The Black-Scholes model is one of the most widely used mathematical models for pricing European-style options and covered warrants. Understanding how to apply this model helps investors evaluate whether a warrant is fairly priced and make more informed investment decisions.
The Black-Scholes Model
The Black-Scholes model, developed by Fischer Black, Myron Scholes, and Robert Merton in 1973, provides a theoretical framework for pricing European-style options and covered warrants. It revolutionized derivatives pricing and earned Scholes and Merton the Nobel Prize in Economics in 1997. The model provides a mathematical formula that estimates the theoretical value of a call warrant based on five key input variables.
Black-Scholes Formula for Call Warrants
C = S₀ × N(d₁) − X × e⁻ʳᵀ × N(d₂)
C
Theoretical price of the call warrant
This is what we're solving for
S₀
Current spot price of the underlying stock
The market price at evaluation time
X
Strike price (exercise price)
The price at which the warrant can be exercised
T
Time to expiration (in years)
Convert days to years: days ÷ 365
r
Risk-free interest rate (annual)
Usually the government bond yield (as decimal, e.g., 0.05 for 5%)
σ (sigma)
Volatility of the underlying stock (annual)
Standard deviation of returns, as decimal (e.g., 0.25 for 25%)
N(d₁) and N(d₂)
Cumulative distribution function of the standard normal distribution
These represent probabilities and are calculated using d₁ and d₂ (see formulas below)
e⁻ʳᵀ
Present value discount factor
e ≈ 2.71828; This discounts the strike price to present value
Understanding the Formula Components
The Black-Scholes formula can be broken down into two main parts:
S₀ × N(d₁)
This represents the expected value of the underlying asset price at expiration, weighted by the probability that the warrant will be in-the-money.
- • N(d₁) is the probability-adjusted delta
- • Higher when warrant is more likely to be in-the-money
- • Ranges from 0 to 1
X × e⁻ʳᵀ × N(d₂)
This represents the present value of the strike price, weighted by the probability of exercising the warrant.
- • e⁻ʳᵀ discounts the strike price to present value
- • N(d₂) is the probability of exercise
- • The cost we expect to pay if exercised
The difference between these two components gives us the theoretical warrant price. Essentially, the formula calculates: Expected payoff from owning the asset minus the expected cost of exercising the warrant.
Calculating d₁ and d₂
Before we can compute the warrant price, we need to calculate two intermediate parameters: d₁ and d₂. These parameters are crucial as they feed into the cumulative normal distribution functions N(d₁) and N(d₂), which represent probabilities used in the main formula.
d₁ Formula
d₁ = [ln(S₀/X) + (r + σ²/2) × T] / (σ × √T)
Component Breakdown:
- • ln(S₀/X): Natural logarithm of the moneyness ratio (how far in/out-of-the-money)
- • (r + σ²/2) × T: Risk-free rate plus half the variance, multiplied by time
- • σ × √T: Volatility adjusted for time (standard deviation of returns over time period)
What it measures:
d₁ measures how many standard deviations the expected stock price is from the strike price, adjusted for the risk-free rate and volatility. It's directly related to the warrant's delta (price sensitivity).
d₂ Formula
d₂ = d₁ − σ × √T
d₂ is simply d₁ minus the volatility-adjusted time component. It's always smaller than d₁.
What it represents:
d₂ represents the probability that the warrant will finish in-the-money. N(d₂) gives the actual probability percentage (between 0 and 1).
Key Insight:
The difference (d₁ - d₂) = σ × √T represents the volatility-adjusted time value. More time or higher volatility increases this gap.
Understanding N(d₁) and N(d₂)
N(d₁) and N(d₂) are values from the cumulative standard normal distribution function. These can be found using statistical tables or calculated using Excel's NORM.S.DIST function or similar tools. Here's what they represent:
N(d₁) - The Delta Factor
- • Range: 0 to 1 (0% to 100%)
- • Meaning: Probability-adjusted sensitivity to underlying price changes
- • Practical use: Approximately equals the warrant's delta (price sensitivity)
- • Example: If N(d₁) = 0.60, the warrant price moves about 60% of the underlying's movement
N(d₂) - The Exercise Probability
- • Range: 0 to 1 (0% to 100%)
- • Meaning: Probability that the warrant will be in-the-money at expiration
- • Practical use: Estimates likelihood of profitable exercise
- • Example: If N(d₂) = 0.40, there's a 40% chance the warrant will be exercised
Try Our Theoretical Value Calculator
Instead of manually calculating Black-Scholes values, use our Theoretical Value Calculator to quickly evaluate warrant prices. Simply input the underlying price, strike price, time to expiration, volatility, and risk-free rate, and get instant theoretical pricing results.
Open CalculatorImportant Note
In practice, you don't need to manually calculate these values. Use our theoretical value calculator, Excel formulas (NORM.S.DIST), or online calculators. The key is understanding what these values represent so you can interpret the results correctly.
Volatility (σ) - A Critical Parameter
σ (sigma): Annualized volatility of the underlying stock, expressed as a decimal (e.g., 0.25 for 25%).
How to Calculate Historical Volatility:
- Collect daily closing prices of the underlying asset over a period (typically 30, 60, or 252 trading days)
- Calculate daily returns: ln(Price_today / Price_yesterday)
- Calculate the standard deviation of these daily returns
- Annualize by multiplying by √252 (number of trading days per year): σ_annual = σ_daily × √252
Why Volatility Matters:
- • Higher volatility → Higher probability of large price swings → Higher warrant premium
- • Volatility is the only input that's not directly observable - it must be estimated
- • You can use historical volatility or implied volatility from current warrant prices
- • Typical stock volatilities range from 15% (stable stocks) to 40%+ (volatile stocks)
Factors Affecting Covered Warrant Prices
Understanding how different factors influence warrant pricing is essential for making informed investment decisions. The Black-Scholes model identifies five key variables that determine warrant value.
1. Current Price of the Underlying Asset
The current price of the underlying asset directly affects the intrinsic value of the warrant and thus the warrant price.
For Call Warrants
As the underlying asset price increases, call warrant prices rise. The relationship is positive and direct.
- • Higher underlying price → Higher call warrant value
- • Lower underlying price → Lower call warrant value
2. Strike Price
The strike price (X) is the predetermined price at which the warrant can be exercised. The relationship between the current underlying price and strike price determines whether the warrant is in-the-money, at-the-money, or out-of-the-money.
In-the-Money
S₀ > X
Has intrinsic value = S₀ - X
At-the-Money
S₀ ≈ X
Only time value, most sensitive to volatility
Out-of-the-Money
S₀ < X
Only time value, highest leverage potential
3. Time to Expiration
Time to expiration (T) is measured in years and represents the remaining lifespan of the warrant. More time provides greater opportunity for the underlying asset to move favorably.
Time Decay (Theta)
As expiration approaches, warrant prices decrease due to time decay. This effect accelerates as expiration nears.
- • Longer time to expiration → Higher warrant premium
- • Shorter time to expiration → Lower warrant premium
- • Time decay is more significant for out-of-the-money warrants
4. Volatility (σ)
Volatility measures the degree of variation in the underlying asset's price over time. It's a critical factor in option pricing because it reflects the uncertainty about future price movements.
Volatility Impact
Higher volatility increases warrant prices because there's a greater probability of large price movements that could make the warrant profitable.
- • High volatility → Higher warrant premiums
- • Low volatility → Lower warrant premiums
- • Volatility is often estimated using historical price data or implied from current market prices
5. Risk-Free Interest Rate
The risk-free interest rate (r) represents the return on a risk-free investment, typically government bonds. It affects warrant pricing through its impact on the present value of the strike price.
Interest Rate Effect
For call warrants, higher interest rates increase warrant prices because the present value of the strike price decreases, making it cheaper to finance the future purchase of the underlying asset.
- • Higher interest rates → Higher call warrant prices
- • Lower interest rates → Lower call warrant prices
- • The impact is generally smaller compared to other factors like volatility and time to expiration
Step-by-Step Calculation Example
Let's work through a practical example to see how the Black-Scholes model works:
Example Scenario
Input Parameters:
- • S₀ = 100,000 VND (current stock price)
- • X = 105,000 VND (strike price)
- • T = 0.25 years (91 days until expiration)
- • r = 0.06 (6% annual risk-free rate)
- • σ = 0.30 (30% annual volatility)
This warrant is:
Out-of-the-money (S₀ < X)
Has only time value, no intrinsic value
Step 1: Calculate d₁
d₁ = [ln(100,000/105,000) + (0.06 + 0.30²/2) × 0.25] / (0.30 × √0.25)
d₁ = [-0.0488 + (0.06 + 0.045) × 0.25] / (0.30 × 0.5)
d₁ = [-0.0488 + 0.02625] / 0.15
d₁ = -0.1503
Step 2: Calculate d₂
d₂ = d₁ − σ × √T
d₂ = -0.1503 − 0.30 × 0.5
d₂ = -0.3003
Step 3: Find N(d₁) and N(d₂)
Using standard normal distribution table or calculator:
N(d₁) = N(-0.1503) = 0.4403 (approximately)
N(d₂) = N(-0.3003) = 0.3821 (approximately)
These values mean there's a 38.21% probability of finishing in-the-money
Step 4: Calculate Present Value Discount Factor
e⁻ʳᵀ = e^(-0.06 × 0.25) = e^(-0.015)
e⁻ʳᵀ = 0.9851
Step 5: Calculate Warrant Price
C = S₀ × N(d₁) − X × e⁻ʳᵀ × N(d₂)
C = 100,000 × 0.4403 − 105,000 × 0.9851 × 0.3821
C = 44,030 − 39,492
C = 4,538 VND
Interpretation: The theoretical price of this warrant is approximately 4,538 VND. If the market price is significantly higher, the warrant may be overpriced. If it's lower, it may be undervalued (though market factors not captured by the model could explain the difference).
Calculate Warrant Prices Instantly
Performing these calculations manually can be time-consuming. Save time and reduce errors by using our Theoretical Value Calculator, which automatically computes d₁, d₂, N(d₁), N(d₂), and the final warrant price using the Black-Scholes model. Perfect for evaluating multiple warrants quickly.
Use Calculator NowUsing the Black-Scholes Model in Practice
While understanding the formula is important, practical application requires several considerations:
Model Assumptions and Limitations
- European vs. American Style: The model assumes European-style exercise (only at expiration), while many warrants allow early exercise.
- Constant Volatility: The model assumes constant volatility, but real markets exhibit volatility clustering and changing patterns.
- Market Factors: Real warrant prices are also influenced by supply and demand, liquidity, issuer credit risk, and market sentiment.
Conclusion
The Black-Scholes model provides a valuable framework for understanding covered warrant pricing and the factors that influence warrant values. While the model has limitations, it remains a fundamental tool for professional investors evaluating warrant opportunities. By understanding how underlying price, strike price, time to expiration, volatility, and interest rates affect warrant pricing, you can make more informed investment decisions and better assess whether a warrant is fairly priced in the market.