Master Element 4 covering option pricing components, The Greeks (Delta, Gamma, Theta, Vega, Rho), Black-Scholes variables, futures pricing, cost of carry, and Put-Call Parity.
Derivatives Element 4: Derivative Pricing (~18%)
This element focuses on the mathematical foundations and variables that determine the fair value of derivative contracts. It covers everything from basic intrinsic value to the complex risk sensitivities known as "The Greeks".
Components of Option Pricing
An option's premium (its market price) consists of two distinct parts:
Intrinsic Value
The "built-in" value of an option if exercised immediately:
- For a Call: Stock Price - Strike Price (if positive)
- For a Put: Strike Price - Stock Price (if positive)
Time Value (Extrinsic Value)
The portion of premium exceeding intrinsic value. It reflects the probability that the option will move further in-the-money before expiration. As expiration nears, time value decays toward zero.
The Six Variables of Option Pricing
Under the Black-Scholes Model (used for European options), six inputs determine an option's theoretical price:
| Variable | Effect on Calls | Effect on Puts |
|---|---|---|
| Underlying Asset Price | Higher = Higher Call value | Higher = Lower Put value |
| Strike Price | Lower strike = Higher Call value | Higher strike = Higher Put value |
| Time to Expiration | More time = Higher value | More time = Higher value |
| Volatility (IV) | Higher = Higher value | Higher = Higher value |
| Risk-Free Interest Rate | Higher = Higher Call value | Higher = Lower Put value |
| Dividends | Expected divs = Lower Call | Expected divs = Higher Put |
The Greeks: Measuring Risk Sensitivities
The Greeks are used by traders and risk managers to measure how an option's price reacts to specific market changes:
Delta (Price Sensitivity)
Measures sensitivity of option price to a $1.00 change in underlying asset:
- Calls have positive Delta (0 to +1.00)
- Puts have negative Delta (0 to -1.00)
- Often used as proxy for probability of expiring ITM
Gamma (Delta's Rate of Change)
Measures the rate of change in Delta for every $1.00 move in stock. Highest for ATM options near expiration.
Theta (Time Decay)
Measures the dollar amount an option loses every day as it approaches expiration:
- Always NEGATIVE for option buyers (time works against you)
- Always POSITIVE for option sellers (time works for you)
Vega (Volatility Sensitivity)
Measures sensitivity to a 1% change in Implied Volatility. If Vega is 0.10, the option price will rise by $0.10 if volatility increases by 1%.
Rho (Interest Rate Sensitivity)
Measures sensitivity to a 1% change in interest rates. Usually the least significant Greek for short-term retail options.
Futures and Forward Pricing
Pricing for futures and forwards is based on the Cost of Carry model:
Futures Price = Spot Price + Carrying Costs - Carrying Benefits
Carrying Costs
- Interest paid to finance the purchase
- Storage and insurance for physical commodities
Carrying Benefits
- Dividends (for stocks)
- "Convenience yield" (for physical commodities)
Market Conditions
- Contango: Futures price HIGHER than spot price (normal for assets with high carrying costs)
- Backwardation: Futures price LOWER than spot price (occurs when there's high convenience yield or immediate shortage)
Put-Call Parity
A fundamental no-arbitrage principle for European options. It states that the price of a Call and a Put (with same strike and expiry) are mathematically linked to the underlying stock price and interest rates. If this relationship breaks, an arbitrage opportunity exists.
Key Exam Tips for Element 4
- Premium = Intrinsic Value + Time Value
- Volatility increases BOTH Call and Put values
- Theta is NEGATIVE for buyers, POSITIVE for sellers
- Delta range: Calls (0 to +1), Puts (0 to -1)
- Gamma highest for ATM options near expiry
- Contango = Futures > Spot; Backwardation = Futures < Spot
- Put-Call Parity creates arbitrage if violated