What is the value of an option?
This is THE QUESTION
for practitioners in the derivatives business. You can invent all kinds of
options - we looked at EUROPEAN CALLS/PUTS, but there are
more of them.
For selling and trading options you need to determine their fair
price, i.e. the price the option is worth.
Let us use our PACHINKO
MODEL OF STOCKS .
The pachinko board tells us the probabilites of the stock prices,
eg, from 100 attempts the stock stays unchanged in 40 cases, it increases
by 10 % in 30 cases and so on. In each event you know the amount of money
you receive - the payoff.
A European Call Option that expires one year from now. The underlying
stock costs $100, the strike price is also
$100. The stock price evolves randomly
over the year in the Pachinko manner. Assume the following Pachinko result
(4 bins) for 100 attempts:
|(a) 10 attempts
||= 10% of attempts
||price + 20%
|(b) 40 attempts
|(c) 30 attempts
||>= 30 %
|(d) 20 attempts
||= 20 %
For each of those cases you can determine the payoff of a CALL option
at expiration date.
(a) Payoff is $120 - $100 = $20
(b) Payoff is $110 - $100 = $10
(c,d) trash your option, payoff = $0
As stated a couple of times option prices reflect not a specific outcome
but the AVERAGE of all.
The AVERAGE is : 0.1
(= 10%) times $20 + 0.4
(= 40%) times $10 + 0.5
(the rest) times $0 = $6
Price of the Option:
One more point: The $6 is the expected return
after the option expired in a year's time. However,
instead of buying the option you could have
put the money into an account and collect interest.
Therefore the price of the option is the amount
of money that will return $6 after one year.
Assuming for instance a risk-free
interest rate of 5% a year, this means - after doing the math
if you start out with $5.71 you will have a total of $6 after
collecting interest for a year.
Price of the Option: $ 5.71
THAT'S IT FOR PART I!