No-arbitrage and replication

What you will be able to do

After this lesson, you should be able to:

• explain why identical dated state-contingent cash flows have the same price under the stated assumptions;
• distinguish exact replication from matching only an expected cash flow;
• calculate a claim value from signed positions in a replicating portfolio.

Compare cash flows, not labels

The law of one price applies when two portfolios produce the same signed-cash-flow at every relevant future date and in every modeled state. The names of the portfolios can differ. What matters is the full set of dated cash flows, from the same perspective.

Suppose the two portfolios had different prices at valuation-time $0$. In the frictionless model of this lesson, a trader could then buy the cheaper portfolio and sell the more expensive one. The future cash flows of the two positions cancel, so the trader keeps the initial difference in price. This arbitrage argument is the reason for the pricing rule. Tuckman and Serrat state the law of one price and connect deviations from it to an arbitrage trade built from a replicating portfolio[1].

Matching only the expected cash flow is not enough. A payment of USD 100 only in an up state is not the same cash flow as USD 100 only in a down state, even if the two states have equal probability.

Value the replicating portfolio

Use the local replication-security-index $\explain{replication-security-index}{i}$ to select one security and the local replication-security-count $\explain{replication-security-count}{n}$ to count them. Let the signed replication-security-quantity be $\explain{replication-security-quantity}{a_{\explain{replication-security-index}{i}}}$ and the current replication-security-value be $\explain{replication-security-value}{V_{0,\explain{replication-security-index}{i}}}$. When this portfolio exactly matches the claim’s future cash flows, the replicated-claim-value is the value of the portfolio:

$$\explain{replicated-claim-value}{V_0^{\mathrm{claim}}}=\sum_{\explain{replication-security-index}{i}=1}^{\explain{replication-security-count}{n}}\explain{replication-security-quantity}{a_{\explain{replication-security-index}{i}}}\explain{replication-security-value}{V_{0,\explain{replication-security-index}{i}}}$$

(1.6.1)

A positive $\explain{replication-security-quantity}{a_{\explain{replication-security-index}{i}}}$ is a long position, and a negative $\explain{replication-security-quantity}{a_{\explain{replication-security-index}{i}}}$ is a short position. The formula uses current values because the portfolio already matches the claim’s cash flows at every future date and in every state.

Suppose a claim is replicated by two units of security A worth USD 12 each and one unit of security B worth USD 7. The value of the replicating portfolio, and therefore of the claim, is:

$$\explain{replicated-claim-value}{V_0^{\mathrm{claim}}}=2(12)+1(7)=\text{USD }31$$

(1.6.2)

Where replication is used next

Forward prices use cash-and-carry replication. Put-call parity compares two portfolios with identical expiry cash flows. A binomial option model constructs a state-by-state replicating portfolio. Risk-neutral valuation is another way to calculate that same no-arbitrage value inside a specified model.

Knowledge check 1.6.1 No-arbitrage and replication

1. DirectQuestion 1Not answered

2. TransferQuestion 2Not answered

3. DirectQuestion 3Not answered

4. TransferQuestion 4Not answered

Model boundary and review note

Real trades can have bid-ask spreads, different borrowing and lending rates, short-sale constraints, collateral, taxes, liquidity differences, and execution risk. These effects can make a textbook arbitrage impossible to execute. This lesson sets up a frictionless pricing benchmark. It does not claim that every observed price difference is a riskless profit.

The source locator, notation, examples, and answer keys remain draft pending independent human review.

References

1. Tuckman & Serrat, Fixed Income Securities: Tools for Today's Markets (4th ed., 2022). Ch. 1 §§1.3-1.4, printed pp. 53-57, law of one price and arbitrage replication. draft ↩