CFA Level II 2025 Derivatives: Black-Scholes vs Binomial—When to Use Which?

CFA Level II 2025 Derivatives: Black-Scholes vs Binomial—When to Use Which?

Portfolio derivatives constitute a core topic in the CFA Level II curriculum, demanding not only mastery of mathematical techniques but also judicious model selection. Two of the most widely taught option-pricing frameworks are the Black–Scholes–Merton (BSM) model and the Cox–Ross–Rubinstein (CRR) binomial model. Each has distinct assumptions, strengths, and limitations. This article examines the theoretical foundations and practical applications of both models, provides copy-friendly formulas, and offers guidance on when to deploy each in 2025’s market environment.

1. Origins and Conceptual Differences CFA Level II 2025 Derivatives, Black-Scholes vs Binomial

• Black–Scholes–Merton (BSM): Introduced in 1973 by Fischer Black, Myron Scholes, and extended by Robert Merton, the BSM model derives a closed-form solution for European calls and puts by solving a stochastic differential equation under the risk-neutral measure.

• Cox–Ross–Rubinstein (CRR) Binomial Model: Developed in 1979, the binomial approach builds a discrete-time recombining lattice in which, at each step, the underlying can move up by factor u or down by factor d. The option’s value is then determined by backward induction under risk-neutral probabilities.

While both techniques rest on the risk-neutral valuation principle, their mechanics—continuous vs. discrete time—drive their respective use cases.

2. Black–Scholes–Merton Model

2.1 Key Assumptions

1. Underlying follows geometric Brownian motion with constant volatility (σ) and drift equal to the risk-free rate (r).

2. No dividends are paid during the option’s life.

3. Continuous trading, no transaction costs or taxes.

4. Constant, known interest rate and volatility.

5. Markets are frictionless and liquid.

2.2 Formula for European Call and Put

Let S₀ = current spot price, K = strike price, T = time to expiration (in years), r = continuously compounded risk-free rate, σ = volatility, N(·) = standard normal CDF. Define:

d₁ = [ln(S₀/K) + (r + ½σ²)T] / (σ√T)
d₂ = d₁ - σ√T

Then:

Call:  C = S₀ N(d₁) – K e^(–rT) N(d₂)
Put:   P = K e^(–rT) N(–d₂) – S₀ N(–d₁)

These closed-form solutions allow instantaneous pricing of European options.

2.3 Limitations in Practice

• Dividends: Absent adjustment, BSM misprices dividend-paying stocks.

• Early Exercise: Cannot value American options with early exercise features.

• Constant σ & r: Real markets exhibit stochastic volatility and shifting rates.

• Jumps & Skews: Heavy tails and volatility smiles in option markets contradict BSM’s lognormal assumption. CFA Level II 2025 Derivatives, Black-Scholes vs Binomial

3. Cox–Ross–Rubinstein Binomial Model

3.1 Lattice Construction

Divide time T into N equal intervals of length Δt = T/N. At each node, the underlying price either:

• Goes up by factor u = e^(σ√Δt)

• Goes down by factor d = 1/u

The risk-neutral probability p is:

p = [e^(rΔt) – d] / (u – d)

Where r is the continuously compounded rate per interval.

3.2 Backward Induction

1. Terminal Payoffs: At node (i, N), compute option payoff:

   • Call: max(S_{i,N} – K, 0)

   • Put: max(K – S_{i,N}, 0)

2. Discount and Roll Back: At node (i, j):

   V_{i,j} = e^(–rΔt) [ p V_{i+1,j+1} + (1–p) V_{i,j+1} ]

3. American Option: At each node, allow early exercise:

   V_{i,j} = max( IntrinsicValue, e^(–rΔt)[p V_{i+1,j+1} + (1–p) V_{i,j+1}] )

This method easily accommodates discrete dividends, early exercise, and path-dependent features.

4. Computational Considerations

Model	Complexity	Features
BSM	O(1)	European options, continuous-time
Binomial (CRR)	O(N²)	American options, dividends, flexibility

• Binomial converges to BSM as N→∞, but at a computational cost.

• For European, BSM’s closed-form wins on speed; for American or exotics, binomial’s flexibility is invaluable.

  
  

5. When to Use Which Model

Scenario	Recommended Model
European, no dividends	Black–Scholes–Merton
Dividend-paying underlyings	Binomial (with discrete adjustments)
American exercise features	Binomial
Barrier / path-dependent options	Binomial
Rapid, real-time pricing of vanilla	Black–Scholes–Merton
Low-liquidity or complex payoffs	Binomial

• BSM excels for vanilla European options in liquid markets where speed is paramount

• Binomial is the go-to for American options, discrete dividends, and bespoke payoffs, despite its higher computational burden

6. Input Parameters in 2025 Markets

• Risk-Free Rate (r): As of July 10, 2025, the 3-month T-bill yield trades at 4.25% (annualized).

• Volatility (σ): Equity implied volatilities remain elevated—averaging 22–25% for major indices—reflecting macro uncertainty.

Accurate inputs are critical. Mis‐estimation of σ or r can skew both BSM outputs and binomial lattices.

7. Illustrative Example

European Call, S₀ = 100, K = 100, T = 1 year, σ = 20%, r = 4.25%:

• Black–Scholes:

  d₁ = [ln(100/100) + (0.0425 + 0.5×0.20²)×1] / (0.20×1) = 0.3313 d₂ = 0.3313 - 0.20 = 0.1313 C = 100 N(0.3313) – 100 e^(–0.0425) N(0.1313) ≈ 100×0.6293 – 95.88×0.5523 ≈ 62.93 – 52.93 = 10.00

• Binomial (N = 3):

  1. Δt = 1/3, u= e^(0.20√(1/3))=1.122, d=0.891

  2. p=(e^(0.0425/3)–0.891)/(1.122–0.891)=0.547

  3. Build tree, compute terminal payoffs, roll back to present value: ≈ 9.95

Even with just three steps, binomial approximates BSM closely; increasing N improves accuracy but adds computation.

8. Common Modeling Pitfalls

1. Ignoring Early Exercise: Valuing American calls on dividend-paying stocks with BSM underestimates value.

2. Discrete Dividends: Failing to adjust S₀ for known dividend amounts skews both models.

3. Parameter Drift: Using historical σ vs. implied σ can lead to inconsistent valuations.

4. Lattice Granularity: Too few steps in binomial produces material pricing bias; too many steps incur latency.

5. Interest Rate Conventions: Mixing discrete vs. continuous compounding for r yields errors if inconsistent between u, d, and discounting.

9. 2025 Market Context & Model Selection

• High-Rate Environment: Elevated r increases the cost of carry, affecting p in binomial and discount factors in BSM.

• Volatility Regimes: In stressed volatility, the binomial model’s ability to incorporate local vol changes can better capture skew and kurtosis.

• Exotic Features: As structured products proliferate, lattice‐based models often replace closed‐form solutions.

Pragmatically, trading desks often use BSM for intra­day quoting, switching to binomial or finite-difference methods for end­of­day valuation and risk reporting.

Black–Scholes–Merton and Cox–Ross–Rubinstein binomial models each hold a firm place in the CFA Level II toolkit:

• Use BSM for speed and elegance when pricing European, non-dividend options in liquid markets.

• Use Binomial for flexibility, handling American features, dividends, and complex payoffs.

Mastery of both methods—and the judgment to select appropriately—ensures accurate valuation, robust risk management, and exam success in 2025’s evolving derivatives landscape.

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