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 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 many 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.

Example:

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% \$120 each (b) 40 attempts = 40% price +10% \$110 each (c) 30 attempts >= 30 % price unchanged \$100 each (d) 20 attempts = 20 % price -10% \$90 each

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: 6 Dollars

 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.

 (Corrected) Price of the Option:   \$ 5.71

THAT'S IT FOR PART I!