Chapter 5: Understanding Options Pricing

Objective

In this chapter we will understand in depth how options pricing is affected by various factors, and we will also see Black-Scholes model which is one of the most popular mathematical models used for options pricing calculation. We will also learn about Implied volatility and historical volatility.

Section 5.1

Factors Affecting Option Pricing

Option pricing is influenced by several key factors, each contributing to the option’s total premium. These factors are critical to understand for anyone trading or analyzing options. Let’s break down each one.

1. Underlying Asset Price

The underlying asset price (the spot price of the asset) is perhaps the most influential factor in determining the price of an option. Here’s how it affects different types of options:

For Call Options: The higher the underlying asset price, the more valuable a call option becomes. This is because the option holder has the right to buy the asset at a lower strike price than its current market price.
For Put Options: Conversely, as the underlying asset price decreases, the value of a put option increases because the option holder has the right to sell the asset at a higher strike price than its current market value.

2. Time to Expiry

The time to expiry of an option refers to how much time remains until the option expires. The more time there is before expiration, the greater the possibility that the option will end up in-the-money, and hence the higher its value.

Time Decay (Theta): As the expiration date approaches, the option loses value, primarily because there is less time for the underlying asset’s price to make significant moves. This phenomenon is known as time decay.
Longer Expiry = Higher Premium: All else being equal, options with longer expiration periods tend to have higher premiums due to the greater time value.

3. Volatility

Volatility is a critical factor that drives option prices. It refers to how much the price of the underlying asset fluctuates over time. There are two main types of volatility:

Historical Volatility (HV): This measures past price fluctuations of the underlying asset. It is based on historical data and is often used to estimate future price movements.
Implied Volatility (IV): Implied volatility is the market’s expectation of future volatility, derived from the price of the option itself. Higher implied volatility increases the probability of larger price movements, making options more expensive.

Impact of Volatility:

Higher Volatility = Higher Option Prices: If the underlying asset is expected to be more volatile, both call and put options tend to become more expensive because the potential for larger moves increases the chances of ending up in-the-money.

4. Interest Rates

Interest rates also influence option pricing, particularly for long-term options. The basic relationship is:

For Call Options: When interest rates rise, the price of call options increases. This is because the present value of the strike price (paid later) decreases, making it more attractive to hold the option instead of the underlying asset.
For Put Options: Conversely, when interest rates rise, put options tend to decrease in value for the same reason—the present value of the strike price increases.

5. Dividends

When the underlying asset (usually stocks) pays a dividend, it affects the price of options:

For Call Options: When a dividend is expected, the underlying asset’s price typically drops by the dividend amount when it goes ex-dividend. This decreases the value of call options, as the underlying asset is worth less.
For Put Options: The value of put options may increase because the underlying asset price drops by the dividend amount, making the strike price more valuable in comparison.

Section 5.2

The Black-Scholes Model

The Black-Scholes model is one of the most widely used methods for pricing European-style options (options that can only be exercised at expiration). The model uses several key inputs (the factors mentioned above) to calculate the theoretical price of an option.

Key Inputs to the Black-Scholes Model:

S₀: Current price of the underlying asset
K: Strike price of the option
T: Time to expiration (in years)
r: Risk-free interest rate (typically the rate of return on government bonds)
σ: Volatility of the underlying asset
N(x): Cumulative normal distribution function (for calculating the probabilities associated with the option’s outcome)

The formula for a European call option is:

$\cdot N(d₁) – K \cdot e^{-rT} \cdot N(d₂)$

Where:

$\frac{\ln(S₀ / K) + (r + 0.5 \cdot σ²) T}{σ \sqrt{T}}$
$\sqrt{T}$

The formula for a put option is similar, with adjustments for the different payoff structure.

Though the Black-Scholes model is powerful, it assumes constant volatility and interest rates, and it’s designed for European options, which can only be exercised at expiration.

Section 5.3

Greeks and Their Significance

The Greeks are key risk management tools in options trading, helping traders understand how different factors (like changes in price or time) affect the price of an option. The main Greeks are:

1. Delta (Δ): Sensitivity to Underlying Price

Delta measures how much the price of an option changes with respect to a change in the price of the underlying asset.
For Call Options: Delta is positive. As the underlying asset price increases, the value of the call option increases.
For Put Options: Delta is negative. As the underlying asset price decreases, the value of the put option increases.

2. Gamma (Γ): Sensitivity of Delta

Gamma measures the rate of change of delta as the price of the underlying asset changes. It helps assess how stable delta is.
Gamma is highest for options that are at-the-money and decreases as the option moves further in- or out-of-the-money.

3. Theta (Θ): Sensitivity to Time Decay

Theta measures the sensitivity of an option’s price to the passage of time. It shows how much the option’s price will decrease as time passes, assuming all other factors remain constant.
Time Decay: Theta is typically negative for both calls and puts, meaning options lose value as expiration approaches.

4. Vega (V): Sensitivity to Volatility

Vega measures the sensitivity of an option’s price to changes in the implied volatility of the underlying asset.
Higher Volatility = Higher Option Prices: If implied volatility increases, the price of options tends to increase, as greater volatility increases the likelihood of the option ending in-the-money.

5. Rho (ρ): Sensitivity to Interest Rates

Rho measures the sensitivity of an option’s price to changes in interest rates.
For Call Options: A higher interest rate generally increases the price of the call option because the present value of the strike price decreases.
For Put Options: A higher interest rate generally decreases the price of the put option.

Section 5.4

Implied Volatility vs Historical Volatility

Understanding the distinction between implied volatility and historical volatility is crucial for options traders.
- Implied Volatility (IV) reflects the market’s expectation of how volatile an asset will be in the future. It is derived from option prices and shows how much the market expects the price of the underlying asset to move.
- Historical Volatility (HV), on the other hand, measures how volatile an asset has been in the past, calculated from historical price data.

Key Differences:

Implied Volatility is forward-looking, based on market sentiment and expectations.
Historical Volatility is backward-looking, based on actual price movements.

Volatility and Option Pricing: Higher implied volatility often leads to higher option premiums, as it increases the potential for larger price moves. Traders often use implied volatility to gauge the level of uncertainty or risk in the market.

Section 5.5

Volatility Smile and Skew

The volatility smile and volatility skew are patterns that describe how implied volatility changes for options with different strike prices and expiration dates.

Volatility Smile

The volatility smile is a graphical representation of implied volatility across different strike prices for options with the same expiration date. Typically, at-the-money options have lower implied volatility, and options that are deep in-the-money or deep out-of-the-money have higher implied volatility, creating a “smile” shape on a chart.

Volatility Skew

The volatility skew (or volatility term structure) refers to how implied volatility varies with different strike prices or expiration dates, and often shows that out-of-the-money puts tend to have higher implied volatility than out-of-the-money calls.
This skew is often seen in equity options due to market participants’ demand for protection against downside risk.

Final Takes

Conclusion

Understanding the factors that affect options pricing, including the Black-Scholes model, the Greeks, and the behavior of implied and historical volatility, is essential for any options trader. By mastering these concepts, traders can make more informed decisions, manage risk more effectively, and develop strategies tailored to market conditions.