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!
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VIII. The pricing of options
Let's get back to the game mentioned in the beginning of the presentation. If we get two rose tokens next to our green one than we will be in the same situation as we have had two yellow ones.
17th illustration
The yellow token worth is 1 (as it is sure that we will get 1 unit of money in a flash). The prices of the yellow and green tokens squarely determine the third token's value, in this case. Outside of this game, we could list several examples where two rates unambiguously influence a third one. (17th example)
If the green token's value is 1 (i.e. the risk premium is 0) then, as for the above-mentioned correspondence, the value of the getting-off option is 0.5.
18th illustration
If the organizers of the game, counting on 400 participants, prepare 200 yellow, 100 green and 100blue tokens then they have to be prepared with 700 or 100 prize units, on the whole. We can easily get rid of this insure situation if we prepare 100 yellows less, 100greens more and we reverse the clearing of the blue tokens: we win with the heads instead of the tales. In this case, the organisers have to prepare 400 units of prize, apart from the outcome of the throws.
19th illustration
The daily run of the yields on the BUX strongly reminds us of the heads and tails game (19th illustration).
As for a next step to illustrate the closing of options, let's imagine that a share's rate is 100 and its value rises or falls with 10 forints with every step. How much is an option worth that allows us to buy the share for K=100 after 2 steps? The answer is: 5 dollars. Why?
If we follow this strategy then we will have the same amount of money at expiration as the option is worth.
At the start the rate is 100 and we own 0.5 piece of share.
If the rate grows from 100 to 110 then we buy another half (- 55 $ ).
If the rate falls to 90 from 100 then we sell our half (+ 45 $ ).
At the start we buy 0.5 piece of share with a credit of 45 $ .
For this we need: 0.5x100 45 = 50 45 = 5 $ , This is the call's starting value.
We follow the delta-strategy: we get the probable final values of the call.
20th illustration
21th illustration
The following strategy illustrates a fork position that provides call and put options for a K = 100 forints rate after 4 steps.
At the start we have 0 shares and 15 $ on deposit.
For this we need. 15 $, which is the call's starting value.
We follow the delta-strategy: we get the probable final values of the call.
If we imagine the rate changes as a continuous function, instead of discreet leaps, then the Brown-movements and its mathematical theory can help us. This was Bachelier's unaccepted thought in 1900. A process characterised with the Brown-movements can be described with randomly given points. It is a random zigzag that has cut-off points in all of its points. Like a forward player's game when he meets more and more defensive players.
As it's also very hard to imagine these kinds of progressions but we can scarely believe that there are closed formulas, the so called Ito-formulas, which help us to define its transforms (i.e. how does the random process accumulates that is the logarithm or the square of the original one). This formula has been known since 1951 and in finance was firstly used by Merton in the 70's, even for pricing the options.
We can follow the process of a share, in 1995, and also the run of its call option's value, which was created in September. The share's rate is a Brown-movement, and the run of the option's value is its transform.
With the help of the Ito-formula we can find the coefficient of the dw, the risk taker amidst the two Brown-movements. With this we can get to know how much of the basic product we need at certain moments to cover our sale commitment. Delta changes from one moment to the other; however, with a continuous selling and buying we can constantly maintain a risk-free position.
There is a function equation, set up by Black and Scholes, which is used to find the options' value for every point of time and for every probable rate (i.e. c(S, t) function). It is a second-rate partial differential function. It describes the connection between the wanted function's curve and gradient towards the rate and between its gradient towards the time and the functions altitude.
According to the equation: we can always cover the risk that comes of selling a piece of option by continuously changing the number of the shares we own. A risk-free portfolio's yield is the interest rate.
The description of options closing and its mathematical apparatus is also available to examine the relations between the future set of the interest rates and interest rates with different durations. And, presumably, this kind of usage should be much more important than using it for pricing options in connection with shares and foreign bills.
22th illustration
23th illustration
For example, the mobility of beginning an investment that depends on the oil price greatly raises the project's value. The left half of the 34 th illustration shows that if the mining had to be started immediately then, according to the present oil prices, it would be loss-making. If the start of the project can be postponed then we can get use of a future oil price rise i.e. due to the price stagnation. In this case, we can begin the exploitation profitably. But the measure of the price fall does not influence us: we do not start exploiting the oil neither when the business is just a little bit unprofitable or when it is making a Great lots.
24th illustration
25th illustration
After the closing the volume of the options has risen with great pace.
