Options

CALL OPTIONS

When you buy a call option you have the right (not the obligation) to buy a stock at a predetermined price, which is called exercise price or strike price, over a specified period of time. We distinguish two kinds of options:

• European call options: when the option can be exercised only when it matures;

• American call options: when you can exercise the option at any time until maturity.

Suppose that a stock is trading at $500, you can buy a call option with an exercise price of $500 and a period of 6 years. This means that, during those 6 years, you can decide to buy that same stock at $500. Now, it not convenient for you to do so if the price of the stock is lower than the exercise price. Suppose that after those 6 years the stock is worth $400, why would you buy that stock for $500? In this case you will prefer to buy the stock at its true price and, the option will not be exercised and therefore will be worthless. An option therefore is valuable only if the stock price becomes higher than the exercise price. In that case, the value of the option will be exactly the market price of the share minus the strike price. However, the value of the option does not exactly represent the profit the owner makes because, when he buys the option, he has to pay a fee. The price of the option decreases as the strike price increases because, the higher the price you choose as strike price, the less probable is that the stock will reach that price. Conversely, the fee of the option increases as the time period increases. This is because it is more probable that a stock increases its price by a large margin in a long period than in a short one. We can represent the relationship between the stock price and the value of the option as in the following picture:

PUT OPTIONS

When you buy a put option, you have the right (not the obligation) to sell a stock at the predetermined exercise price over a specified period of time. A put option is profitable when the price of the stock (during that specified period of time) is lower than the exercise price. Imagine that a you buy a put option on a stock with exercise price of $500 with a period of 6 years. If the price turns out to be higher than the exercise price, you will not exercise the option. This is because, you could earn more money by selling the stock yourself on the market. If instead, during these 6 years, the price of the stock is $400. In that case you are willing to exercise your option and you will sell a stock valued on the market $400 at a price of $500. So, the value of the option is: exercise price minus the market price of the share. Obviously, this is not your true profit because you paid the fees to buy the option at first, so your profit will be $100-fee. This price will depend on the time horizon of the option and on the exercise price. As the time horizon increases, the fee increases for the same reason of the call options. However, the fee for a put option will increase as the exercise price increases. This is because it is less probable that the stock will have a sharp increase in price over the same period of time. The relationship between the stock price and the value of the option can be depicted by the following picture:

THE SELLER’S PERSPECTIVE

Obviously, if you can buy an option, there is someone willing to sell or “write” it. When you are selling a call, you are giving the buyer the right to buy from you your shares at the exercise price within a specific time frame. In case the option is exercised, you have the obligation to sell your shares. You will benefit from the sale of the call option only if the price of the option will not rise above the exercise price. If this happens, the buyer will exercise the call, he will make a profit and you will incur in a loss equal to the exercise price minus the market price of the share. Suppose that you wrote an option promising to sell shares at a price of $500 to an investor if he is willing to do so. If the price of the shares rises until $600, the buyer will exercise the option and you will have to sell $600 worth of shares at a price of $500. So you incur in a loss of -$100. If instead you are selling a put option, you are giving the buyer the right to sell you shares at the exercise price within a specific time frame. Again, if the option is exercised, you will have the obligation to buy those shares. You will benefit from the sale of your option only if the price of the option stays above the price of the share. If this happens, the buyer will exercise the call, he will make a profit and you will incur a loss equal to the market price minus the exercise price. Suppose that you wrote an option promising to buy shares at a price of $500 to an investor if he is willing to do so. If the price of the shares is $400 and the buyer exercises the option, you will be obliged to buy $400 worth of shares at a price of $500. Therefore you will incur in a loss of -$100. The value of the options for both sellers is represented below.

DO NOT FORGET ABOUT THE FEES

Now that we introduced the essential concept, we have to modify our diagrams to take into account the fees a buyer pays when buying an option. If we leave the graphs as represented above, buying an option will always be a sure investment. This is because at worst the option will not be exercised and it will be worth 0. Selling an option would instead mean a financial suicide: the seller in the cases depicted above loses or at best makes 0 profits. Why should then people sell options? The answer is because, when a buyer acquires an option he pays the fee to the seller. This means that, in order for a call option to be really profitable for the buyer, the stock price must be equal to the exercise price plus the fee. A put option in instead really profitable when the price of the stock is lower than the exercise price minus the fee. The seller of a call option will earn money as long as the stock price is equal to the exercise price plus the fee. The seller of a put option will earn money until the stock price is equal to the exercise price less the fee. Suppose that the fee is $50, we have to modify our graphs as follows:

COMBINING FINANCIAL ASSETS

If you buy a stock, your payoff will exactly be represented by the 45˚ line in the graph.

This is because, when the stock price falls by $1 you exactly lose $1 while, when the stock price rises by $1 you earn $1. Now, imagine a situation in which you can earn money when the stock price rises but you will not risk to lose money when the stock price falls.

Now imagine the opposite situation: a situation in which you will lose money when the stock price decreases but you will earn nothing when the stock price increases.

Both of these strange outcomes are possible. How? By combining various financial assets. The second imaginary scenario can be obtained in reality by buying a stock at a certain price and selling a call at exactly the same price. If the price of the stock decreases, you will lose money because you own the share while, if the stock price increases, you will not earn any money because the buyer of the call will exercise it thus buying from you the stocks at the exercise price. The following graph shows that combination graphically:

The first situation instead will be the result of the you buying the share of the company and buying a put having exercise price equal to the shar price. If the share price increases, you will earn money thanks to your share and you will not exercise the option. If instead the price falls, you will exercise your option and sell the shares at the exercise price anyway. The sum of the financial assets is graphically represented by:

There us another way in which you can obtain the same result: by buying a call and by setting the same quantity of money as the exercise price in a bank account. So, if the price of the shares rises above the exercise price, you will exercise the call thus making a profit. If instead the price of the stock is below the exercise price, you will not exercise the call but you will still have the money in your bank account. Once again the combination is represented as follows:

Because these two strategies give the same result, we can set them equal to each other obtaining the fundamental relationship for European options:

Value of call + exercise price(present value) = value of put + share price

So the payoffs of:

Buy call + invest the present value of exercise price in a safe asset

Is equal to:

Buy a put option + buy the share

This relationship is called put-call parity and you can twist it around to obtain:

Value of put = value of call + exercise price (present value) – share price

So the payoff of buying a put is equal to:

Buy a call option + invest the present value of exercise price in a safe asset + sell the share

Also this relationship can depicted graphically as follows:

HOW TO DECOMPOSE A PATTERN

When you want to decompose a certain payment scheme in financial investments, you can do so by drawing a diagram representing it and then decompose it. Suppose that a firm wants to give a bonus to the CEO of $10,000 for every dollar that the stock price of the company exceeds the current price of $24. However, the maximum bonus the company is willing to pay is $500,000. This means that to obtain the maximum bonus, the stock price will have to be: 500,000 = 10,000(P-24), where P is 74. The CEO will have therefore nothing if the price is less than $24, 24*(P-24) if the stock is priced between 24 and 74 and will have 500,000 when the stock price is 74 or more. We can graphically represent the situation as follows:

You can imagine this payment scheme as the sum of the purchase and the sale of a call option. You can imagine of purchasing a call option with exercise price of 24 so that you won’t have to pay anything if P<24 but you will earn profit if P>24. Remember that the bonus the CEO can earn is maximum when P=74 so, after that figure, he will not earn anything. You can think of that as the sale of a call option with exercising price of exactly 74. This is because remember that this kind of options are valueless when P<exercise price and will start losing money when P>exercise price. So we have successfully described the payment scheme of our CEO using financial assets. In general any payoff scheme depending on the value of an asset can be decomposed in options on that same asset. By combining financial assets you can create new financial instruments. The field of finance that studies this processes is called financial engineering.

OPTION VALUES

Let’s consider once again a call option with exercise price of $500. If the price of the stock ends up being less than $500, the option will be worthless while, if the price of the stock ends up being more than $500, the option will be worth P-500. This relationship indicates the minimum price of the option. This is because, imagine that the option has some value because P>500, let’s say $550 but that is traded at a price less than its value P-500, let’s say $30. It would be convenient for investors to buy the option, exercise it and sell the stocks. In our case, this would mean a revenue of $550-500=$50 at a cost of just $30 giving therefore a profit of $20. This is an arbitrage opportunity that would not last long on the market. The line depicting the value of the option is therefore the lower bound of the option price. The upper bound of the option’s price is instead the diagonal representing the stock’s performance. This is because the option can not be worth more than the stock on which is based. The price of the option lies on the red curve represented in the shaded area. This line starts at 0 and then rises until it becomes parallel to the lower bound. The line starts at 0 because in that point also the stock is valued 0. If the stock is valued 0, it means that it will never be profitable again and therefore buying an option on that stock makes no sense therefore its price is also 0.

The line becomes parallel and closer to the lower bound because, as the price of the share increases, the probability that the option is exercised is so high that it can be considered as certain. Furthermore, when the price is so high compared to the exercise price, the probability that the owner of the option does not exercise it until the price of the stock becomes again lower than its exercise value is very low. Therefore in a situation like that you could consider yourself as the owner of the stock with the only difference that you have not paid for the stock yet. You will do so when you exercise your right on the stock. Investors buying stocks via option are therefore buying those stocks on credit. Because you are fixing the price at which you will buy the stock in the future, the option will be more valuable, when it’s a long term one and when the interest rates are high. Option prices will always exceed their values because there is always hope that the stock price will get higher than the exercise price. One of the factors influencing the height of the price curve is the variability of the stock considered. If a stock is more volatile, its price will move substantially making it more likely to exceed the exercise price. The buyer of an option will always prefer to buy a high volatile option because the options payoffs are not symmetric: if the price of the stock is lower than exercise price the option is worthless and he will not pay anything (apart from the cost of the option) while, if the price of the stock turns out to be higher than the exercise price, you will receive positive payoffs. The probability that a stock changes its price substantially is higher when considering more volatile (σ^2) stocks and when the time period (t) is longer. Therefore the value of the options depends on the cumulative variability t* σ^2. The time premium of an option is defined as the price of the option minus its value.