Volatility and Greeks

Volatility

Volatility can be a very important factor in deciding which options to buy or sell. Volatility tells investors how much the stock price will fluctuate over a period of time. The formal mathematical value of volatility is defined as “the annualized standard deviation of the daily price change of a stock.”

There are two types of volatility: statistical volatility and implied volatility.

Statistical volatility: measures the change in actual asset prices over a specific time frame.

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Implied Volatility: For option prices, measures the amount that the “market” expects the asset price to change. This is the volatility implied by the market itself.

The calculation of volatility is a difficult problem in mathematics.

In the Black-Scholes model, volatility is defined as the annual standard error of the stock price. Option strategists have a way of using the market to calculate volatility. This is called using implied volatility, which means the volatility that the market itself implies. This is similar to the efficient market hypothesis. If an option has enough volume trading near its price near the at-the-money option price, the option is usually priced appropriately.

Black-Scholes formula

The Black-Scholes formula is the first widely adopted option pricing model. This formula uses the current stock price, expected dividends, option strike price, expected interest rate, time remaining until expiration, and expected stock volatility to calculate the theoretical value of an option. Although the Black-Scholes model does not perfectly describe the real-world option market, it is still frequently used in option valuation and trading.

The variables in the Black-Scholes formula include:

  • Stock Price
  • Strike Price
  • The time remaining to maturity expressed as a percentage of the year
  • Current risk-free interest rate
  • Volatility expressed as annual standard error.

Greek letters

Greeks are statistics expressed as percentages. Investors can use them to get a better overall understanding of a stock’s performance. These statistics can help investors decide which options strategy is best to pursue. Investors should remember that statistics only show trends based on a stock’s past performance. It does not guarantee that a stock’s future performance will be based on past performance. These trends can change significantly based on new stock performance.

Beta: Measures how closely the movements of an individual stock follow the movements of the overall stock market.

Delta: Delta measures the relationship between an option’s price and the price of its underlying stock. For a call contract, a delta of 0.5 means that for every dollar increase in the underlying stock, the premium will increase by half a point. For a put contract, a decrease in the stock price will cause the premium to increase. As options approach expiration, the delta of an out-of-the-money contract approaches 1.

In this example, the delta for XYZ stock is 0.50. When the stock price changes by $2.00, the option price changes by 50 cents for every dollar the stock price changes. Therefore, the change in the option price is (0.50 * 2) = 1.00. The call option price increases by $1.00. The put option price decreases by $1.00. The delta is not a fixed percentage value. Changes in the stock price and changes in the time remaining until expiration affect the value of the delta.

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Gamma: Gamma is a measure of the sensitivity of the delta to a unit change in the underlying stock. Gamma represents the absolute change in the delta. For example, a change of 0.150 in gamma means that if the price of the underlying stock rises or falls by 1.0, the delta will rise by 0.150. This result may not be completely accurate due to rounding.

Lambda: Lambda is a measure of leverage. It is the expected percentage change in option value for a one percent change in the underlying stock value.

LoVu: LoVu measures the change in option value relative to the interest rate. LoVu represents the absolute change in option value for a one percentage change in interest rates. For example, a LoVu value of 0.060 indicates that if interest rates decrease by one percentage point, the theoretical value of the option increases by 0.060. This result may not be completely accurate due to rounding.

Fee: Fees measure the change in option value with respect to time. The fee represents the absolute change in option value for each “unit of time” decrease in the time until expiration. The Option Calculator assumes that “one unit of time” is seven days. For example, a fee of -250 means that the theoretical value of the option will change by -0.250 for every seven days that the time until expiration decreases. This result may not be completely accurate due to rounding. Note: If the time until expiration is seven days or less, the 7-day fee changes to a 1-day fee. (See Time Weakening).

Vega (Kappa, Omega, Set): Vega measures the change in option value relative to volatility. Vega represents the absolute change in option value relative to a one percent change in volatility. For example, a Vega of 0.090 means that if volatility increases by one percent, the option’s theoretical value will increase by 0.090; if volatility decreases by one percent, the option’s theoretical value will decrease by 0.090. This result may not be completely accurate due to rounding.