No-Arbitrage Replication

Price a payoff by building a portfolio that copies it; equal payoffs must have equal prices.

What is it?

If a portfolio of traded assets matches a derivative's payoff in every state, the derivative must cost the same as the portfolio.

Forward pricing, put-call parity and binomial trees all come from this single idea; trading interviews test it directly.

Use it when asked for a fair price of a derivative or whether prices allow a free profit.

Think of this method when…

Match the payoff in every state with stock and bond; the replication cost is the price. Equivalently, discount risk-neutral expected payoffs.

Δ = \frac{V _{u} - V _{d}}{S _{u} - S _{d}}, V_{0} = \frac{q V _{u} + ( 1 - q ) V _{d}}{1 + r}, q = \frac{( 1 + r ) S _{0} - S _{d}}{S _{u} - S _{d}}

A stock at 100 goes to 120 or 80 in one period; the interest rate is 0. Price a call with strike 100.

1
Payoffs: 20 in the up state, 0 in the down state.
2
Delta: $Δ = 20/40 = 0.5$ shares. Borrow 40 so the down state pays $0.5 \cdot 80 - 40 = 0$ .
3
Cost today.
$0.5 \times 100 - 40 = 10$
Answer: $10$

A forward on a non-dividend stock at 50 with 1 year to expiry and 4% rate: do you need a model of the stock's drift?

18 QuantSolve questions are tagged with this method.