Binomial & Black-Scholes: Pricing Models & Applications

TAKEAWAYS:

• The binomial model prices options by building a recombining tree of stock prices, allowing for early exercise in American options and discrete time steps.
• The Black-Scholes model assumes no dividends, constant volatility, and continuous trading—its formula $C=S_{0}N(d_1)-K{e}^{-rT}N(d_2)$ requires $d_1$ and $d_2$ calculations.
• American vs. European options: Binomial models handle early exercise, while Black-Scholes assumes European-style (no early exercise).
• Volatility (σ) and risk-free rate (r) are critical inputs—higher volatility increases option premiums, while higher rates favor calls.
• Real-world tie: NEPSE-listed companies (e.g., Nabil Bank) use Black-Scholes to price employee stock options; eSewa could theoretically use binomial trees for dynamic pricing of digital services.
• Margin of error: Both models simplify reality—binomial trees are flexible but computationally intensive; Black-Scholes is elegant but sensitive to assumptions.

1. The Binomial Option Pricing Model: A Step-by-Step Tree

The binomial model is the only model that can price American options (options exercisable anytime before expiry). It works by:

1. Building a stock price tree with up (u) and down (d) moves over discrete periods.
2. Calculating option payoffs at each node.
3. Discounting back using the risk-neutral probability $q=\frac{e^{r\Delta t}-d}{u-d}$.

Key Components

Worked Example: Pricing a Call Option on Himalayan Bank Stock

Given:

• Current stock price ($S_0$) = Rs 400
• Strike price ($K$) = Rs 380
• Risk-free rate ($r$) = 6% per annum
• Volatility ($\sigma$) = 20% per annum
• Time to expiry ($T$) = 6 months (1 period)
• Stock split: 2-for-1 (adjust strike price to Rs 190).

Step 1: Build the Price Tree

graph TD
    A["S₀ = 400"] -->|"u = 1.18"| B["S₁ = 472"]
    A -->|"d = 0.85"| C["S₁ = 340"]
    B -->|"Call Payoff"| D["Max(472-190, 0) = 282"]
    C -->|"Call Payoff"| E["Max(340-190, 0) = 150"]

Step 2: Calculate Risk-Neutral Probability $$q=\frac{e^{0.06\times 0.5}-0.85}{1.18-0.85}\approx 0.52$$

Step 3: Compute Option Value $$C=e^{-0.06\times 0.5}[0.52\times 282+(1-0.52)\times 150]=0.97\times (146.64+72)=\text{Rs 214.50}$$

Interpretation: The call option on Himalayan Bank stock (adjusted for the 2-for-1 split) is worth Rs 214.50. If the stock rises to Rs 472, the option’s intrinsic value is Rs 282.

2. Multi-Period Binomial Model: American Options

For American options, we must check early exercise at every node. The process:

1. Work backward from expiry.
2. Compare intrinsic value vs. continuation value at each step.
3. Use put-call parity to price puts if needed.

Example: 2-Period American Put on Ncell Stock

Given:

• $S_0=2500$, $K=2400$, $r=0.05$, $\sigma =0.25$, $T=1$ year (2 periods).
• Dividend: Rs 100 paid at $t=0.5$.

Step 1: Build the Tree (with Dividend Adjustment)

graph TD
    A["S₀ = 2500"] -->|"u = 1.22"| B["S₁ = 3050"]
    A -->|"d = 0.82"| C["S₁ = 2050"]
    B -->|"u"| D["S₂ = 3703"]
    B -->|"d"| E["S₂ = 2486"]
    C -->|"u"| F["S₂ = 2541"]
    C -->|"d"| G["S₂ = 1682"]

Step 2: Calculate Put Payoffs at $t=1$

• At $D$: $\max (2400-3703,0)=0$
• At $E$: $\max (2400-2486,0)=0$
• At $F$: $\max (2400-2541,0)=0$
• At $G$: $\max (2400-1682,0)=718$

Step 3: Backward Induction (Check Early Exercise) At $t=0.5$:

• For $S_1=3050$:
  • Continuation value: $e^{-0.05\times 0.5}\times 0=0$
  • Intrinsic value: $\max (2400-3050,0)=0$ → No early exercise.
• For $S_1=2050$:
  • Continuation value: $e^{-0.025}[0.5\times 0+0.5\times 718]=352.60$
  • Intrinsic value: $\max (2400-2050,0)=350$
  • Exercise early (Rs 350 > Rs 352.60? No, but close—this is a borderline case).

Final Put Value at $t=0$: $$P=e^{-0.05\times 0.5}[0.5\times 0+0.5\times 350]=\text{Rs 170.50}$$

Real-World Tie: Ncell uses binomial trees to dynamically price prepaid mobile data options (e.g., "Rs 500 for 1GB valid 30 days"). The model helps set strike prices based on expected volatility in data usage.

3. The Black-Scholes-Merton Model: Assumptions & Formula

The Black-Scholes model is a closed-form solution for European options, derived under strict assumptions:

Assumptions (and Their Violations in Reality)

The Black-Scholes Formula for a Call Option

$$C=S_{0}N(d_1)-K{e}^{-rT}N(d_2)$$ where: $$d_1=\frac{\ln (S_0/K)+(r+{\sigma }^2/2)T}{\sigma \sqrt{T}},d_2=d_1-\sigma \sqrt{T}$$

Worked Example: Pricing a Call on Daraz Stock

Given:

• $S_0=1200$, $K=1100$, $r=0.07$, $\sigma =0.35$, $T=0.5$ years.

Step 1: Calculate $d_1$ and $d_2$ $$d_1=\frac{\ln (1200/1100)+(0.07+0.35^2/2)\times 0.5}{0.35\sqrt{0.5}}=\frac{0.0953+0.0663}{0.2475}\approx 0.67$$ $$d_2=0.67-0.35\times 0.707\approx 0.43$$

Step 2: Look Up $N(d_1)$ and $N(d_2)$ from Standard Normal Table

• $N(0.67)\approx 0.7486$
• $N(0.43)\approx 0.6664$

Step 3: Plug into Black-Scholes $$C=1200\times 0.7486-1100\times e^{-0.07\times 0.5}\times 0.6664=898.32-715.50=\text{Rs 182.82}$$

Interpretation: The call option on Daraz stock is worth Rs 182.82. If Daraz’s stock rises to Rs 1300, the option’s intrinsic value is Rs 200.

4. Comparing Binomial and Black-Scholes Models

When to Use Which?

• Binomial: Pricing American options, employee stock options (e.g., at Nabil Bank), or when volatility is not constant.
• Black-Scholes: Quick valuation for European options on highly liquid stocks (e.g., Ncell, Himalayan Bank).

5. Real-World Applications in Nepal

Example 1: NEPSE Index Futures (NIFTY 50)

• Idea Used: Black-Scholes for pricing index futures.
• How: NEPSE uses a modified Black-Scholes to set margin requirements for index futures contracts. If the NIFTY 50 index is volatile (high σ), margins increase.
• Worked Example:
  • Suppose NIFTY 50 = 10,000, $K=10,200$, $T=3$ months, $r=0.06$, $\sigma =0.25$.
  • Calculate call premium using Black-Scholes, then set initial margin = 1.5 × premium.

Example 2: eSewa’s Dynamic Pricing

• Idea Used: Binomial trees for real-time pricing adjustments.
• How: eSewa could use a 2-period binomial model to adjust transaction fees based on:
  • Up move (u): High demand (e.g., Dashain) → higher fees.
  • Down move (d): Low demand → discounts.
• Example:
  • Current fee = Rs 5, $u=1.2$, $d=0.9$, $r=0.05$.
  • If demand rises to Rs 6, the "option" to charge more is valued using the tree.

Example 3: Nabil Bank’s Loan Hedging

• Idea Used: Black-Scholes for interest rate swaps.
• How: Nabil Bank uses Black-Scholes on zero-coupon bonds to hedge against rising interest rates.
• Example:
  • Bank issues a 5-year loan at 8% but fears rates will rise.
  • It enters a swap where it pays fixed (8%) and receives floating (LIBOR + 1%).
  • The swap’s value is priced using Black-Scholes with $S_0=$ current bond price, $K=$ strike rate.

6. Common Pitfalls and Exam Traps

1. Ignoring Dividends:

   • Mistake: Applying Black-Scholes directly to a dividend-paying stock.
   • Fix: Adjust $S_0$ to $S_0-PV(\text{dividend})$.
   • Exam Tip: If a question mentions dividends, always adjust the stock price.

2. American vs. European:

   • Mistake: Using Black-Scholes for an American option.
   • Fix: Use binomial model or put-call parity for early exercise.
   • Example: A question asks for the price of an American put—binomial is the only correct method.

3. Volatility Misinterpretation:

   • Mistake: Thinking higher volatility lowers option prices.
   • Truth: Higher σ increases both call and put prices.
   • Exam Tip: If σ doubles, the option premium rises significantly.

4. Stock Splits:

   • Mistake: Forgetting to adjust $K$ and $S_0$ after a split.
   • Rule: For a 2-for-1 split, new $K=\text{old }K/2$, new $S_0=\text{old }{S}_0/2$.
   • Example: If $K=25$ and a 3-for-1 split occurs, new $K=25/3\approx 8.33$.

5. Risk-Neutral Probability:

   • Mistake: Using the real-world probability (not $q$) in binomial pricing.
   • Fix: Always use $q=\frac{e^{r\Delta t}-d}{u-d}$.

Exam Tip: How to Score Full Marks

1. Show All Steps:

   • For binomial trees, draw the tree (even if not required) and label all nodes.
   • For Black-Scholes, write down $d_1$ and $d_2$ formulas before plugging numbers.

2. Handle Units Carefully:

   • If $T$ is in years but $r$ is annual, ensure $\Delta t$ matches (e.g., 6 months = 0.5 years).
   • Example: If $T=90$ days, use $T=0.25$ years.

3. Adjust for Dividends/Splits:

   • Dividend: Subtract $PV(\text{dividend})$ from $S_0$.
   • Split: Divide both $S_0$ and $K$ by the split ratio.

4. Compare with Intrinsic Value:

   • For calls: $\text{Intrinsic}=\max (S_0-K,0)$.
   • For puts: $\text{Intrinsic}=\max (K-S_0,0)$.
   • Exam Tip: If the model price is less than intrinsic, you’ve made a mistake.

5. Use Real-World Examples:

   • If the question is about Nepal’s stock market, tie your answer to NEPSE, Ncell, or Nabil Bank.
   • Example: "The binomial model is used by Nepal Investment Bank to price employee stock options, where early exercise is allowed."

6. Margin of Error:

   • If asked about approximations, state:
     • Binomial ≈ Black-Scholes as periods → ∞.
     • Black-Scholes fails if dividends or volatility is not constant.

Final Visual Summary

flowchart TD
    A["Option Pricing Models"] --> B["Binomial Model"]
    A --> C["Black-Scholes Model"]
    B --> D["Discrete Time Steps\nHandles Early Exercise\nFlexible for Dividends"]
    C --> E["Continuous Time\nNo Early Exercise\nAssumes Constant σ"]
    D --> F["Used for:\n- American Options\n- Employee Stock Options\n- Illiquid Markets"]
    E --> G["Used for:\n- European Options\n- Liquid Markets\n- Quick Valuation"]
    H["Real-World Use in Nepal"] --> I["NEPSE Futures\nNcell Options\nNabil Bank Hedging"]