Introduction to Options

Options represent contracts that give the holder the right, but not the obligation, to buy or sell an underlying asset at a specified price on or before a specified date. Understanding options is essential for comprehending the most versatile and widely used derivative instruments, which provide opportunities for hedging, speculation, and income generation. Options have grown significantly in importance, with trading volumes in the billions of contracts annually across various asset classes. The optionality embedded in options makes them valuable for risk management and speculation, as they provide asymmetric payoff profiles that allow investors to limit losses while maintaining unlimited profit potential.

The key feature of options is the right, but not the obligation, to exercise the contract. The holder of an option has the choice to exercise the option, depending on the market conditions at the time. This feature distinguishes options from forward and futures contracts, which create obligations to buy or sell the underlying asset. The holder will only exercise the option if it is profitable to do so, meaning that the option holder’s loss is limited to the premium paid, while the potential profit can be substantial. This asymmetric risk-return profile makes options attractive for various investment and risk management purposes.

Options are classified as call options or put options. A call option gives the holder the right to buy the underlying asset at the strike price on or before the expiration date. A put option gives the holder the right to sell the underlying asset at the strike price on or before the expiration date. The buyer of a call option profits if the price of the underlying asset rises above the strike price, while the buyer of a put option profits if the price of the underlying asset falls below the strike price. The seller of an option, also known as the writer, receives the premium and has the obligation to fulfill the contract if the holder exercises the option.

Options are also classified as European options or American options. European options can only be exercised on the expiration date, while American options can be exercised at any time before the expiration date. The classification affects the value of the option, as American options are generally more valuable due to their flexibility in allowing early exercise. The majority of options traded on exchanges are American options, while many over-the-counter options are European options. The distinction between European and American options is important for understanding option pricing and valuation.

Call Options: Characteristics and Payoffs

Call options give the holder the right to buy the underlying asset at the strike price on or before the expiration date. Understanding call options is essential for comprehending how investors can profit from rising prices and how companies can manage their exposure to price increases. Call options are the most common type of option, with applications in various markets and activities, including stock options, index options, and commodity options.

When an investor purchases a call option, they pay a premium to the seller for the right to buy the underlying asset. The premium represents the price of the option and is determined by various factors, including the price of the underlying asset, the strike price, the time to expiration, the volatility of the underlying asset, and the risk-free interest rate. The buyer’s maximum loss is limited to the premium paid, regardless of how far the underlying price falls. The buyer’s profit potential is theoretically unlimited, as the underlying price can rise indefinitely. This asymmetric payoff profile is the primary attraction of call options for speculators and hedgers.

The payoff of a call option at expiration depends on the relationship between the underlying price and the strike price. If the underlying price is below the strike price, the option is out-of-the-money and the holder will not exercise it, resulting in a loss equal to the premium paid. If the underlying price is above the strike price, the option is in-the-money and the holder will exercise it, resulting in a payoff equal to the difference between the underlying price and the strike price, minus the premium paid. The breakeven point for the call buyer is the strike price plus the premium paid.

Example of Call Option Payoff:

Suppose an investor purchases a call option on Company XYZ stock with a strike price of $100 and a premium of $5 per share. The option gives the investor the right to buy XYZ shares at $100 per share on or before the expiration date. Let us examine the payoff at expiration under different scenarios:

• **Scenario 1: XYZ price at expiration is $90.** The option is out-of-the-money because the market price is below the strike price. The investor will not exercise the option, as they could buy the shares cheaper in the market. The investor loses the entire $5 premium. Payoff = -$5 per share.

• **Scenario 2: XYZ price at expiration is $100.** The option is at-the-money. The investor is indifferent between exercising and not exercising, as the market price equals the strike price. In either case, the investor loses the $5 premium. Payoff = -$5 per share.

• **Scenario 3: XYZ price at expiration is $105.** The option is in-the-money by $5. The investor exercises the option, buying the shares at $100 and immediately selling them in the market at $105, earning a $5 profit. However, the investor paid a $5 premium, so the net payoff is $0. This is the breakeven point. Payoff = $105 – $100 – $5 = $0.

• **Scenario 4: XYZ price at expiration is $115.** The option is in-the-money by $15. The investor exercises the option, buying the shares at $100 and selling them at $115, earning a $15 profit. After subtracting the $5 premium, the net profit is $10. Payoff = $115 – $100 – $5 = $10 per share.

Formula for Call Option Payoff at Expiration:

The payoff of a call option at expiration can be expressed mathematically as:

Call Payoff = Max(S – K, 0)

Where:

• S = The price of the underlying asset at expiration

• K = The strike price of the option

Formula for Call Option Profit at Expiration:

The profit of a call option at expiration, considering the premium paid, is:

Call Profit = Max(S – K, 0) – C

Where:

• S = The price of the underlying asset at expiration

• K = The strike price of the option

• C = The premium paid for the call option

Formula for Call Option Breakeven Point:

The breakeven point for a call option is:

Breakeven = K + C

Where:

• K = The strike price of the option

• C = The premium paid for the call option

The seller of a call option, also known as the writer, receives the premium from the buyer and has the obligation to sell the underlying asset at the strike price if the option is exercised. The seller’s maximum profit is limited to the premium received, while the loss potential is theoretically unlimited if the underlying price rises significantly. The seller’s payoff at expiration is the premium received minus the maximum of zero and the difference between the underlying price and the strike price. The breakeven point for the call seller is also the strike price plus the premium received.

Put Options: Characteristics and Payoffs

Put options give the holder the right to sell the underlying asset at the strike price on or before the expiration date. Understanding put options is essential for comprehending how investors can profit from falling prices and how companies can manage their exposure to price decreases. Put options are widely used for hedging and speculation, providing protection against downside risk and enabling investors to profit from market declines.

When an investor purchases a put option, they pay a premium to the seller for the right to sell the underlying asset. The buyer’s maximum loss is limited to the premium paid, regardless of how far the underlying price rises. The buyer’s profit potential is limited to the strike price minus the premium, as the underlying price cannot fall below zero. The asymmetric payoff profile of put options makes them valuable for hedging against downside risk and for speculating on market declines.

The payoff of a put option at expiration depends on the relationship between the underlying price and the strike price. If the underlying price is above the strike price, the option is out-of-the-money and the holder will not exercise it, resulting in a loss equal to the premium paid. If the underlying price is below the strike price, the option is in-the-money and the holder will exercise it, resulting in a payoff equal to the difference between the strike price and the underlying price, minus the premium paid. The breakeven point for the put buyer is the strike price minus the premium.

Example of Put Option Payoff:

Suppose an investor purchases a put option on Company ABC stock with a strike price of $50 and a premium of $4 per share. The option gives the investor the right to sell ABC shares at $50 per share on or before the expiration date. Let us examine the payoff at expiration under different scenarios:

• **Scenario 1: ABC price at expiration is $60.** The option is out-of-the-money because the market price is above the strike price. The investor will not exercise the option, as they could sell the shares for more in the market. The investor loses the entire $4 premium. Payoff = -$4 per share.

• **Scenario 2: ABC price at expiration is $50.** The option is at-the-money. The investor is indifferent between exercising and not exercising, as the market price equals the strike price. In either case, the investor loses the $4 premium. Payoff = -$4 per share.

• **Scenario 3: ABC price at expiration is $46.** The option is in-the-money by $4. The investor exercises the option, buying the shares in the market at $46 and selling them at $50, earning a $4 profit. However, the investor paid a $4 premium, so the net payoff is $0. This is the breakeven point. Payoff = $50 – $46 – $4 = $0.

• **Scenario 4: ABC price at expiration is $40.** The option is in-the-money by $10. The investor exercises the option, buying the shares in the market at $40 and selling them at $50, earning a $10 profit. After subtracting the $4 premium, the net profit is $6. Payoff = $50 – $40 – $4 = $6 per share.

Formula for Put Option Payoff at Expiration:

The payoff of a put option at expiration can be expressed mathematically as:

Put Payoff = Max(K – S, 0)

Where:

• S = The price of the underlying asset at expiration

• K = The strike price of the option

Formula for Put Option Profit at Expiration:

The profit of a put option at expiration, considering the premium paid, is:

Put Profit = Max(K – S, 0) – P

Where:

• S = The price of the underlying asset at expiration

• K = The strike price of the option

• P = The premium paid for the put option

Formula for Put Option Breakeven Point:

The breakeven point for a put option is:

Breakeven = K – P

Where:

• K = The strike price of the option

• P = The premium paid for the put option

The seller of a put option receives the premium from the buyer and has the obligation to buy the underlying asset at the strike price if the option is exercised. The seller’s maximum profit is limited to the premium received, while the loss potential is limited to the strike price minus the premium, as the underlying price cannot fall below zero. The seller’s payoff at expiration is the premium received minus the maximum of zero and the difference between the strike price and the underlying price. The breakeven point for the put seller is also the strike price minus the premium received.

Option Pricing: Key Factors and Intrinsic vs Time Value

Option pricing is determined by various factors, including the price of the underlying asset, the strike price, the time to expiration, the volatility of the underlying asset, the risk-free interest rate, and any income or costs associated with holding the underlying asset. Understanding the factors that affect option pricing is essential for comprehending how options are valued and how changes in market conditions affect option prices. These factors interact in complex ways to determine the premium that buyers are willing to pay and sellers are willing to accept.

The option premium consists of two components: intrinsic value and time value. Intrinsic value represents the amount by which the option is in-the-money, reflecting the immediate profit that would be realized if the option were exercised. For a call option, intrinsic value is the maximum of zero and the difference between the underlying price and the strike price. For a put option, intrinsic value is the maximum of zero and the difference between the strike price and the underlying price. Intrinsic value is the minimum price at which an option can trade, as no rational investor would sell an option for less than its intrinsic value.

Time value represents the portion of the option premium that exceeds the intrinsic value, reflecting the possibility that the option may become more valuable before expiration. Time value is influenced by the time to expiration, the volatility of the underlying asset, and the level of interest rates. Time value is highest for at-the-money options, as these options have the greatest potential to move into or out of the money. Time value decreases as the option approaches expiration, with the rate of decay accelerating in the final weeks before expiration.

Example of Intrinsic Value and Time Value:

Consider a call option on Company XYZ stock with a strike price of $100. The stock is currently trading at $105, and the option premium is $8.

• Intrinsic Value: The option is in-the-money by $5 ($105 – $100 = $5). The intrinsic value is $5.

• Time Value: The option premium is $8, and the intrinsic value is $5, so the time value is $3 ($8 – $5 = $3).

The time value of $3 represents the premium that investors are willing to pay for the possibility that the stock price will rise further before expiration, increasing the option’s value.

Now consider a put option on Company ABC stock with a strike price of $50. The stock is currently trading at $45, and the option premium is $7.

• Intrinsic Value: The option is in-the-money by $5 ($50 – $45 = $5). The intrinsic value is $5.

• Time Value: The option premium is $7, and the intrinsic value is $5, so the time value is $2 ($7 – $5 = $2).

The time value of $2 represents the premium that investors are willing to pay for the possibility that the stock price will fall further before expiration, increasing the option’s value.

Formula for Intrinsic Value:

For a call option:
Intrinsic Value = Max(S – K, 0)

For a put option:
Intrinsic Value = Max(K – S, 0)

Where:

• S = The current price of the underlying asset

• K = The strike price of the option

Formula for Time Value:

Time Value = Option Premium – Intrinsic Value

Where:

• Option Premium = The market price of the option

• Intrinsic Value = The amount by which the option is in-the-money

Factors Affecting Option Prices

Several key factors affect the price of an option, with each factor having a different impact on call options and put options. Understanding these factors is essential for comprehending how option prices change in response to market conditions and for making informed investment decisions.

The price of the underlying asset is the most important factor affecting option prices. Higher underlying prices increase call option prices, as the option becomes more likely to be in-the-money at expiration. Higher underlying prices decrease put option prices, as the option becomes less likely to be in-the-money at expiration. The relationship between the underlying price and the option price is not linear, as the option’s delta changes with the underlying price.

Effect of Underlying Price:

The strike price is the second most important factor affecting option prices. Higher strike prices decrease call option prices, as the option becomes more difficult to be in-the-money. Higher strike prices increase put option prices, as the option becomes easier to be in-the-money. The relationship between the strike price and the option price is also non-linear.

Effect of Strike Price:

The time to expiration affects option prices, with longer time to expiration generally increasing option prices for both calls and puts. The relationship between time to expiration and option price is captured by the theta of the option, which measures the sensitivity of the option price to the passage of time. Options lose value as they approach expiration, with the rate of time decay accelerating in the final weeks.

Effect of Time to Expiration:

The volatility of the underlying asset affects option prices, with higher volatility increasing option prices for both calls and puts. Volatility reflects the uncertainty about future price movements, with higher uncertainty increasing the probability that the option will be in-the-money at expiration. The relationship between volatility and option price is captured by the vega of the option, which measures the sensitivity of the option price to changes in volatility.

Effect of Volatility:

The risk-free interest rate affects option prices, with higher interest rates increasing call option prices and decreasing put option prices. The relationship between interest rates and option prices is captured by the rho of the option, which measures the sensitivity of the option price to changes in interest rates. The effect of interest rates is generally smaller than the effect of other factors, particularly for shorter-term options.

Effect of Risk-Free Interest Rate:

The Black-Scholes Option Pricing Model

The Black-Scholes option pricing model, developed by Fischer Black and Myron Scholes in 1973, provides a rigorous framework for pricing European options, based on the current price of the underlying, the strike price, the time to expiration, the risk-free rate, and the volatility of the underlying. Understanding the Black-Scholes model is essential for comprehending how options are valued and how option prices relate to market conditions. The Black-Scholes model revolutionized option trading and is widely used in practice, providing a theoretical foundation for option pricing that is the basis for most option trading and risk management activities.

The Black-Scholes model is based on several key assumptions: the underlying asset follows a lognormal random walk, meaning that returns are normally distributed; there are no transaction costs or taxes; the risk-free rate is constant and known; the volatility of the underlying is constant and known; trading is continuous and there are no trading restrictions; the markets are frictionless and there are no arbitrage opportunities; and the options are European, meaning they can only be exercised at expiration. These assumptions simplify the model but also limit its applicability in practice. Despite these limitations, the Black-Scholes model provides a valuable framework for option pricing and is widely used by practitioners.

The Black-Scholes Formula for a European Call Option:

The Black-Scholes formula for a European call option is:

C = S × N(d₁) – K × e^(-rT) × N(d₂)

Where:

• C = The theoretical price of the call option

• S = The current price of the underlying asset

• K = The strike price of the option

• r = The risk-free interest rate (expressed as a decimal)

• T = The time to expiration (in years)

• e = The mathematical constant approximately equal to 2.71828

• N() = The cumulative standard normal distribution function (this gives the probability that a normally distributed random variable will be less than a specified value)

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

d₂ = d₁ – σ × √T

Where:

• σ = The volatility of the underlying asset (expressed as a decimal)

• ln = The natural logarithm (logarithm to base e)

• √T = The square root of the time to expiration

The Black-Scholes Formula for a European Put Option:

The Black-Scholes formula for a European put option is:

P = K × e^(-rT) × N(-d₂) – S × N(-d₁)

Where:

• P = The theoretical price of the put option

• All other symbols are the same as defined for the call option formula

Explanation of the Symbols:

Step-by-Step Example: Calculating a Call Option Price Using Black-Scholes

Let us work through a complete example to demonstrate how to use the Black-Scholes model.

Given Information:

• Current stock price (S) = $100

• Strike price (K) = $105

• Time to expiration (T) = 6 months = 0.5 years

• Risk-free rate (r) = 5% = 0.05

• Volatility (σ) = 20% = 0.20

Step 1: Calculate d₁

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

First, calculate ln(S/K):

• S/K = 100/105 = 0.95238

• ln(0.95238) = -0.04879

Next, calculate (r + σ²/2) × T:

• σ² = 0.20² = 0.04

• σ²/2 = 0.04/2 = 0.02

• r + σ²/2 = 0.05 + 0.02 = 0.07

• 0.07 × T = 0.07 × 0.5 = 0.035

Add these together:

• -0.04879 + 0.035 = -0.01379

Calculate σ × √T:

• √T = √0.5 = 0.7071

• σ × √T = 0.20 × 0.7071 = 0.14142

Now calculate d₁:

• d₁ = -0.01379 / 0.14142 = -0.0975

Step 2: Calculate d₂

d₂ = d₁ – σ × √T

• d₂ = -0.0975 – 0.14142 = -0.2389

Step 3: Calculate N(d₁) and N(d₂)

N(d₁) and N(d₂) represent the cumulative standard normal distribution values. We need to find the probability that a standard normal random variable is less than the d values. In practice, these are found using statistical tables or computer functions.

For d₁ = -0.0975, the standard normal table gives approximately 0.4612.
For d₂ = -0.2389, the standard normal table gives approximately 0.4056.

Step 4: Calculate the Call Option Price

C = S × N(d₁) – K × e^(-rT) × N(d₂)

First, calculate e^(-rT):

• rT = 0.05 × 0.5 = 0.025

• e^(-0.025) = 0.9753

Now calculate:

• S × N(d₁) = 100 × 0.4612 = 46.12

• K × e^(-rT) × N(d₂) = 105 × 0.9753 × 0.4056 = 41.56

Therefore:

• C = 46.12 – 41.56 = 4.56

Result: The theoretical call option price is approximately $4.56 per share.

Step-by-Step Example: Calculating a Put Option Price Using Black-Scholes

Using the same inputs as above, we can calculate the put option price.

Given Information:

• Current stock price (S) = $100

• Strike price (K) = $105

• Time to expiration (T) = 6 months = 0.5 years

• Risk-free rate (r) = 5% = 0.05

• Volatility (σ) = 20% = 0.20

We have already calculated:

• d₁ = -0.0975

• d₂ = -0.2389

• N(d₁) = 0.4612

• N(d₂) = 0.4056

Step 1: Calculate N(-d₁) and N(-d₂)

For a standard normal distribution, N(-x) = 1 – N(x):

• N(-d₁) = N(0.0975) = 1 – 0.4612 = 0.5388

• N(-d₂) = N(0.2389) = 1 – 0.4056 = 0.5944

Step 2: Calculate the Put Option Price

P = K × e^(-rT) × N(-d₂) – S × N(-d₁)

We already calculated e^(-rT) = 0.9753

Now calculate:

• K × e^(-rT) × N(-d₂) = 105 × 0.9753 × 0.5944 = 60.84

• S × N(-d₁) = 100 × 0.5388 = 53.88

Therefore:

• P = 60.84 – 53.88 = 6.96

Result: The theoretical put option price is approximately $6.96 per share.

Verification Using Put-Call Parity:

The put-call parity relationship states that C + K × e^(-rT) = P + S

Let us verify:

• C + K × e^(-rT) = 4.56 + (105 × 0.9753) = 4.56 + 102.41 = 106.97

• P + S = 6.96 + 100 = 106.96

The slight difference of $0.01 is due to rounding. This verifies that our calculations are consistent.

Interpreting Black-Scholes Model Outputs

The Black-Scholes model provides several important outputs beyond the option price, including the Greeks, which are measures of the option’s sensitivity to various factors. Understanding these outputs is essential for managing option positions and for making informed investment decisions.

The delta of a call option is N(d₁), which in our example is 0.4612. This means that for a $1 increase in the stock price, the call option price will increase by approximately $0.46. The delta of a put option is N(d₁) – 1, which equals 0.4612 – 1 = -0.5388. This means that for a $1 increase in the stock price, the put option price will decrease by approximately $0.54.

The gamma of the option represents the rate of change of delta with respect to the underlying price. Gamma is higher for at-the-money options and decreases as the option moves in-the-money or out-of-the-money. Gamma is important for understanding how delta changes as the underlying price moves, and for managing the risk of large price movements.

The theta of the option represents the rate of time decay. For a call option, theta is typically negative, meaning that the option loses value as time passes. In our example, the theta of the call option would be approximately -$0.35 per day, meaning that the option loses about $0.35 in value each day, all else being equal.

The vega of the option represents the sensitivity of the option price to changes in volatility. In our example, the vega of the call option would be approximately $0.39, meaning that a 1% increase in volatility (from 20% to 21%) would increase the option price by approximately $0.39.

Limitations of the Black-Scholes Model:

The Black-Scholes model has several important limitations that practitioners must consider. First, the assumption of constant volatility is not realistic, as volatility changes over time. Second, the assumption of continuous trading is not realistic, as trading is discrete. Third, the model does not account for transaction costs or taxes, which can affect option prices. Fourth, the model is only applicable to European options, with different models required for American options. Fifth, the model assumes that returns are normally distributed, but actual returns often have fatter tails than the normal distribution. Despite these limitations, the Black-Scholes model is widely used in practice and provides a valuable framework for option pricing.

Option Greeks: Delta and Gamma

The option Greeks represent measures of the sensitivity of option prices to changes in various factors, providing a framework for understanding option risk and for managing option portfolios. Understanding the option Greeks is essential for comprehending how options behave in different market conditions and how to manage option risk. The option Greeks are fundamental tools for option trading and risk management, used by traders to hedge their positions and by risk managers to assess the risk of option portfolios.

Delta (Δ): Delta measures the sensitivity of the option price to changes in the price of the underlying asset, providing a measure of the option’s exposure to the underlying price. The delta of a call option is positive, ranging from 0 to 1, while the delta of a put option is negative, ranging from -1 to 0. The delta provides a measure of the option’s equivalent position in the underlying asset, with a delta of 0.5 indicating that the option behaves like half a share of the underlying.

Formula for Delta:

• Call Option Delta = N(d₁)

• Put Option Delta = N(d₁) – 1

Where N(d₁) is the cumulative standard normal distribution function evaluated at d₁.

Example of Delta Calculation:

Using our previous Black-Scholes example, with d₁ = -0.0975:

• Call Option Delta = N(-0.0975) = 0.4612

• Put Option Delta = 0.4612 – 1 = -0.5388

This means that if the stock price increases by $1, the call option price increases by approximately $0.46, while the put option price decreases by approximately $0.54.

Gamma (Γ): Gamma measures the sensitivity of the option’s delta to changes in the price of the underlying asset, providing a measure of the curvature of the option’s payoff. Gamma is positive for long option positions and negative for short option positions. The gamma provides a measure of the option’s convexity, with higher gamma indicating greater convexity.

Formula for Gamma:

Γ = N'(d₁) / (S × σ × √T)

Where:

• N'(d₁) is the standard normal probability density function evaluated at d₁

• The standard normal probability density function is N'(x) = (1/√(2π)) × e^(-x²/2)

Example of Gamma Calculation:

Using our previous Black-Scholes example:

• d₁ = -0.0975

• N'(d₁) = (1/√(2π)) × e^(-(-0.0975)²/2)

• N'(d₁) = 0.3970 (calculated using the standard normal density function)

• Γ = 0.3970 / (100 × 0.20 × √0.5)

• Γ = 0.3970 / (100 × 0.20 × 0.7071)

• Γ = 0.3970 / 14.14

• Γ = 0.0281

This means that for a $1 increase in the stock price, the delta will increase by approximately 0.0281. For example, if the stock price increases by $1, the call option delta would increase from 0.4612 to approximately 0.4893.

Interpretation of Delta and Gamma:

The delta and gamma are used together in option portfolio management to manage risk. Delta hedging involves taking positions in the underlying asset to offset the delta exposure of the option portfolio, reducing the sensitivity to underlying price changes. Gamma hedging involves taking positions in options to offset the gamma exposure of the portfolio, reducing the convexity of the portfolio’s exposure. The combination of delta and gamma hedging provides a comprehensive approach to option risk management.

Example of Delta Hedging:

Suppose you have a long position in 100 call options with a delta of 0.4612. Your total delta exposure is 100 × 0.4612 = 46.12 shares equivalent. To delta hedge your position, you need to sell 46.12 shares of the underlying stock. If the stock price falls, the loss on the call options will be offset by the gain on the short stock position. If the stock price rises, the gain on the call options will be offset by the loss on the short stock position.

Option Greeks: Theta, Vega, and Rho

Theta (Θ): Theta measures the sensitivity of the option price to the passage of time, providing a measure of the time decay of the option. Theta is typically negative for long option positions, as options lose value over time, and positive for short option positions, as options gain value over time.

Formula for Theta:

For a call option:
Θ = -(S × σ × N'(d₁)) / (2 × √T) – (r × K × e^(-rT) × N(d₂))

For a put option:
Θ = -(S × σ × N'(d₁)) / (2 × √T) + (r × K × e^(-rT) × N(-d₂))

Example of Theta Calculation:

Using our previous Black-Scholes example for a call option:

• S = 100, σ = 0.20, √T = 0.7071, N'(d₁) = 0.3970

• r = 0.05, K = 105, e^(-rT) = 0.9753, N(d₂) = 0.4056

First term: -(S × σ × N'(d₁)) / (2 × √T)

• -(100 × 0.20 × 0.3970) / (2 × 0.7071)

• -(7.94) / (1.4142)

• -5.61

Second term: -(r × K × e^(-rT) × N(d₂))

• -(0.05 × 105 × 0.9753 × 0.4056)

• -(5.25 × 0.9753 × 0.4056)

• -2.08

Theta = -5.61 – 2.08 = -7.69

This means that the option loses approximately $7.69 per year in time value, or about $0.021 per day ($7.69 / 365). For a one-day passage of time, all else being equal, the call option price would decrease by approximately $0.021.

Vega (V): Vega measures the sensitivity of the option price to changes in the volatility of the underlying asset, providing a measure of the option’s exposure to volatility risk. Vega is positive for long option positions, as higher volatility increases option prices, and negative for short option positions, as higher volatility decreases option prices.

Formula for Vega:

V = S × √T × N'(d₁)

Example of Vega Calculation:

Using our previous Black-Scholes example:

• S = 100, √T = 0.7071, N'(d₁) = 0.3970

• V = 100 × 0.7071 × 0.3970

• V = 28.08

This means that for a 1% increase in volatility (σ increases from 20% to 21%), the option price increases by approximately 0.01 × 28.08 = $0.281. For a 1% decrease in volatility (σ decreases from 20% to 19%), the option price decreases by approximately $0.281.

Rho (ρ): Rho measures the sensitivity of the option price to changes in the risk-free interest rate, providing a measure of the option’s exposure to interest rate risk. Rho is typically small compared to other Greeks and is often ignored for short-term options.

Formula for Rho:

For a call option:
ρ = K × T × e^(-rT) × N(d₂)

For a put option:
ρ = -K × T × e^(-rT) × N(-d₂)

Example of Rho Calculation:

Using our previous Black-Scholes example for a call option:

• K = 105, T = 0.5, e^(-rT) = 0.9753, N(d₂) = 0.4056

• ρ = 105 × 0.5 × 0.9753 × 0.4056

• ρ = 20.78

This means that for a 1% increase in the risk-free rate (r increases from 5% to 6%), the call option price increases by approximately 0.01 × 20.78 = $0.208. For a 1% decrease in the risk-free rate, the option price decreases by approximately $0.208.

Summary of the Black-Scholes Model Components

The Option Greeks Summary: