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Volatility and Normal Distribution

Volatility and Normal Distribution in Trading​

Volatility and normal distribution are foundational concepts in financial markets, particularly in options trading. Asset price movements often follow a statistical pattern resembling a normal distribution curve. This bell-shaped and symmetrical curve shows most price movements clustering around the mean (average price), while extreme price movements are less frequent.

Understanding Normal Distribution​

In statistics, a normal distribution is a probability distribution where data points are symmetrically distributed around the mean. The majority of values fall within specific standard deviations (σ) from the mean. In financial terms, this distribution helps traders estimate how much an asset's price is likely to fluctuate in a given period.

Key Characteristics of Normal Distribution:

  • 68% of values fall within ±1σ of the mean.
  • 95% of values fall within ±2σ.
  • 99.7% of values fall within ±3σ.

This implies that extreme price movements are rare, while smaller, more frequent price changes are the norm.

Volatility and Its Relation to Normal Distribution​

Volatility measures an asset’s price fluctuation over time and is represented as the standard deviation in a normally distributed dataset.

  • Higher Volatility: Indicates larger price movements.
  • Lower Volatility: Suggests smaller, steadier price fluctuations.

This relationship helps traders predict the range within which an asset’s price will move.

Example:
If a stock is trading at ₹100 with an annual volatility of 20%, traders expect the price to stay within ₹80 to ₹120 (±20%) for about 68% of the time, assuming a normal distribution. This helps traders estimate the potential risk and reward of holding or trading the asset.

Volatility in Options Pricing​

Normal distribution plays a critical role in pricing options. The Black-Scholes Model, a widely used options pricing model, assumes that stock prices follow a normal distribution of returns.

Impact of Volatility on Option Pricing:

  • Low Volatility: Results in lower option premiums due to the reduced likelihood of significant price movements.
  • High Volatility: Leads to higher option premiums, as larger price swings increase the chance of options expiring in the money.

The Significance of Fat Tails​

While normal distribution provides a framework for predicting price movements, financial markets often experience events that deviate from this model. These rare, extreme events are known as fat tails.

In a perfect normal distribution, the probability of price movements beyond ±3σ is extremely low. However, real markets frequently encounter extreme events, such as crashes or surges, more often than the model predicts. Traders must account for these fat-tail events when assessing market risk.

Applications for Traders​

Understanding the relationship between volatility and normal distribution helps traders in various ways:

  • Risk Assessment: Predicting the likelihood of price movements within a range to gauge potential risks.
  • Options Pricing: Using volatility to estimate how far an asset’s price might move, affecting option premiums.
  • Portfolio Management: Evaluating an asset’s price stability to balance risk and reward.

However, traders should remember that markets do not always follow perfect statistical models. Outliers, or fat-tail events, occur more frequently than expected, making it essential to remain cautious about unexpected market swings.

Conclusion​

Volatility and normal distribution are vital tools for traders, especially in options trading. These concepts allow traders to predict price movements, price options accurately, and manage risk effectively. However, the market’s tendency to produce fat-tail events means that traders must always be prepared for extreme outcomes beyond normal distribution predictions.