m10-binomial-model

M10 – Binomial Model: CFAI Practice Problems

Source: CFAI CFA1 Derivatives Practice 2026 – Volume 7 Back to module: m10-binomial-model

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Exhibit: SAPP

Sarah Park (SAPP) introduces the one-period binomial option pricing model to junior analysts. She explains how to construct a replicating portfolio, calculate risk-neutral probabilities, and price options using the binomial framework. She presents several scenarios involving changes in model inputs and asks analysts to evaluate the impact on option values.

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Question 1

In a one-period binomial model, all else being equal, if the size of the down move increases (i.e., the down factor $d$ decreases), the value of a European put option will most likely:

• A. Decrease
• B. Increase
• C. Remain unchanged

Answer

A. Decrease

When the down factor $d$ decreases (larger down move), the risk-neutral probability of an up move (${\pi }_u$) increases (since ${\pi }_u=\frac{(1+r)-d}{u-d}$ and a smaller $d$ increases the numerator). A higher probability of an up move means less weight on the down-state payoff, which is where the put has value. This decreases the expected payoff of the put under risk-neutral probabilities, reducing its value.

📖 Giải thích chi tiết

Ôn lại khái niệm: Binomial model — Risk-neutral probability: ${\pi }_u=\frac{(1+r)-d}{u-d}$ ${\pi }_d=1-{\pi }_u$

Option value: $p_0=\frac{{\pi }_u\cdot p_u+{\pi }_d\cdot p_d}{1+r}$

Phân tích khi $d$ giảm:

1. ${\pi }_u=\frac{(1+r)-d}{u-d}$ — tử số $(1+r)-d$ tăng khi $d$ giảm
2. Mẫu số $u-d$ cũng tăng, nhưng tử số tăng nhanh hơn → ${\pi }_u$ tăng
3. ${\pi }_d=1-{\pi }_u$ giảm
4. Put có payoff cao ở down state → ${\pi }_d$ giảm → expected payoff giảm → put value giảm

Tại sao A đúng: $d$ giảm → ${\pi }_u$ tăng → ít trọng số vào down state (nơi put ITM) → put value giảm.

Tại sao B sai: Mặc dù down move lớn hơn tạo payoff put cao hơn trong down state, risk-neutral probability ${\pi }_d$ giảm đủ để net effect là put value giảm. Tại sao C sai: Thay đổi $d$ ảnh hưởng trực tiếp đến risk-neutral probabilities và do đó ảnh hưởng option value.

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Question 2

In the one-period binomial model, option values are determined by:

• A. Actual (real-world) probabilities of up and down moves discounted at the expected return
• B. Risk-neutral probabilities of up and down moves discounted at the risk-free rate
• C. Risk-neutral probabilities of up and down moves discounted at the expected return on the underlying

Answer

B. Risk-neutral probabilities of up and down moves discounted at the risk-free rate

The binomial model uses risk-neutral pricing: expected payoffs are calculated using risk-neutral probabilities (${\pi }_u$ and ${\pi }_d$) and then discounted at the risk-free rate. This approach gives the same result as the replicating portfolio method and does not require knowledge of actual probabilities or expected returns.

📖 Giải thích chi tiết

Ôn lại khái niệm: Risk-neutral pricing principle: $V_0=\frac{{\pi }_u\cdot V_u+{\pi }_d\cdot V_d}{1+r_f}$

Trong đó:

• ${\pi }_u,{\pi }_d$: Risk-neutral probabilities (không phải actual probabilities)
• $r_f$: Risk-free rate (không phải expected return)
• $V_u,V_d$: Option payoffs in up/down states

Tại sao B đúng: Binomial model sử dụng risk-neutral framework:

1. Tính risk-neutral probabilities từ $u,d,r_f$
2. Tính expected payoff dưới risk-neutral measure
3. Discount bằng risk-free rate

Ưu điểm: Không cần biết actual probabilities hay risk preferences của investors.

Tại sao A sai: Actual probabilities + expected return là real-world pricing — phức tạp hơn và cho cùng kết quả. Binomial model dùng risk-neutral approach đơn giản hơn. Tại sao C sai: Risk-neutral probabilities phải discount bằng risk-free rate, không phải expected return. Sử dụng expected return sẽ “double count” risk adjustment.