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Replicating a Call in a Two-Period Binomial Model

Article Quant Q&A · Author: Markus
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Summary

The document presents a question about constructing a self-financing portfolio that replicates a call option in a two-period binomial tree. The investor has option payoffs at the terminal date, arbitrage-free option prices at the intermediate nodes, and the call price at the initial date, along with the stock prices and risk-free return.

It asks how to find the stock and risk-free asset holdings at the initial time and how to rebalance them at the intermediate time. The question frames the task around working through a concrete example rather than relying on an unfamiliar theorem. However, the document contains no answer or derivation, so it does not provide the hedge ratios, calculations, or a completed replication method. Its value is as a clear statement of the inputs and rebalancing problem in discrete-time option replication.

Key ideas

  • A call can be replicated by dynamically holding the underlying stock and risk-free asset in a binomial model.
  • The terminal payoffs and node prices provide the data needed to determine the portfolio holdings.
  • A self-financing strategy may require rebalancing when the tree reaches its intermediate date.
  • The document asks for a worked construction but does not provide the solution.

Tags

OptionsDerivatives pricingPortfolio construction
Full text
# Replicating portfolio: initial portfolio?


# Replicating portfolio: initial portfolio?












I have a bit of trouble understanding how to determine the replicating portfolio of a call using just a stock and the riskfree asset.

I have times $t = 0,1,2$, and at time $2$, we have $3$ payoffs ($(69, 4, 0)$), at time $1$ the arbitrage-free prices are $18$ and $0.8$, and at time $0$, the arbitrage-free price of the call is $4$. (Note that this is just a two-period binomial tree).

How do I determine the replicating self-financing portfolio, both what is needed to be held at time $0$ and what is needed to be held at time $1$? All the other values (stock prices, rate of return on riskfree asset, and equivalent martingale measures) are available as well. I would prefer an answer based on this simple example as opposed to an answer using some theorem which I have not heard of.

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