# Replication of the portfolio in single step binomial model

I would be grateful if anyone would comment how to construct this:

Assume $S_{i}^k$ is a stock price at time level $i$ and at price level $k$. Assume option is written on $S$ with a a payoff $f_{T}^{k}$ at maturity. Let $f_{i}^{k}$ be the value of the payoff at time level $i$. $B_{i} = B_{0}e^{rt_{i}}$ is the bond price at $t=t_{i}$.

Construct a portfolio of a stocks and b bonds at $t=0$ according to $\Pi_{0} = aS_{0}^{0} + bB_{0}$. Evaluate a and b such that the value of the portfolio replicates the payoff of the option.

• Do I understand correctly: there is one stock and one time step? You have $k$ price levels. How many are these? 2 ? – Ric Apr 20 '15 at 9:38

I read the question as follows: You have one stock $S_0$ and after one period it either goes up to $S^+$ where the option takes the value $f^+$ or it goes down to $S^-$ where the option takes the value $f^-$. The bond grows from $B_0$ to $B_1 = B_0 \exp(r)$. Then you need to solve $$a S^+ + b B_1 = f^+ \\ a S^- + b B_1 = f^-$$ for $a,b$ which are $2$ equation in $2$ unknowns which has a solution - $a$ the number of shares to buy and $b$ the investment in the bond.