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Replicating Derivative Payoffs in a One-Period Binomial Model

Article Quant Q&A · Author: Matheus Diógenes Andrade
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Summary

The document explains why a portfolio of stock and a money market account is chosen to match a derivative’s payoff in a one-period, two-outcome binomial model. The derivative has specified payoffs for the up and down outcomes, and the investor selects the initial cash position and stock holding so that the portfolio reproduces both amounts at the end of the period.

The rationale is the no-arbitrage principle: assets with identical future payoffs in every possible state should have the same current value. Once the portfolio replicates the derivative, its cost provides the derivative’s no-arbitrage price. The explanation is conceptual and does not work through a numerical example or derive the portfolio weights. Its setup is limited to a one-period model with two outcomes and assumes the relevant payoffs can be replicated using the available stock and cash account.

Key ideas

  • A derivative in the one-period model is defined by its payoff in each possible outcome.
  • Choose stock and cash holdings so the portfolio matches the derivative payoff in every state.
  • No-arbitrage implies that a replicating portfolio and the derivative should have the same current price.
  • The explanation assumes a two-outcome model and does not derive the hedge positions.

Tags

Derivatives pricingOptionsArbitrage
Full text
# What is a "derivative security"?


# What is a "derivative security"?












I am reading the book Stochastic Calculus for Finance I The Binomial Asset Pricing Model, from Steven E. Shreve. And, at the first chapter "The Binomial No-Arbitrage Pricing Model", in page 5, it says:

> In the general one-period model, we define a derivative security to be a security that pays some amount $V_1 (H)$ at time one if the coin toss results in head and pays a possibly different amount $V_1 (T)$ at time one if the coin toss results in tail.

And then, two paragraphs later, it is said:

> We want to choose $X_0$ and $\Delta_0$ so that $X_1(H) = V_1(H)$ and $X_1(T) = V_1(T)$. (Note here that $V_1(H)$ and $V_1(T)$ are given quantities, the amounts the derivative security will pay off depending on the outcome of the coin tosses. At time zero, we know what the two possibilities $V_1(H)$ and $V_1(T)$ are; we do not know which of these two possibilities will be realized.)

Here,

- $X_0$ refers to the value of our portfolio of stock and money market account at time one $0$;

- $X_1(H)$ the value at time $1$ if event $H$ happens;

- $X_1(T)$ the value at time $1$ if event $T$ happens; And

- $\Delta_0$ the percentage of shares of stock bought at time zero.

My question is: Why do we want to choose $X_0$ and $\Delta_0$ so that $X_1(H) = V_1(H)$ and $X_1(T) = V_1(T)$?

## Answer by Sane (score 4, accepted)

https://quant.stackexchange.com/a/80733

The principle of no arbitrage states that two assets with the same payoff must have the same price. By setting up your portfolio to exactly replicate the payoffs of the derivative security in both scenarios (heads and tails), you ensure that your portfolio will have the same value as the derivative security at time 1, regardless of the outcome.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.