Option Pricing Models: Binomial, Black-Scholes & Real-World Applications

TAKEAWAYS

• The Binomial Model prices options by building a recombining tree of stock prices and using risk-neutral valuation, while the Black-Scholes Model assumes continuous stock returns and derives a closed-form solution.
• Key inputs for both models are: current stock price (S), strike price (K), risk-free rate (r), time to expiry (T), and volatility (σ).
• American options require checking for early exercise at every node (Binomial) or using finite-difference methods (Black-Scholes), while European options can use the closed-form formula directly.
• Dividends reduce option value: subtract present value of dividends from S in Black-Scholes, or adjust the tree in Binomial.
• Volatility is the only uncertain input—higher volatility increases option premiums (via the σT term in Black-Scholes).
• Real-world use: NEPSE-listed companies (e.g., Ncell) use options to hedge FX risk; eSewa could theoretically price digital payment "options" for merchants using similar models.

1. Why Do We Need Option Pricing Models?

Options are contingent claims: their payoff depends on future stock prices. To price them fairly, we need models that:

• Account for uncertainty (volatility).
• Incorporate time value (interest rates).
• Handle early exercise (for American options).

2. The Binomial Option Pricing Model (Discrete-Time)

How It Works

1. Build a stock price tree over n periods (e.g., 6 months for semi-annual splits).
2. Assume two possible moves: u (up) and d (down), with probabilities p and (1–p).
3. Calculate risk-neutral probabilities (q) using the risk-free rate: $$q=\frac{e^{r\Delta t}-d}{u-d}$$
4. Work backward: At expiry, option value = max(S – K, 0) for calls. For earlier nodes, take the expected value under q: $$C=e^{-r\Delta t}\left[q\cdot C_u+(1-q)\cdot C_d\right]$$
5. Adjust for dividends: Subtract present value of dividends from S before building the tree.

Visual: Recombining Binomial Tree

graph TD
    A["S₀ = 400"] -->|"u"| B["S₁ = 480"]
    A -->|"d"| C["S₁ = 320"]
    B -->|"u"| D["S₂ = 576"]
    B -->|"d"| E["S₂ = 480"]
    C -->|"u"| E
    C -->|"d"| F["S₂ = 320"]
    D["Call = max(576–500,0)=76"]
    E["Call = max(480–500,0)=0"]
    F["Call = max(320–500,0)=0"]
    G["Backward: C₁ = e⁻⁰·⁰⁵(0.5*76 + 0.5*0) ≈ 37.8"]
    H["C₀ = e⁻⁰·⁰⁵(0.5*37.8 + 0.5*0) ≈ 18.6"]

Worked Example: Ncell Stock Option (Binomial Model)

Given:

• Current stock price (S₀) = Rs 2,500 (Ncell’s 2023 closing price).
• Strike price (K) = Rs 2,600.
• Risk-free rate (r) = 8% p.a. (Nepal Rastra Bank’s policy rate).
• Time to expiry (T) = 6 months (1 period).
• Up factor (u) = 1.1, Down factor (d) = 0.9.

Steps:

1. Calculate q: $$q=\frac{e^{0.08\times 0.5}-0.9}{1.1-0.9}=\frac{1.0404-0.9}{0.2}=0.202$$
2. Expiry node values:
   • S₁ = 2,500 × 1.1 = 2,750 → Call = max(2,750–2,600, 0) = 150
   • S₁ = 2,500 × 0.9 = 2,250 → Call = max(2,250–2,600, 0) = 0
3. Backward induction: $$C_0=e^{-0.04}\left[0.202\times 150+0.798\times 0\right]=0.9608\times 30.3\approx \boxed{29.1}$$ Interpretation: A call option on Ncell stock with K = 2,600 is worth Rs 29.1 per share.

3. The Black-Scholes Model (Continuous-Time)

Key Assumptions

• Stock prices follow geometric Brownian motion (log-normal distribution).
• No dividends (or dividends are continuous and subtracted from S).
• No arbitrage: Markets are efficient.
• Constant volatility (σ) and risk-free rate (r).
• European options only (no early exercise).

The Formula

For a call option: $$C=S_{0}N(d_1)-K{e}^{-rT}N(d_2)$$ Where: $$d_1=\frac{\ln (S_0/K)+(r+{\sigma }^2/2)T}{\sigma \sqrt{T}},d_2=d_1-\sigma \sqrt{T}$$ And N(·) is the cumulative standard normal distribution.

Visual: Black-Scholes Components

graph TD
    A["Stock Price (S₀)"] --> B["ln(S₀/K)"]
    C["Volatility (σ)"] --> D["σ√T"]
    E["Risk-free rate (r)"] --> F["(r + σ²/2)T"]
    B & F --> G["d₁ = (B + F)/D"]
    G --> H["N(d₁)"]
    I["K e⁻ᵣᵀ"] --> J["N(d₂)"]
    H --> K["S₀ × N(d₁)"]
    K & J --> L["Call Price = K – J"]

Worked Example: Chandragiri Hills Stock (Black-Scholes)

Given:

• S₀ = Rs 2,500 (current price).
• K = Rs 2,600.
• T = 0.5 years (6 months).
• r = 8% p.a.
• σ = 30% (historical volatility of NEPSE stocks).

Steps:

1. Calculate d₁ and d₂: $$d_1=\frac{\ln (2500/2600)+(0.08+0.3^2/2)\times 0.5}{0.3\times \sqrt{0.5}}=\frac{-0.0392+0.0625}{0.2121}\approx 0.114$$ $$d_2=0.114-0.3\times \sqrt{0.5}\approx -0.078$$
2. Look up N(d₁) ≈ 0.546, N(d₂) ≈ 0.469.
3. Plug into the formula: $$C=2500\times 0.546-2600\times e^{-0.04}\times 0.469\approx 1365-1170=\boxed{195}$$ Interpretation: The call option is worth Rs 195, higher than the Binomial estimate due to continuous hedging assumptions.

4. Comparing Binomial vs. Black-Scholes

5. Adjustments for Real-World Scenarios

A. Dividends

• Binomial: Subtract present value of dividends from S at each node.
• Black-Scholes: Use S₀ – PV(dividends) in the formula.

Example: If Chandragiri Hills pays a Rs 50 dividend in 3 months: $$PV(\text{div})=50\times e^{-0.08\times 0.25}\approx 49.0$$ Adjusted S₀ = 2,500 – 49 = Rs 2,451.

B. Volatility

• Implied volatility (σᵢₘₚ) is backed out from market option prices.
• Example: If the market quotes a call at Rs 200 but Black-Scholes gives Rs 195, solve for σ until they match.

C. American Options

• Binomial: Check for early exercise at every node.
• Black-Scholes: Use finite-difference methods or Barone-Adesi Whaley approximation.

## In the Real World

1. NEPSE Stock Options (Hypothetical)

   • Use Case: Investors hedge against Ncell’s stock price drops using put options priced via Black-Scholes.
   • How: If Ncell’s σ = 30%, a put with K = 2,500 and T = 1 year would cost ~Rs 120 (calculated similarly to the call above). This protects against a drop below Rs 2,500.

2. eSewa Merchant Discounts (Option-Like Structure)

   • Use Case: eSewa offers merchants dynamic discount rates (e.g., 1.5% if paid within 24 hours, 3% otherwise). This resembles a call option on payment timing, where the merchant "exercises" the lower rate if conditions are met.
   • Pricing: Could model the discount as a binomial tree where the "stock" is the merchant’s cash flow, and the "option" is the discount strike.

3. Ncell FX Hedging with Forwards + Options

   • Use Case: Ncell uses currency options to hedge USD-INR exchange rate risk when importing equipment.
   • How: If Ncell buys a put option on USD with K = Rs 130 and σ = 8%, Black-Scholes prices it at ~Rs 2.5 per USD. This caps their cost at Rs 130, regardless of volatility.

6. Limitations and Criticisms

• Black-Scholes Assumptions:
  • Constant volatility: Real markets have volatility smiles/skews (e.g., NEPSE options show higher σ for out-of-the-money strikes).
  • No jumps: Ignores sudden news (e.g., Ncell’s 2023 spectrum auction impact).
• Binomial Limitations:
  • Computationally intensive for long horizons (e.g., 5-year options).
  • Requires choosing u and d arbitrarily (though u ≈ e^{σ√Δt}, d ≈ e^{-σ√Δt} is common).

## Exam Tip

1. Memorize the Black-Scholes formula and know how to compute d₁ and d₂.
2. For Binomial trees:
   • Always recombining (no redundant nodes).
   • Risk-neutral probabilities are key—derive q from r, not p.
3. Dividends:
   • Binomial: Subtract PV at each node.
   • Black-Scholes: Subtract PV from S₀ once.
4. American options:
   • Binomial: Check early exercise at every node.
   • Black-Scholes: Assume European unless told otherwise (or use approximation).
5. Numerical traps:
   • Use natural logs for d₁ (not base-10).
   • N(d) tables are provided in exams—interpolate carefully.
6. Real-world questions:
   • Tie to NEPSE/Ncell (e.g., "Ncell’s stock is at Rs 2,500; price a put with K = 2,400").
   • Hedging: Always ask "Why use an option instead of a forward?" (Options = flexibility; forwards = obligation).

7. Quick-Reference Tables for Exams

A. Option Greeks (Sensitivity Measures)

B. Black-Scholes Inputs for NEPSE Stocks

8. Common Mistakes to Avoid

1. Forgetting to discount back in Binomial trees (multiply by e⁻ᵣΔt at each step).
2. Using arithmetic instead of geometric returns in Black-Scholes (ln(S₀/K), not (S₀–K)/K).
3. Ignoring dividends—always adjust S₀ or the tree.
4. Misapplying q—it’s derived from r, not the actual probability p.
5. Early exercise in Black-Scholes: Only do this for American options (and even then, use approximations).