A `two-step' model, an illustration of portfolio switching

How about if we allow time to grow? Let us look at figure :

  

Figure: The `two-step' binomial model for stock market

Let the stock price at box i be s_i. If, when the derivative matures¹⁸ at time 2T, it has a payout of f(i), i=3, 4, 5, 6 respectively.¹⁹ What is the price of this derivative at time zero? Answer: It is easy to handle this problem with the redefined probability q. We let $q_{\alpha}$ be this `probability' from 0 to 2; $q_{\beta}$ be the `probability' from 2 to 6; $q_{\gamma}$ be the `probability' from 1 to 4. Doing the thing backwards, we have:

$$(q_{\gamma}f(6)+(1-q_{\gamma})f(5))\exp(-rT)$$ f(2) = (9)

$$(q_{\beta}f(4)+(1-q_{\beta})f(3))\exp(-rT)$$ f(1) = (10)

where f(2) and f(1) are the prices of that derivative at time T if the stock prices are s₂ and s₁ respectively. With f(2) and f(3), we can get f(1) which is...

$$f(0)\equiv p=(q_{\alpha}f(2)+(1-q_{\alpha})f(1))\exp(-rT)$$ (11)

Note that f(2), f(3), f(4) are generally different, so in most cases if we create a hedging portfolio at time zero, we have to look at how the stock behaves (whether it is at s₁ or s₂) and changes the content of the portfolio at time T to make the cost of the content of the portfolio at time T be f(1) or f(2) respectively.