price must rise to 105 SEK to reach break-even. Break-even for a call warrant is calculated in the fol-lowing way.

*Break-even = Exercise price + (parity * price of the warrant)*

Considering a put warrant the break-even is calculated in a slight different way. The break-even price is here showing what price the underlying asset must decrease to before the in-vestment is profitable. That is why the equation for calculating break-even for put warrants looks like following.

*Break-even = Exercise price – (parity * price of the warrant)*

Premium is the percentage an underlying asset must increase to reach break-even. The premium is a measure which can be compared to the investors believes about how much the underlying asset will increase. If the investor believes that the underlying asset will rise, in percentage terms, more than the premium an investment will take place. The premium is calculated in the same way for both call and put warrants.

*Premium = ((Break-even / spot price of underlying asset) -1) * 100</em
*

**Leverage and elasticity**

**Leverage and elasticity**

*One of the most important concepts to why investor invests in warrants is an effect called leverage. This tool is used in the way that it gives the warrant a high exposure in the under-lying asset. Exposure in this sense is how much higher return on invested capital the inves-tor will gain compared to investing in the underlying asset. A leverage of 4 gives the inves-tor 4 times higher return per invested SEK. Although mostly investors thinks in the way that a leverage on 4 leads to that he or she only needs to invest one fourth of the capital to make the exact same amount of money. The leverage is continuously changing and the formula used to calculate it is illustrated below.
Leverage = Spot price / (Price of the warrant * Parity)
The elasticity is well connected to the leverage; it helps the investor to analyze how sensi-tive a warrant is to price changes in the underlying asset. The elasticity describes how many percent the warrant should change in value if the underlying asset is changing by one per-cent. Leverage multiplied with delta becomes elasticity.
Elasticity = Leverage * Delta
*

**Intrinsic value and time value**

*The aspect of time is important when investors are considering an investment in warrants. For every day that passes by, the expiration date is getting closer and for every day the un-derlying asset is not moving in the right direction the chance of reaching breakeven is re-duced. The price of a warrant in general has two elements namely, the time value and the real value. Real value is also called the intrinsic value. What is discussed in the rest of this section concerns call warrants.
The intrinsic value is the financial gain that is received if the warrant is exercised. For all time when the price of the underlying asset is below the exercise, meaning when the war-rant is out-of-the-money, the intrinsic value will be zero. The intrinsic value rises linear when the price of the underlying asset increases, no matter how long time it is to expira-tion. At the date of maturity the warrants value will consist of one hundred percent intrin-sic value and zero percent time value (Eiteman, Stonehill & Moffett, 2004).
The second element of the value is the time value. This value only exists because the price of the underlying asset has a potential to move, closer to or further into the money. The time value is supposed to gain or loose the same value if the underlying asset is moving in either direction from the exercise price. This is only a proof of that the warrant price is cal-culated from models with principal based on an expected distribution of possible outcomes around the exercise price. The time value is by far the most discussed and debated one among investors (Eiteman, Stonehill & Moffett, 2004).
Calculating these two different elements might be difficult sometimes. The best way to show it is by illustrating it with graphs and formulas. The intrinsic value is the sum of money the investor would receive if converting his warrants today instead of waiting for the expiration date (Although that is not possible for European warrants). Resulting in the following formula for call warrants;
*

*Intrinsic value = (Spot Price – Exercise Price) / Parity*And the formula for the time value is;

*Time value = Price of the warrant – Intrinsic Value*Illustrated in figure 2.1 is what happens if the same warrant is compared at two different occasions. The graph to the right shows what has happened to the time value and intrinsic value e.g. one month after the graph on the left side, all else kept constant. This is called the convexity, or more general “the ice hockey stick”. The grey line, the one that is curved, is the total value of the warrant. The black line, which is straight until it reaches the exercise price and then rises linearly, is the intrinsic value line. The two dashed lines in the graphs above shows how big part of the value of the warrant that belongs to the two different elements, time value and intrinsic value. As was discussed earlier in the chapter, the intrinsic value is zero if the spot price is lower than the exercise price. This is illustrated with the two black lines (X) in figure 2.1. After the spot price rises above the exercise price one can se a linear increase in the intrinsic value. If the exercise price of the warrant is higher than then spot price, the time value is one hundred percent of the total value of the warrant. When moving further on in time, from the left to the right graph, the grey line is moving closer to the black line. The reason for this is simply that the closer to maturity, the smaller is the chance that the spot price will increase. Therefore it looks like the shape of an ice hockey stick which is called the convexity principle. X is indi-cating the intrinsic value and the time value is showed by the letter Y. If everything else is kept constant and the only change is that in the right graph the maturity date is closer than it is in the left graph it will result in a decrease in the time value but the intrinsic value re-mains unchanged. The time value will decrease from Y1 to Y2. However, the right graph is showing that a warrant still has a value even if it is far out-of-the-money and close to ma-turity .

*1 Background
1.1 Problem discussion
1.2 Purpose
2 Theoretical framework
2.1 Covered Warrants
2.2 Key ratios of warrants.
2.3 History of the warrant market.
2.4 Risk
2.5 Differences and similarities with options.
2.6 The Issuers.
2.7 Pricing.
2.8 Black & Scholes option pricing model.
2.9 Volatility.
2.10 Previous research
3 Method
3.1 The pre study.
3.2 Introduction to the method of the main study.
3.3 Method of choice.
3.4 Sample
3.5 Time of study and measurement intervals
3.6 Black & Scholes.
3.7 Correlation tests.
3.8 Interviews.
3.9 Validity
3.10 Reliability..
4 Implied volatility findings.
4.1 H&M implied volatility diagrams.
4.2 H&M correlation tests.
4.3 H&M warrants correlation to the stock.
4.4 Ericsson implied volatility diagrams
4.5 Ericsson correlation tests.
4.6 Ericsson warrants correlation to the stock
5 Analysis
5.1 Question 1.
5.2 Question 2
6 Conclusions
7 Reflections and suggestions to further studies.
References
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Covered Warrants How the implied volatility changes over time
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