Replicating a One-Step Binomial Option with Stock and Bond Positions

Summary

The document explains how to construct a portfolio of stock and a risk-free bond that matches an option's payoff in a one-period, two-outcome binomial model. In the up state, the portfolio's stock and bond values must equal the option value; the same condition applies in the down state. These two equations determine the stock holding and bond holding when the two stock outcomes differ.

The response interprets the question as a single stock with an up and down terminal price, and a bond that grows at the risk-free rate. Solving the two payoff-matching equations gives the hedge positions, with the stock quantity representing shares and the bond quantity the financing position. This is a basic replication argument, not a numerical worked example. It assumes the stated two-state setup and does not discuss transaction costs, early exercise, changing hedge positions over multiple periods, or whether the market permits the required trades.

Key ideas

• Match the portfolio value to the option payoff in each possible terminal state.
• The up-state and down-state conditions form two equations for the stock and bond holdings.
• The bond grows at the stated risk-free rate over the period.
• The explanation applies to a one-period, two-outcome model and does not cover practical trading frictions.

Tags

# Replication of the portfolio in single step binomial model


# 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.

## Answer by Richi Wa (score 2)

https://quant.stackexchange.com/a/17443

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.

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