I. Physics, Mathematics and finance: Bachelier and the Brown-movements » More!
II. Game: How much does the” FAIN IT!” token worth? » More!
III. Options » More!
IV. Options positions » More!
V. Price fluctuation on the financial markets » More!
VI. Buying volatility, sending volatility » More!
VII. Covering the options undertaking » More!
VIII. The pricing of options » More!
IX. Some edification » More!
Language / Sprache
VII. Covering the options undertaking
The call option protects against a possible price rise, the put option protects against a possible price fall. But why is it good to become committed to an option? “Who will write out an option here?”
For somebody who is entitled to a call or put option the loss is limited and the profit is unlimited. His situation is alike to a lottery player's: there is a great chance that he will lose a little sum (option price) but he has a little chance to win a great amount of money.
The committed party of the option (who writes out the option) is in a controversial position: his maximal profit is the option price but his potential losses are unlimited.
Let's see how he can cover himself against the undertaken risk.
• an insurance company that sells us a car insurance
• an investor who sells shares or foreign bills within the confines of a forward business
• an investor who undertakes a sale commitment (i.e. he sold a call option against the option price)
a) The insurance company
On one hand, the company creates a damage community amidst the insured clients and he believes in the fact that not everybody's car is damaged and home burns down at the same time.
And on the other, the company collects data about certain damages that happen and their actuaries (insurance-mathematicians) carry out static calculations for the expected damages that should be paid. Firstly, they calculate what the chances are of our insured client that he having a car crash within one year. Secondly, in case of a car crash, they have to calculate how many times of 1000 is it followed by these or those kind of damages (how sever the crash is). So after these calculations the expected measure of the damage payment can be given. To this calculated sum they also add the company's costs and the anticipated profit. After this, they calculate the present value of the given sum, which shows approximately how much the fee will be.
b) Forward sale
If we have sold a share as a forward business then it is very easy to cover the undertaken position: we buy a share now. From this point we aren't interested in anything. We know the present purchasing price, we know for how much we have made the forward sale, we have no risk and nothing to do.
c) Call sale
12th illustration
The situation is different if, instead of a fixed sale, we have undertaken a sale commitment, i.e. we have sold a call option, for a rate of ex. K=100. At expiration we are covered if we have exactly the same amount of shares that is being bought from us. So, if the rate is over K=100 we will need one piece of share. But if the rate is under K=100 they won't call the option, and we do not prefer to keep the share, neither (12th illustration).
But how can we know now if the exchange rate will be under or above K=100 at the expiration date of the option? And how many shares should we keep meanwhile to make sure our position is covered?
This is not an easy question. But not more complicated than a basketball match, either. In the beginning of the match the teams have equal chances to win. But during the game the more one team gains advantage over the other the bigger chances it gets to win. The present state of the match has a greater effect on the final score if it's near to end of the game. If one party has great advantage over the other just before the end of the game then the match is decided, as a matter of fact. Excitement occurs at the end when it is unsure who will win until the last minute.
The same situation is present in the stock market. However, here the final score is not due to the baskets but to the information that make the exchange rate go up and down. If we knew the possible values of the option for every point of time before the expiration and for every possible exchange rate then we would know how the option's value would react to the changes of the rate and so we could tell how many shares we would need to cover our sell commitment. So, let us assume that we know the K=100 call option's value for every possible point of time, and for every possible exchange rate.
13th illustration
Here we have illustrated the value of the call option for different probable exchange rates as it is T-t = 1 and a half year before expiration.
We have earmarked a few points and for that we have given the rise of the curve, as well. The rise is called delta by option dealers. Its value shows that if we if the optional object's (here the share) price grows by 1 forint then how much will the option's value change. When it is sure that the rate will be over 100 then delta = 1. If the rate is deep under K=100 then the delta is 0.
It gives us the key to resolve the problem. If we always own delta amount of shares then we will be always covered against the little change of the rate. Ex. if the option's delta is 0.5 then we lose 1x0.5 forints on our options position due to the rate rise but we win 1x0.5 forints for our 0.5 piece of shares. If, after a time, the rate changes in a way that the delta rises to 0.7 then we have to buy another 0.2 piece of share. We always have to buy or sell the same amount of shares as the delta changes!
As for the rate, the chances of going up or down are 50:50. The 14th illustration shows how much shares we should own at each point of time at different rate changes.
14th illustration
15th illustration
16th illustration
In theory it may sound a little bit complicated but let's examine the goalkeeper the water polo teem during the final of the Olympics.
He acts exactly the same way as we have seen it before and what is called the dynamic delta hedge on the options market. If the ball flies towards the left side then he moves a little to the left side, as well, preferably he closes the angle. (Maybe the goalkeeper's duty, pushing the water with his feet, is a little bit harder than it is for a broker, who commends a computer program to make a continuous portfolio correction.)
In case of the aforementioned exchange guarantee we can follow the dynamic delta hedge on the 15 th illustration.
