What is the Black-Scholes Model? A Complete Guide to Option Pricing

The Black-Scholes model is a mathematical equation used in finance to calculate the theoretical fair value of European option contracts using six main variables.

How do options traders determine if a premium is fairly priced? Or have you heard some traders saying "IV is high," or "The fair value should be lower"? All this comes down to understanding the Black–Scholes model. 

The Black-Scholes model is one of the most widely used mathematical frameworks for pricing options. The model is used to find the theoretical price of the options. It uses different inputs such as stock price, strike price, volatility, time to expiry, and interest rates. Let's dive deep into it. 

Key Takeaways

• The Black-Scholes model helps in determining the theoretical fair price of European call and put
• options.
• The model uses multiple inputs, but volatility is the most important input.
• One of the biggest uses is to reverse engineer the implied volatility from the market
• premium.
• Other Greeks such as Delta, Theta and Gamma are also derived from the same model.

What Is The Black-Scholes Model?

The Black-Scholes model is a mathematical model that is employed to find the theoretical price of any European option. It can be used to find the fair price of both call and put options. 

The formulas used are as follows:

Black-Scholes call option formula,

C = S × N(d1) - K × e^(-rt) × N(d2)

Black-Scholes put option formula,

P = K × e^(-rt) × N(-d2) - S × N(-d1)

d1 formula,

d1 = [ln(S/K) + (r + σ²/2) × t] / (σ × √t)

d2 formula,

d2 = d1 - σ × √t

Where:

S = Current stock price
K = Strike price
r = Risk-free interest rate
t = Time to expiry (in years)
σ = Volatility of the underlying asset
N() = Cumulative normal distribution function
ln = Natural logarithm
e = Euler’s number (~2.718)

Inputs Used in the Black-Scholes Model

Before understanding the Black-Scholes model, let's understand all the inputs: 

Stock Price

The current market price of the stock or underlying asset.

If the stock price rises:

• Call option value tends to rise
• Put option value tends to fall

Strike Price

The strike price is the predetermined price at which the option can be exercised. This is the option we are trying to value using the model.

Time to Expiry

More time means more opportunity for the stock to move. As we know, as expiry increases, the option premium generally rises. Also, the time value increases.

Volatility

Volatility measures expected price movement. This is the implied volatility of the options. This is one of the most important variables in Black-Scholes.

The price of the option is positively correlated with volatility. This is because larger expected moves increase the probability that the option becomes profitable.

For example, let's say we have two stocks, A and B. Stock A moves 1% daily, whereas Stock B moves 8% daily. Even at the same strike and expiry, Stock B's options will usually be more expensive.

Volatility heavily influences option pricing.

Risk-Free Rate

The model assumes money has a time value. This is usually the yield on government treasury bills matching the option's expiration. (The risk-free rate in the Indian market typically fluctuates between 5.2% and 7.5%)

Dividends

Dividend-paying stocks affect option pricing. This is because if a stock is expected to pay dividends, then the call values may decrease and the put values may increase. This is usually because the stock price typically adjusts downward after dividends.

Example

Let us assume that we have these inputs:

Stock Price = ₹100

Strike Price = ₹100

Time to Expiry = 30 days

Volatility = 20%

Risk-Free Rate = 6%

Now, based on the formula, 

d1 = 0.1147

d2 = 0.0573

Call Value = ₹2.54

Put Value = ₹2.04

Now, we check the actual market price. Let's say the price of the call option is ₹4.5. So the option is expensive, and we can consider shorting the call option.

Key Assumptions Behind Black-Scholes

As with all financial models, Black-Scholes also has a lot of assumptions, which try to simplify the reality of trading: 

1. Markets Are Efficient
2. Volatility Is Constant
3. No Transaction Costs
4. Continuous Trading
5. Lognormal Price Movement
6. European Option Exercise

How Black-Scholes Connects to Implied Volatility

One of the most important uses of Black-Scholes is to find the volatility using reverse engineering.

For example, suppose the market premium is ₹10 for an option. We can create a loop to find the correct volatility in the formula that will give us the call price of ₹10. That implied volatility becomes the market's expectation of future movement.

So, the main use of the model is to find implied volatility. This is very important for traders as 

High IV:

• Expensive premiums
• Bigger expected move

Low IV:

• Cheaper premiums
• Lower expected move

How Traders Use Black-Scholes in Practice

Black-Scholes model is extremely useful for traders. Some of the uses are:

1. Fair Value Comparison

The model gives us the theoretical value of the options. Traders can compare this value with the actual market premium. This helps judge the relative richness or cheapness of the option.

Traders can take arbitrage trades using this information.

2. Implied Volatility Analysis

Many traders use Black-Scholes primarily to calculate IV. Once the IV is known, traders can create strategies easily.

For example, if the IV is high, then option selling strategies may become attractive. And if the IV is low, then option buying becomes relatively cheaper.

3. Greeks Estimation

Delta, gamma, theta, and vega are derived using Black-Scholes-based frameworks. All the Greeks are used in different ways to help in creating option strategies. These Greeks can help in estimating:

• Directional sensitivity
• Time decay
• Volatility sensitivity

4. Risk Management

Portfolio managers estimate exposure using model-driven pricing and Greeks.

Black-Scholes and Greeks

The Black-Scholes model also helps in getting values of different Greeks. Here are the most common Greeks used by traders:

• Delta: This measures how much the option may move, based on the movement of the underlying
• Gamma: This measures how the delta moves, based on the movement of the underlying
• Theta: This measures the time decay
• Vega: This measures the option's sensitivity to volatility
• Rho: This measures sensitivity to interest rate changes

Conclusion

The Black-Scholes model changed options trading forever. The model helps traders by providing a lot of information about options, such as IV, delta, theta, gamma, etc. However, the model is based on many assumptions, which might not always be true in the live market.