CFA Level I Exam · Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives

Replication and Risk-Neutral Pricing in Derivatives

Updated 7 October 2026 · Fact-checked

Replication means building the payoff of a derivative using the underlying asset and a risk-free bond. By no-arbitrage, the derivative must cost the same as that portfolio. Risk-neutral pricing gets the same price by using risk-neutral probabilities and discounting the expected payoff at the risk-free rate.

Understand Replication and Risk-Neutral Pricing

A derivative gets its value from an underlying asset. If you can copy the derivative's payoff exactly with the underlying and a risk-free bond, the derivative must cost the same as the copy. If it did not, you could buy the cheap one, sell the expensive one and lock in a risk-free profit. This is the law of one price and the no-arbitrage rule.

Take a forward contract. Its payoff at expiry is S(T) − F0(T). Buy the underlying today and borrow the present value of F0(T). At expiry you owe F0(T) and hold the asset worth S(T). That is the same payoff as a long forward. So the long forward is replicated by a long asset position plus a short (borrowed) zero-coupon bond. The reverse, short forward, is a short asset position plus lending. Rearranging gives the forward price: F0(T) = S0 × (1 + r)^T for an asset with no income or carry costs.

For options, the replication in a one-period binomial tree is a static portfolio. It would be rebalanced each period in a multi-period tree. You hold h units of the underlying and a bond position B so that the portfolio matches the option payoff in both the up state and the down state. The number of units h is the hedge ratio (delta). Because the replicating portfolio and the option have the same payoffs in every state, the option value equals h × S0 + B.

There is a shortcut. Risk-neutral probabilities are the probabilities that make the underlying's expected return equal to the risk-free rate. Under them, the derivative's value is the expected payoff discounted at the risk-free rate. They are not real-world probabilities. They are a pricing device. Real-world probabilities reflect investors' beliefs and risk aversion. The price does not depend on the real-world probabilities or on risk preferences, which is why the method works.

Key formulas to remember

Forward price (no income or carry cost): F0(T) = S0 × (1 + r)^T; Comes from replicating a long forward with the asset bought using borrowed money.
Replicated long forward: Long forward = Long underlying + Borrow PV of F0(T); A short forward is the reverse: short the underlying and lend.
Up and down factors: u = S1(up) ÷ S0 ; d = S1(down) ÷ S0; Used in the one-period binomial tree.
Risk-neutral probability of an up move: π = (1 + r − d) ÷ (u − d); The down probability is 1 − π. Here r is the per-period risk-free rate and there is no income on the underlying.
Hedge ratio (delta): h = (Vu − Vd) ÷ (Su − Sd); Units of underlying held to replicate the option. Vu and Vd are option values in the up and down states.
Option value by replication: V0 = h × S0 + B, where B = (Vu − h × Su) ÷ (1 + r); B is the bond position. A negative B means borrowing.
Option value by risk-neutral pricing: V0 = [π × Vu + (1 − π) × Vd] ÷ (1 + r); Gives the same value as replication.

How to solve Replication and Risk-Neutral Pricing questions

Use this method for any replication or risk-neutral pricing question. Both routes give the same answer, so pick the one that is quicker.

1Write down S0, the up price Su, the down price Sd, and the one-period risk-free rate r. Check whether the underlying pays income.
2Compute u = Su ÷ S0 and d = Sd ÷ S0 if you need the factors.
3Compute the option payoff in each state. For a call: max(S − X, 0). For a put: max(X − S, 0).
4For risk-neutral pricing, compute π = (1 + r − d) ÷ (u − d), then 1 − π.
5Compute the expected payoff using π and 1 − π, and discount one period at 1 + r.
6For replication, compute h = (Vu − Vd) ÷ (Su − Sd), then B = (Vu − h × Su) ÷ (1 + r), then V0 = h × S0 + B.
7Check the answer: the value must be non-negative. A call value is at most S0, and a European put value is at most X ÷ (1 + r). If the options are close, test with the other method.

Quickest way: Risk-neutral shortcut for a one-period tree

When to use it: Use it when the question asks only for the option value and not for the hedge ratio or the bond position.

Find π = (1 + r − d) ÷ (u − d).
Compute the payoffs Vu and Vd.
Compute [π × Vu + (1 − π) × Vd] ÷ (1 + r).
Keep π in memory on the calculator to avoid rounding errors. On the TI BA II Plus, press STO 1 after computing π, and RCL 1 when you need it.
Eliminate options that are negative, options above S0 for a call, and options above X ÷ (1 + r) for a European put.

Common mistakes in Replication and Risk-Neutral Pricing

Using real-world probabilities to price the derivative.
The question gives an up probability, so it looks like it should be used.
Fix: Real-world probabilities are a distractor for pricing. Use π from (1 + r − d) ÷ (u − d).
Forgetting to discount the expected payoff at the risk-free rate.
Students stop after computing the expected payoff.
Fix: Always divide by (1 + r) per period. The discount rate is the risk-free rate, not the required return on the underlying.
Getting the sign of the bond position wrong in replication.
B is computed as a number and its meaning is not checked.
Fix: A long call is replicated by buying h units of the underlying and borrowing. So B is usually negative for a call. A positive B means lending.
Mixing up u and d, or using u − 1 instead of u.
Up move of 20% is read as u = 0.20.
Fix: u = 1.20 for a 20% rise, and d = 0.80 for a 20% fall. Use the factor, not the percentage.
Thinking risk-neutral probabilities mean investors are risk neutral.
The name is misleading.
Fix: Investors can be risk averse. Risk-neutral probabilities are a mathematical tool that gives the same price as replication.

Worked examples

Example 1

A stock trades at $50. In one period it will be either $60 or $42. The risk-free rate is 5% per period. The stock pays no income. What is the value of a one-period European call with a strike of $50? Options: A) $4.76 B) $5.56 C) $5.71

Show the solution

u = 60 ÷ 50 = 1.20 and d = 42 ÷ 50 = 0.84.
π = (1.05 − 0.84) ÷ (1.20 − 0.84) = 0.21 ÷ 0.36 = 0.58333.
Call payoffs: Vu = max(60 − 50, 0) = 10 and Vd = max(42 − 50, 0) = 0.
Expected payoff = 0.58333 × 10 + 0.41667 × 0 = 5.8333.
Discount: 5.8333 ÷ 1.05 = 5.5556.
Check by replication: h = (10 − 0) ÷ (60 − 42) = 0.5556. B = (10 − 0.5556 × 60) ÷ 1.05 = (10 − 33.333) ÷ 1.05 = −22.222. V0 = 0.5556 × 50 − 22.222 = 27.778 − 22.222 = 5.556.

Answer: The call is worth $5.56 by both methods. The answer is B.

Example 2

A stock is at $100. In one period it will be $125 or $80. The risk-free rate is 4% per period and the stock pays no income. What is the hedge ratio (delta) for a one-period call with a strike of $100, and how many units of the stock does the replicating portfolio hold? Options: A) 0.22 B) 0.44 C) 0.56

Show the solution

Call payoffs: Vu = 125 − 100 = 25 and Vd = max(80 − 100, 0) = 0.
Hedge ratio h = (Vu − Vd) ÷ (Su − Sd) = (25 − 0) ÷ (125 − 80) = 25 ÷ 45.
h = 0.5556.
The replicating portfolio holds 0.5556 units of the stock, financed partly by borrowing.
B = (25 − 0.5556 × 125) ÷ 1.04 = (25 − 69.444) ÷ 1.04 = −42.735. The value is 0.5556 × 100 − 42.735 = 12.82, which is positive and below $100.

Answer: h = 0.5556, which rounds to 0.56. The answer is C.

Exam tips

If a question gives a real-world up probability and asks for a derivative price, ignore it for pricing. It is usually a trap.
Know both routes. If a question asks for the hedge ratio or the borrowing amount, use replication. If it asks only for value, use risk-neutral pricing.
For forwards, remember the replication in words: long forward equals long asset plus borrowing the present value of the forward price.
Sanity-check every answer. A call cannot be worth more than the underlying, and a value must not be negative.
Convert percentage moves to factors before computing π, and keep unrounded π in the calculator.

Practice questions from Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives

Replication and Risk-Neutral Pricing: frequently asked questions

How do you replicate a forward contract with a risk-free bond?

For a long forward, buy the underlying and borrow the present value of the forward price by issuing a zero-coupon bond. At expiry you hold the asset and repay the forward price, which matches the long forward payoff of S(T) − F0(T). For a short forward, short the asset and lend the present value of the forward price.

What is the difference between risk-neutral and real-world probabilities?

Real-world probabilities describe how likely each outcome actually is, and they reflect investors' risk aversion. Risk-neutral probabilities are those that make the underlying earn the risk-free rate on average. You use them only for pricing, discounting at the risk-free rate.

Why does risk-neutral pricing give the same price as replication?

The replicating portfolio matches the derivative's payoff in every state, so no-arbitrage forces the same price. Risk-neutral probabilities are built so that the underlying is priced correctly under them. Applying them to the derivative's payoffs therefore gives the same value.

Does the real-world probability of an up move affect the option price?

No, not in the replication framework. The option value depends on the up and down prices, the strike, and the risk-free rate. The real-world probability affects the expected return, but not the arbitrage-free price.