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I don't get the intuition behind the construction of long option + short 1/2 stock portfolio for finding the value of an option using binomial model. In Paul Wilmott Introduces Quantitative Finance, he uses the binomial asset model to find the option value. There are three assumption of his model

  • We will have a stock, and a call option on that stock expiring tomorrow.
  • The stock can either rise or fall by a known amount between today and tomorrow.
  • Interest rates are zero.

And the underlying is currently worth \$100 and can rise to \$101 or fall to \$99 between today and tomorrow, where the probability of a given underlying to rise or fall are 0.6 and 0.4, respectively. Based on this simple model, we have to find the value of such an option.

Usually as a beginner in quant finance we will think the value of an option should be 0.6 since the net return is 1 if the stock rises and 0 if the stock falls. Then the expectation values of the value of the option is 0.6. However, Paul Wilmott says this is not correct and the correct option price should be 0.5.

To show the correct option value, he constructs a portfolio to long an option and short 1/2 stock. After some simple maths, he shows that by the risk-neutrality and no-arbitrage principle, the correct option value is 0.5. However, where does the intuition of constructing such a portfolio come from? Why is 1/2 instead of other numbers? Why do we have to short the stock instead of long?

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    $\begingroup$ Because that is what replicates the option. $\endgroup$
    – Arshdeep
    Commented May 19 at 9:39
  • $\begingroup$ 1/2 is a rough approximation for the Delta of a near ATM or ATMF option. In general the Delta of a call varies between 0 and 1, but 1/2 can be considered "typical" in this sense. $\endgroup$
    – nbbo2
    Commented May 19 at 12:37

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This is a classic interview question, btw.

Probabilities you mention are "subjective" hence are irrelevant for option pricing. You need to use risk neutral stock migration probabilities, such that expectation of discounted values of the stock equal the present value. Since rates are zero, expected value of stock should equal the present value. Denoting $p$ the probability of going to 101,

$101p + 99(1-p) = 100$

gives $p=0.5$.

You can now use it to price the option.

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