Black-Scholes Model Deep Dive

Level 2: Intermediate | Module 2.1 | Time: 3 hours

🎯 Learning Objectives

By the end of this module, you will:

• Understand the Nobel Prize-winning Black-Scholes formula
• Master each input variable (S, K, T, σ, r)
• Learn the assumptions and limitations
• Calculate option prices by hand and conceptually
• Understand when the model works and when it fails

Prerequisites: Range Accruals

Introduction: The Nobel Prize Formula

In 1997, Myron Scholes and Robert Merton won the Nobel Prize in Economics for a formula that revolutionized finance.

The Historical Context

Before Black-Scholes (pre-1973):

Options trading: Mostly intuition and guesswork
Pricing: "Whatever someone will pay"
Risk management: Crude estimates
Market making: Extremely risky business

After Black-Scholes (1973-present):

Options trading: Mathematical precision
Pricing: Standard formula used globally
Risk management: Quantified Greeks
Market making: Sophisticated algorithms
Explosion of derivatives markets

The Impact: Enabled a multi-trillion dollar derivatives industry.

The Formula

For European Call Options

C = S × N(d₁) - K × e^(-rT) × N(d₂)

Where:
d₁ = [ln(S/K) + (r + σ²/2)T] / (σ√T)
d₂ = d₁ - σ√T

Components:
C = Call option price
S = Current stock/asset price (Spot)
K = Strike price
T = Time to expiration (in years)
σ = Volatility (annualized standard deviation)
r = Risk-free interest rate (annualized)
N(x) = Cumulative standard normal distribution
e = Euler's number (≈2.71828)
ln = Natural logarithm

Don’t panic! We’ll break this down piece by piece.

For European Put Options

P = K × e^(-rT) × N(-d₂) - S × N(-d₁)

Or using Put-Call Parity:
P = C - S + K × e^(-rT)

Understanding Each Input

Input 1: S (Spot Price)

Definition: The current market price of the underlying asset.

Example:

Bitcoin currently trading at $50,000
S = $50,000

Impact on Option Price:

• Higher S → Call worth MORE, Put worth LESS
• Lower S → Call worth LESS, Put worth MORE

Intuition: If Bitcoin is expensive now, calls are more valuable (already closer to profitability).

Input 2: K (Strike Price)

Definition: The price at which you can buy (call) or sell (put) the underlying.

Example:

Option to buy Bitcoin at $55,000
K = $55,000

Impact on Option Price:

• Higher K → Call worth LESS, Put worth MORE
• Lower K → Call worth MORE, Put worth LESS

Intuition: Calls with lower strikes are more valuable (easier to be ITM).

Input 3: T (Time to Expiration)

Definition: Time remaining until option expires, expressed in years.

Calculation:

Days to expiration: 30 days
T = 30 / 365 = 0.0822 years

Days to expiration: 90 days
T = 90 / 365 = 0.2466 years

Impact on Option Price:

• More time → Option worth MORE (both calls and puts)
• Less time → Option worth LESS (time decay)

Intuition: More time = more opportunity for favorable price movement.

This is Theta decay (we’ll cover in Greeks module).

Input 4: σ (Volatility - Sigma)

Definition: The annualized standard deviation of the underlying asset’s returns. Measures uncertainty.

This is the most important and complex input!

Typical Values:

Asset          | Annual Volatility (σ)
---------------|---------------------
US Bonds       | 5-10%
Stocks (SPY)   | 15-25%
Tech Stocks    | 30-50%
Bitcoin        | 60-100%
Altcoins       | 80-200%

Calculation (Historical Volatility):

1. Take daily returns for past N days
2. Calculate standard deviation of returns
3. Annualize: σ_annual = σ_daily × √252 (trading days)

Example:
Daily returns std dev: 3.2%
Annual volatility: 3.2% × √252 = 50.8%

Impact on Option Price:

• Higher σ → Options worth MORE (both calls and puts)
• Lower σ → Options worth LESS

Intuition: More volatility = more chance of large price swings = options more valuable.

Input 5: r (Risk-Free Rate)

Definition: The theoretical return on a risk-free investment (typically government bonds).

Examples:

US Treasury (1-year): 4.5%
→ r = 0.045

European Government Bond: 2%
→ r = 0.02

Impact on Option Price:

• Higher r → Call worth slightly MORE, Put worth slightly LESS
• Lower r → Call worth slightly LESS, Put worth slightly MORE

Intuition: Higher rates make future cash flows worth less today (discounting effect).

Note: For short-term options (< 3 months), interest rate impact is minimal.

Step-by-Step Calculation Example

Let’s price a Bitcoin call option by hand!

Given Information

Underlying: Bitcoin
S = $50,000 (current BTC price)
K = $55,000 (strike price)
T = 90 days = 90/365 = 0.2466 years
σ = 80% = 0.80 (annualized volatility)
r = 4% = 0.04 (risk-free rate)

Calculate: Call option price (C)

Step 1: Calculate d₁

d₁ = [ln(S/K) + (r + σ²/2)T] / (σ√T)

First, calculate each component:

ln(S/K) = ln(50,000/55,000) = ln(0.9091) = -0.0953

(r + σ²/2) = 0.04 + (0.80²/2) = 0.04 + 0.32 = 0.36

(r + σ²/2)T = 0.36 × 0.2466 = 0.0888

σ√T = 0.80 × √0.2466 = 0.80 × 0.4966 = 0.3973

Now assemble:
d₁ = (-0.0953 + 0.0888) / 0.3973
d₁ = -0.0065 / 0.3973
d₁ = -0.0164

Step 2: Calculate d₂

d₂ = d₁ - σ√T
d₂ = -0.0164 - 0.3973
d₂ = -0.4137

Step 3: Find N(d₁) and N(d₂)

These are cumulative normal distribution values. Use a table or calculator:

N(d₁) = N(-0.0164) ≈ 0.4935
N(d₂) = N(-0.4137) ≈ 0.3395

Intuition: These represent probabilities in the Black-Scholes framework.

Step 4: Calculate Present Value of Strike

K × e^(-rT) = $55,000 × e^(-0.04 × 0.2466)
             = $55,000 × e^(-0.00986)
             = $55,000 × 0.9902
             = $54,461

Step 5: Calculate Call Price

C = S × N(d₁) - K × e^(-rT) × N(d₂)
C = $50,000 × 0.4935 - $54,461 × 0.3395
C = $24,675 - $18,490
C = $6,185

Result: The call option is worth $6,185

Verification

Intrinsic Value = max(S - K, 0) = max($50k - $55k, 0) = $0 (OTM)
Time Value = $6,185 - $0 = $6,185

Makes sense! It's an OTM option with 90 days and high volatility,
so all value is time/volatility value.

Intuitive Understanding: What the Formula Does

The Two Parts of the Formula

Part 1: S × N(d₁)

• S: Current asset price
• N(d₁): Probability-adjusted delta
• Meaning: Expected value of the underlying if the option finishes ITM

Part 2: K × e^(-rT) × N(d₂)

• K × e^(-rT): Present value of the strike price
• N(d₂): Risk-neutral probability of finishing ITM
• Meaning: Expected cost of exercising the option

The Formula: Expected benefit - Expected cost = Option value

The Magical N(d₁) and N(d₂)

N(d₂): Probability the option finishes in-the-money

Example: N(d₂) = 0.3395 = 33.95%

Interpretation: ~34% chance this $55k call finishes ITM

N(d₁): The option’s delta (hedge ratio)

Example: N(d₁) = 0.4935 = 49.35%

Interpretation: For every $1 Bitcoin rises, this option gains ~$0.49

Why two different probabilities?

• N(d₂): Uses risk-neutral probability (theoretical)
• N(d₁): Adjusts for the asset’s expected drift
• Both are needed for accurate pricing

Sensitivity Analysis

Let’s see how changing each input affects our call price:

Changing Spot Price (S)

Original: S = $50,000 → C = $6,185

S = $45,000 → C ≈ $3,500 (lower spot = cheaper call)
S = $50,000 → C = $6,185 (baseline)
S = $55,000 → C ≈ $10,500 (higher spot = more expensive call)
S = $60,000 → C ≈ $16,000 (now ITM, has intrinsic value)

Conclusion: Call price increases with spot price (positive delta)

Changing Time (T)

Original: T = 90 days → C = $6,185

T = 7 days   → C ≈ $2,000 (little time = less value)
T = 30 days  → C ≈ $4,000
T = 90 days  → C = $6,185 (baseline)
T = 180 days → C ≈ $9,000 (more time = more value)
T = 365 days → C ≈ $12,500

Conclusion: More time = higher option value (positive theta... wait, negative theta decay!)

Changing Volatility (σ)

Original: σ = 80% → C = $6,185

σ = 30% → C ≈ $1,800 (low vol = cheap options)
σ = 50% → C ≈ $3,500
σ = 80% → C = $6,185 (baseline)
σ = 100% → C ≈ $8,000 (high vol = expensive options)
σ = 150% → C ≈ $12,000 (extreme vol = very expensive)

Conclusion: Higher volatility = higher option prices (positive vega)

Changing Strike (K)

Original: K = $55,000 → C = $6,185

K = $45,000 → C ≈ $9,500 (ITM call, more valuable)
K = $50,000 → C ≈ $7,800 (ATM call)
K = $55,000 → C = $6,185 (baseline, OTM)
K = $60,000 → C ≈ $4,200 (further OTM, less valuable)
K = $65,000 → C ≈ $2,800

Conclusion: Higher strike = lower call value

Changing Interest Rate (r)

Original: r = 4% → C = $6,185

r = 0% → C ≈ $6,120 (small decrease)
r = 2% → C ≈ $6,150
r = 4% → C = $6,185 (baseline)
r = 6% → C ≈ $6,220 (small increase)
r = 10% → C ≈ $6,300

Conclusion: Interest rate has minimal impact on short-term options

The Critical Assumptions

Black-Scholes makes several assumptions. Real markets violate most of them!

Assumption 1: Constant Volatility

Assumption: Volatility (σ) remains constant over the option’s life.

Reality: Volatility changes constantly!

Example: Bitcoin volatility
Pre-crash: σ = 50%
During crash: σ = 120%
Post-crash: σ = 60%

Black-Scholes can't handle this changing volatility.

Impact: Model misprices options during volatility regime changes.

Assumption 2: Log-Normal Distribution

Assumption: Asset prices follow a log-normal distribution (returns are normally distributed).

Reality: Fat tails! Real markets have more extreme events than normal distribution predicts.

Normal Distribution Prediction:
-3σ event (crash > 15%): Should happen once every 370 years

Reality in crypto:
-3σ events: Happen multiple times per year!

"Black Monday 1987": -20% drop = 20σ event
Under normal distribution: Should never happen in universe's lifetime
Reality: It happened.

Impact: Black-Scholes underprices far OTM options (tail risk).

Assumption 3: No Dividends

Assumption: The underlying asset pays no dividends.

Reality:

• Stocks pay dividends
• Crypto can have staking yields
• Bonds have coupons

Solution: Black-Scholes-Merton adjustment for dividend yield (q):

Replace S with: S × e^(-qT)

Example: Stock pays 2% dividend
q = 0.02
Adjusted S = $50,000 × e^(-0.02 × 0.2466) = $49,754

Assumption 4: European Exercise Only

Assumption: Option can only be exercised at expiration.

Reality: American options can be exercised anytime.

Impact:

• American calls on non-dividend assets ≈ European calls (rarely exercised early)
• American puts often worth MORE than European puts (early exercise valuable)

Solution: Use binomial models for American options.

Assumption 5: No Transaction Costs

Assumption: Trading is frictionless (no fees, no slippage).

Reality:

• Exchange fees (0.05-0.5%)
• Bid-ask spreads
• Market impact
• Gas fees (crypto)

Impact: Real-world prices deviate from theoretical prices by transaction costs.

Assumption 6: Continuous Trading

Assumption: Markets trade 24/7 with perfect liquidity.

Reality:

• Stock markets: 6.5 hours/day, 5 days/week
• Gaps over weekends and holidays
• Liquidity varies

Impact: Weekend gaps and overnight moves create risk not captured by the model.

When Black-Scholes Works Well

✅ Ideal Conditions

1. Liquid, mature markets

Examples:
- S&P 500 index options ✅
- Major FX pairs ✅
- Large-cap stocks ✅
- Bitcoin/Ethereum (most of the time) ✅

2. Near-the-money options

ATM options with 30-90 days to expiration:
- Model very accurate
- Assumptions least violated
- Market makers rely on it

3. Normal volatility regimes

When σ is stable (not spiking or crashing):
- Model predictions reliable
- Greeks accurate
- Hedging works as expected

4. Short to medium term

Options with 1 week to 6 months:
- Time assumptions reasonable
- Interest rate impact small
- Volatility somewhat stable

When Black-Scholes Fails

❌ Poor Conditions

1. Extreme volatility events

COVID crash March 2020:
- Volatility: 30% → 120% in days
- Constant volatility assumption destroyed
- Model severely mispriced options

2. Far out-of-the-money options

Deep OTM puts (crash insurance):
- Black-Scholes underprices them
- Doesn't capture fat tail risk
- Market prices show "volatility skew"

3. Very long-dated options (LEAPS)

2-3 year options:
- Volatility will definitely change
- Interest rates may change
- Dividends uncertain
- Model less reliable

4. Illiquid markets

Obscure altcoins, small-cap stocks:
- Wide bid-ask spreads
- Gaps and discontinuities
- Model assumes perfect liquidity

5. Around major events

Binary events (earnings, FDA approval, regulation):
- Returns NOT log-normal
- Jumps and gaps
- Model fails badly

Alternative Models

When Black-Scholes fails, professionals use:

1. Binomial/Trinomial Models

Advantages:
- Handle American options
- Can incorporate dividends
- Flexible for exotic options

Disadvantages:
- Computationally intensive
- Still assumes constant volatility

2. Stochastic Volatility Models (Heston)

Allows volatility to change randomly

Heston Model adds:
- Volatility as a second random variable
- Mean-reverting volatility process

Better for:
- Long-dated options
- Volatility trading

3. Jump-Diffusion Models (Merton)

Adds discrete jumps to price process

Better captures:
- Fat tails
- Crash risk
- Event-driven moves

4. Local Volatility Models

Volatility varies by strike and time

Calibrated to:
- Match market prices exactly
- Capture volatility smile/skew

Used by market makers

Practical Application: Using Black-Scholes

Use Case 1: Fair Value Check

Scenario:
Market price of call: $7,000
Black-Scholes value: $6,185

Analysis:
Market is pricing in ~13% more value

Possible reasons:
1. Higher implied volatility than you estimated
2. Supply/demand imbalance (more buyers)
3. Upcoming event not in your model
4. Dividend adjustment needed

Action:
If you think market is wrong → sell the call (overpriced)
If market is right → recalibrate your inputs (especially σ)

Use Case 2: Implied Volatility Extraction

Known:
- Market call price: $7,000
- S = $50,000, K = $55,000, T = 0.2466, r = 0.04

Unknown: What volatility is the market using?

Process (iterative):
1. Try σ = 80% → C = $6,185 (too low)
2. Try σ = 90% → C = $7,500 (too high)
3. Try σ = 85% → C = $6,850 (close)
4. Try σ = 86.5% → C = $7,000 ✅

Result: Implied volatility = 86.5%

Market expects higher volatility than historical 80%!

Use Case 3: Structuring Products

Building a Principal Protected Note:

Given:
- $100,000 to invest
- 1 year term
- r = 5%

Step 1: Bond cost
PV = $100,000 / (1.05) = $95,238

Step 2: Option budget
$100,000 - $95,238 = $4,762

Step 3: Price ATM call using Black-Scholes
C = $8,000 per Bitcoin (at $50k strike)

Step 4: Calculate participation
$4,762 / $8,000 = 59.5%

Now you know exactly what participation rate you can offer!

Practice Exercise: Price an Option

Given

Asset: Ethereum
S = $3,000
K = $3,200
T = 60 days = 60/365 = 0.1644 years
σ = 70% = 0.70
r = 4% = 0.04

Calculate: Call option price

Step 1: Calculate d₁

ln(S/K) = ln(3,000/3,200) = ln(0.9375) = -0.0645

(r + σ²/2)T = (0.04 + 0.70²/2) × 0.1644
            = (0.04 + 0.245) × 0.1644
            = 0.285 × 0.1644 = 0.0468

σ√T = 0.70 × √0.1644 = 0.70 × 0.4055 = 0.2838

d₁ = (-0.0645 + 0.0468) / 0.2838
   = -0.0177 / 0.2838
   = -0.0624

Step 2: Calculate d₂

d₂ = d₁ - σ√T = -0.0624 - 0.2838 = -0.3462

Step 3: Find N(d₁) and N(d₂)

N(d₁) = N(-0.0624) ≈ 0.4751
N(d₂) = N(-0.3462) ≈ 0.3646

Step 4: PV of Strike

K × e^(-rT) = $3,200 × e^(-0.04 × 0.1644)
            = $3,200 × 0.9935
            = $3,179

Step 5: Call Price

C = S × N(d₁) - K × e^(-rT) × N(d₂)
C = $3,000 × 0.4751 - $3,179 × 0.3646
C = $1,425 - $1,159
C = $266

Answer: The call option is worth approximately $266

Key Takeaways

1. Black-Scholes revolutionized options pricing

• Nobel Prize-winning formula
• Industry standard since 1973
• Enabled derivatives explosion

2. Five inputs determine option price

• S (spot), K (strike), T (time), σ (volatility), r (rate)
• Volatility is the most important and hardest to estimate
• Interest rate matters less for short-term options

3. The model makes strong assumptions

• Constant volatility (violated constantly)
• Log-normal returns (fat tails exist)
• No dividends (adjustable)
• European exercise (limits use)
• Frictionless markets (not reality)

4. Works well in specific conditions

• Liquid markets
• ATM options
• Short to medium term
• Normal volatility regimes

5. Know when it fails

• Extreme volatility
• Far OTM options
• Around major events
• Very long-dated options

6. Alternative models exist for complex scenarios

• Binomial for American options
• Stochastic vol for long-dated
• Jump-diffusion for crashes

What’s Next?

You’ve mastered option pricing fundamentals! You now understand:

• ✅ The Black-Scholes formula and its components
• ✅ How to calculate option prices
• ✅ Sensitivity to different inputs
• ✅ Assumptions and limitations
• ✅ When to trust and when to adjust

Ready to learn simulation-based pricing?

Continue to: Monte Carlo Simulation →

Learn how to price exotic options and path-dependent structures using simulation.

Tools & Resources

Interactive Tools:

• Black-Scholes Calculator - Price options instantly
• Implied Vol Calculator - Extract IV from prices
• Sensitivity Analyzer - See Greeks in action

Code Examples:

# Python implementation
from scipy.stats import norm
import numpy as np
 
def black_scholes_call(S, K, T, r, sigma):
    d1 = (np.log(S/K) + (r + sigma**2/2)*T) / (sigma*np.sqrt(T))
    d2 = d1 - sigma*np.sqrt(T)
    call = S*norm.cdf(d1) - K*np.exp(-r*T)*norm.cdf(d2)
    return call
 
# Example
price = black_scholes_call(S=50000, K=55000, T=0.25, r=0.04, sigma=0.80)
print(f"Call price: ${price:,.2f}")

Next Module: Monte Carlo Simulation →

Related Topics:

• The Greeks - Understand option sensitivities
• Volatility - Master the most important input
• Implied Volatility Surface - Advanced pricing