Beginner’s Guide to Time Decay & Implied Volatility-Chapter 3

3.1 What is Time Decay (Theta) in Options Trading?

Time Decay

Time decay, also known as “Theta,” is a concept primarily associated with options trading. It refers to the rate at which the value of an option decreases as it approaches its expiration date. Options have a time premium, which represents the extra value traders are willing to pay for the possibility that an option may become profitable before it expires. As time passes, the likelihood of big price movements diminishes, causing the time premium to gradually erode. This erosion is called time decay.

How Time Decay Works

Time decay, often referred to as “theta” in options trading, represents how the value of an option decreases over time. This reduction primarily affects the extrinsic value (time value) of the option, leaving its intrinsic value unaffected. Extrinsic value is influenced by factors such as volatility and time remaining until expiration.

Why Time Decay Occurs

• Options lose extrinsic value because time is a limited resource—less time means fewer chances for the underlying asset to make significant price movements.
• The closer the expiration date, the faster the extrinsic value erodes. This process accelerates in the final 30 days before expiration, known as the “time decay curve.”

Intrinsic Value in Options – Meaning & Formula

Intrinsic value refers to the actual value of an option based on the underlying asset’s current price, regardless of time or implied volatility. It represents the immediate “profitability” of exercising the option.

For Call Options:

A call option has intrinsic value when the underlying asset’s price is above the strike price.

Formula: Intrinsic Value = Current Price of Underlying Asset – Strike Price

If the stock is trading at ₹120 and the strike price of the call is ₹100, the intrinsic value is ₹20. This means if the buyer exercises the call, they can purchase the stock at ₹100 and potentially sell it at ₹120 in the market, realizing a profit of ₹20.

For Put Options:

A put option has intrinsic value when the underlying asset’s price is below the strike price.

Formula: Intrinsic Value = Strike Price – Current Price of Underlying Asset

If the stock is trading at ₹80 and the strike price of the put is ₹100, the intrinsic value is ₹20. This means if the buyer exercises the put, they can sell the stock for ₹100 while it is worth ₹80 in the market, realizing a profit of ₹20.

Extrinsic (Time) Value of Options Explained

Extrinsic value refers to the portion of an option’s price above its intrinsic value, and it reflects factors like time until expiration, implied volatility, and market sentiment. It’s also known as the time value of an option.

Time Until Expiration:

Extrinsic value decreases as the option approaches expiration (known as time decay). The longer the time to expiration, the greater the chance of price movement in the underlying asset, and thus, the higher the extrinsic value.

Implied Volatility:

Higher implied volatility increases extrinsic value, as it suggests greater uncertainty and a wider range of potential price movements for the underlying asset.

Formula for Total Option Price:

Total Option Price (Premium) = Intrinsic Value + Extrinsic Value

If a call option is priced at ₹50, and its intrinsic value is ₹20, the remaining ₹30 represents the extrinsic value.

Time Decay: How It Affects Buyers vs Sellers

Time decay is a double-edged sword in options trading:

• For Buyers: It reduces the value of the option as expiration approaches, which can lead to losses if the underlying asset doesn’t move significantly.
• For Sellers: It creates opportunities to profit by selling options and benefiting from their gradual erosion in value.

How Theta Impacts Options Prices Over Time

The effect of time decay on pricing is measured by theta, which quantifies the rate at which an option’s price decreases each day. Let’s break this down:

Impact by Option Type

Out-of-the-Money (OTM):

• These options have no intrinsic value and rely entirely on extrinsic value.
• Time decay affects them most significantly, often rendering them worthless as expiration nears.

At-the-Money (ATM):

• These options experience rapid time decay because they heavily depend on extrinsic value.
• Their value erodes faster than ITM options but slower than OTM options.

In-the-Money (ITM):

Time decay impacts these options less, as their intrinsic value offers a cushion against the loss of extrinsic value.

Pricing Dynamics

Time decay reduces the premium paid for options. For example, a call option priced at ₹50 may lose ₹1 each day due to theta decay if the underlying asset’s price remains constant. As expiration approaches, this rate may increase, leading to sharp declines in the option’s value.

Time Decay Benefits

Time decay offers a strategic advantage for option sellers, also known as “writers.” Here’s how they benefit:

Key Advantages

Profit from Premiums: Sellers collect premiums upfront when selling options. As time decay erodes the extrinsic value, the likelihood of the option being exercised decreases, allowing sellers to profit if the option expires worthless.

High-Probability Trades: Strategies such as selling credit spreads or writing short straddles thrive on time decay, as traders aim for stable underlying asset prices within specific ranges.

For Buyers

Buyers can mitigate losses from time decay by exiting positions early or using shorter expiration periods to minimize exposure.

Difference Between Time Decay and Moneyness

Time decay and Moneyness are interconnected but distinct concepts in options trading:

Time Decay (Theta)

A measure of how much the extrinsic value of an option reduces over time.

Impact depends on factors like expiration date and volatility, irrespective of the option’s profitability (Moneyness).

Moneyness

Indicates the profitability of an option based on the strike price and underlying asset’s current price:

• In-the-Money (ITM):The strike price is favorable, and the option has intrinsic value.
• At-the-Money (ATM):The strike price equals the asset’s current price, relying entirely on extrinsic value.
• Out-of-the-Money (OTM):The strike price is unfavorable, containing no intrinsic value.

Differences in Impact

• Time decay affects extrinsic value, while moneyness determines intrinsic value.
• ATM and OTM options are more vulnerable to time decay than ITM options.

Example of Time Decay in Options with reference to Power Sector

Imagine there’s a company that supplies electricity, and its stock price is ₹100. You buy a call option (a type of financial contract) for this stock with a strike price of ₹110. This option lets you buy the stock at ₹110 before the expiration date, and you pay ₹10 as a fee (called the premium).

• Early on (30 days left): There’s still a lot of time for the stock price to go above ₹110. The premium stays around ₹10 because the option has potential.
• Midway (15 days left): The stock price hasn’t moved much—it’s still around ₹100. Now there’s less time for the stock to rise above ₹110, so the premium might reduce to ₹6.
• Final days (2 days left): The stock price is still ₹100, and there’s very little time left for it to rise above ₹110. The option becomes almost worthless, and the premium could drop to ₹2.

Why does the Premium Reduce?

• Time Decay (Theta): As the expiration date approaches, there’s less time for the stock price to move in a way that benefits the option buyer. This reduces the time value of the option, which is a key component of the premium.
• Stock Price Movement: If the stock price doesn’t move closer to the strike price (or go above it, for call options), the option becomes less attractive because it’s less likely to be profitable.
• Volatility: Options rely on volatility (how much the stock price fluctuates). If the market becomes calmer, the chances of big price movements decrease, making the option less valuable.
• Intrinsic Value: If the stock price is far below the strike price (for call options) or far above the strike price (for put options), the option has no intrinsic value. This also causes the premium to drop.
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3.2 What is Implied Volatility (IV) in Options?

Implied volatility (IV) measures the market’s expectations of future price movements of the underlying asset. It is derived from the price of an option and reflects how volatile traders expect the asset to be during the option’s lifespan. Higher implied volatility means higher option prices, as there is a greater chance of significant price movements, while lower Implied Volatility suggests less expected volatility and, thus, lower option prices.

How Implied Volatility Works?

• Implied volatility (IV) is a measure of how much the market believes the price of a stock or other underlying asset will move in the future. It is a key factor in determining the price of an options contract. When traders buy or sell options they are gaining exposure on how much the price will fluctuate before the option expires.
• Unlike historical volatility which measures past price fluctuations, implied volatility is forward-looking and derived from the current market price of an option. Implied volatility isn’t directly observable in market. It must be calculated using an options pricing models like Black-Scholes. Implied Volatility is used to gauge whether options prices are relatively cheap or expensive. Options with higher implied volatility will be more expensive than an option with low implied volatility.
• Some traders try to profit from changes in implied volatility. The trader might buy options when implied volatility is low, expecting it to rise, or sell options when implied volatility is high , expecting it to fall. Implied volatility is a key input into many risk management models that traders and institutions use to manage their options portfolios.

3.3 How IV Affects Call and Put Option Prices

Options pricing is influenced by various factors (known as “the Greeks”), and implied volatility is a critical one. Here’s how it plays a role:

How IV Affects Both Call and Put Options Prices

Implied volatility (IV) directly influences the premium (price) of options. Both call options (which give the buyer the right to buy the underlying asset at a predetermined price) and put options (which give the buyer the right to sell the asset at a predetermined price) become more expensive as IV rises.

This increase happens because high IV implies greater uncertainty and a wider range of possible outcomes for the underlying asset’s price. With more uncertainty, the likelihood of reaching a favorable strike price—whether for a call or a put—rises.

Why Option Prices Rise with Higher IV

The higher the IV, the more the market expects significant price swings for the underlying asset. Even if the stock price remains unchanged, the greater probability of large movements increases the value of time in the option’s pricing (known as the “time value”).

As a result, both call and put option holders are paying for the possibility that these larger price swings could lead to profitable positions.

Probability of Reaching a Favorable Strike Price

For Call Options

Higher IV increases the chance that the stock price could rise above the strike price (the level at which the call option holder can buy the stock). Even if the stock price moves unpredictably, the larger range of potential price movements makes it more likely to favor the buyer.

For example, if a stock is currently priced at ₹100, and the strike price of a call option is ₹120, high IV could suggest that the stock might swing upward and cross ₹120 before expiration. This possibility makes the call option premium more expensive.

For Put Options

Similarly, higher IV boosts the probability that the stock price could fall below the strike price of the put option, making it valuable to the buyer.

For example, if the stock price is ₹100 and the put option strike price is ₹80, increased volatility means the stock might swing downward past ₹80, thereby increasing the put premium.

Time Value Component

The portion of an option’s price attributed to time value grows significantly when IV rises. This is because traders and investors expect the price to have enough room to make a big move—whether upward (for calls) or downward (for puts)—within the remaining time to expiration. Options with longer expiration dates are generally more sensitive to IV because they have more time to realize potential price movements.

Volatility Skew:

Different strike prices may have varying implied volatilities, leading to a phenomenon called the volatility skew. This occurs because market participants perceive different probabilities of price movement across different strike prices.

The “Vega” Greek:

• Vega measures an option’s sensitivity to changes in implied volatility. Options with higher Vega experience larger price changes for a given change in IV.
• Vega is typically highest for at-the-money options and decreases for options that are deep in-the-money or out-of-the-money.

3.4 Computing Implied Volatility

Computing implied volatility (IV) involves using an option pricing model to match the theoretical price of an option with its market price. Since IV is not directly observable, it is determined through an iterative process. Let me guide you through the steps:

Step 1: Understand the Inputs

To calculate implied volatility, you need the following:

• Market price of the option
• Current price of the underlying asset
• Strike price of the option.
• Time to expiration
• Risk-free interest rate .
• Dividend yield .

Step 2: Use an Option Pricing Model

The Black-Scholes Model is the most widely used formula for pricing options. It calculates the theoretical price of an option based on several inputs, including volatility. However, the implied volatility is not a direct input—it’s the unknown value we’re solving for.

Step 3: Iterative Calculation Process

Since implied volatility cannot be directly calculated, the process involves trial and error:

• Guess Initial Volatility: Start with an assumed volatility value (e.g., 20% or 0.20).
• Calculate Theoretical Price: Plug all the known inputs, along with the guessed volatility, into the Black-Scholes Model. Compute the theoretical price of the option.
• Compare Prices: Compare the theoretical price of the option to its actual market price. If they match, the guessed volatility is the implied volatility. If not, proceed to the next step.
• Adjust Volatility: Adjust the guessed volatility value higher or lower, depending on whether the theoretical price is below or above the market price.
• Repeat Until Convergence: Continue iterating until the theoretical price converges closely with the market price within an acceptable tolerance.

The Black-Scholes Model

The Black-Scholes Model, developed by Fischer Black, Myron Scholes, and later refined by Robert Merton, is one of the most popular mathematical models used for pricing options. It provides a theoretical framework for determining the fair market value of European-style options (options that can only be exercised at expiration). Here’s a detailed explanation:

Underlying Assumptions

The model operates under specific assumptions:

1. Efficient Markets: The market is efficient, meaning prices reflect all available information.
2. Lognormal Distribution: The stock prices follow a lognormal distribution (they cannot become negative).
3. Constant Volatility: The volatility of the underlying asset remains constant over the life of the option.
4. No Arbitrage: There’s no opportunity for risk-free profits by combining assets and options.
5. Risk-Free Rate: A constant risk-free interest rate is assumed.
6. No Dividends: The underlying asset pays no dividends during the option’s life (though extensions of the model account for dividends).
7. European Options: It applies only to European options, which can only be exercised at expiration, not American options (which can be exercised anytime).

The Black-Scholes Formula

The formula for the price of a European call option is:

C= S₀ N(d₁)−Xe^−rT N(d₂)

For a European put option:

P=Xe^−rT N(−d₂) −S₀N(−d₁)

Where:

• C: Call option price
• P: Put option price
• S₀: Current price of the underlying asset
• X: Strike price of the option
• T: Time to expiration (in years)
• r: Risk-free interest rate
• N(d): Cumulative standard normal distribution function
• e: Euler’s number, used for discounting
• d₁and d₂ are intermediary variables defined as:

d1=ln(S₀/X) + (r+σ²/2) T/σ√T

d2=d1−σ √T

 σ: Volatility of the underlying asset

Key Components Explained

1. Current Stock Price (S₀): Reflects the current market price of the underlying asset.
2. Strike Price (X): The price at which the option holder can buy (call) or sell (put) the underlying asset.
3. Time to Expiration (T): Measured in years. The longer the time, the more potential for price movements, affecting the option’s price.
4. Risk-Free Rate (r): A hypothetical return from a risk-free investment, such as a government bond.
5. Volatility (σsigma): Measures the standard deviation of the underlying asset’s returns. Higher volatility increases the option price due to greater uncertainty.
6. Normal Distribution Function (N(d)N(d)): Represents the probability that a standard normal random variable is less than or equal to dd. It helps estimate the likelihood of the option finishing in the money.

How the Model Works

• The model assumes that the underlying stock price moves according to a geometric Brownian motion with constant drift and volatility.
• It uses risk-neutral valuation, meaning that all investors are indifferent to risk. The expected return of the stock is replaced with the risk-free rate in the pricing formula.
• By plugging the known values (stock price, strike price, time to expiration, risk-free rate, and volatility) into the formula, you can calculate the theoretical option price.

Strengths of the Black-Scholes Model

1. Simplicity: The formula is easy to use for calculating theoretical prices.
2. Insight: It provides a clear relationship between option price and its inputs.
3. Standardization:It’s widely adopted, making it a benchmark in the options market.

Limitations of the Black-Scholes Model

1. Assumption of Constant Volatility: Real-world volatility often changes over time (volatility smile/skew).
2. No Dividends: The basic model doesn’t account for dividends, but extensions do.
3. European Options: It applies to European-style options only, limiting its use for American options.
4. Efficient Markets: In practice, markets may not always be perfectly efficient.

3.5 Real-World Example: NIFTY Options & IV Fluctuations

Imagine it’s a week before an important event, such as the Union Budget announcement, and the NIFTY index is trading at 18,000. Market participants expect significant price movements due to anticipated policy changes, leading to higher implied volatility in NIFTY options.

Scenario 1: High Implied Volatility

Suppose a trader is analyzing a NIFTY call option with a strike price of 18,200 (slightly out-of-the-money).

Due to the uncertainty around the budget announcement, implied volatility spikes to 30%. This increases the premium of the call option, say from ₹120 to ₹200.

For Buyers:

• The buyer pays a higher premium because the market expects large price movements. If NIFTY surges to 18,500 post-budget, the buyer gains significantly.
• However, if NIFTY stays stable or moves slightly, the buyer incurs a loss due to the high premium paid

For Sellers:

• The seller collects a higher premium upfront due to the elevated IV, but faces a higher risk if NIFTY moves drastically post-event.

Scenario 2: Volatility Crush

After the budget announcement, the uncertainty is resolved, and implied volatility drops to 15%. Option premiums fall, say from ₹200 back to ₹120.

For Buyers:

Buyers suffer from the “volatility crush” if they purchased options during high IV but the market doesn’t move as expected.

For Sellers:

Sellers benefit from the drop in IV, as they can buy back the options at lower premiums to close their positions.

Other  Examples

Stock-Specific Events:

Consider options on Reliance Industries. Before the company’s quarterly earnings report, implied volatility often rises, reflecting the market’s anticipation of potential surprises in financial results.

Traders adjust their strategies based on whether they expect volatility to increase further or revert post-event.

Election Results:

• National or state elections (e.g., Lok Sabha elections) can impact IV in broad-based indices like NIFTY or Bank NIFTY.
• Higher IV reflects market uncertainty about election outcomes, while IV typically drops once results are announced.