Chapter 6: Pricing of Options Contracts and Greek Letters
Overview & Fundamentals of Option Pricing
Option prices in secondary markets are determined by the dynamics of supply and demand. An option gives the buyer the right, but not the obligation, to exercise a trade, capping the buyer's potential loss to the premium paid while leaving the upside potentially unlimited. Because this optionality has monetary value, quantitative financial models such as the Binomial Option Pricing Model and the Black-Scholes Merton Model are utilized to calculate the fair theoretical value of option contracts.
6.1 Variables Affecting Option Pricing
The price of an option contract is determined by six primary market variables:
- Spot Price of the Underlying Asset (S)
- Strike / Exercise Price (X)
- Volatility of the Spot Price (σ)
- Time to Expiration (T)
- Risk-Free Interest Rate (r)
- Anticipated Dividends (D)
Impact of Variables on Call and Put Options
- Spot Price (S): An increase in the spot price increases the value of a call option (as the option is more likely to be exercised) and decreases the value of a put option.
- Strike Price (X): An increase in the strike price decreases the value of a call option (as the payoff S - X decreases) and increases the value of a put option.
- Volatility (σ): Higher volatility in the underlying stock price increases the price of both call and put options because wider price swings expand potential gains while downside risk remains capped.
- Time to Expiration (T): Longer expiration periods increase the value of American options, as there is a higher probability of the option moving into-the-money before maturity. Option prices decay as they approach expiration.
- Risk-Free Rate (r): An increase in the risk-free rate increases the value of call options and decreases the value of put options.
- Dividends (D): Higher anticipated dividends decrease call option prices and increase put option prices.
Summary: Impact of Variables on Option Prices
| Variable | Symbol | Call Option Price | Put Option Price |
|---|---|---|---|
| Spot Price | S | Increases | Decreases |
| Strike Price | X | Decreases | Increases |
| Volatility | σ | Increases | Increases |
| Time to Expiration | T | Increases (American) | Increases (American) |
| Risk-Free Rate | r | Increases | Decreases |
| Dividends | D | Decreases | Increases |
Price Limits of Option Contracts
Option prices operate within definite upper and lower boundaries:
- Maximum Call Price: A call option on a stock can never be worth more than the spot price of the underlying asset (Maximum Call Price = S) for both European and American options.
- Minimum European Call Price: The lower bound for a European call option on a non-dividend-paying stock is S - X * e^(-r * T).
- Maximum Put Price: A put option can never be worth more than the present value of the strike price (Maximum Put Price = X * e^(-r * T)) for both European and American options.
- Minimum European Put Price: The lower bound for a European put option on a non-dividend-paying stock is X * e^(-r * T) - S.
6.2 The Black-Scholes Merton Model for Option Pricing (BSO)
Developed in 1973 by Fischer Black, Myron Scholes, and Robert Merton, the Black-Scholes Merton (BSO) model established a framework for pricing and hedging options.
Core Assumptions & Premise
- Stock price changes are random and log-normally distributed.
- Technical analysis does not influence option pricing.
- A riskless portfolio can be constructed combining the underlying stock and option, eliminating price uncertainty and earning the risk-free interest rate instantaneously.
Model Formulae (Simple Text Format)
Annualized Volatility Conversion
Sigma_annual = Sigma_daily * sqrt(Trading_Days_Per_Year)
(Where the number of trading days per year is taken as 250 on average).
European Call Option Price Formula
c = S * N(d1) - X * e^(-r * T) * N(d2)
European Put Option Price Formula
p = X * e^(-r * T) * N(-d2) - S * N(-d1)
Calculation of d1 and d2
d1 = [ ln(S / X) + (r + (sigma^2) / 2) * T ] / (sigma * sqrt(T))
d2 = d1 - sigma * sqrt(T)
Key Terms & Parameters:
- c / p: Price of European Call / Put option
- S: Current spot price of the underlying stock
- X: Strike / exercise price
- r: Continuously compounded risk-free interest rate
- T: Time to expiration measured in years
- sigma: Annualized volatility (standard deviation of continuously compounded returns)
- N(x): Cumulative distribution function for a standardized normal distribution
- N(d2): Probability that the option will be exercised in a risk-neutral world
- S * N(d1) * e^(r * T): Expected value of a variable equaling spot price ST if ST > X and 0 otherwise in a risk-neutral world
Limiting Conditions & Extreme Scenarios
- Large Spot Price (S -> ∞):
- A call option becomes virtually certain to be exercised, behaving like a forward contract with delivery price X.
- Both N(d1) and N(d2) approach 1.0.
- Call price formula simplifies to c = S - X * e^(-r * T).
- Put option value approaches 0 as N(-d1) and N(-d2) approach 0.
- Volatility Approaches Zero (sigma -> 0):
- d1 and d2 tend to infinity, causing N(d1) and N(d2) to reach 1.0.
- Call price equals Max(S - X * e^(-r * T), 0).
6.3 The Greeks
Option Greeks measure specific risk dimensions of an option position or portfolio, enabling traders to manage exposures effectively.
6.3.1 Delta (Δ)
- Definition: The rate of change of the option price with respect to the price of the underlying asset (Delta = Change in Option Price / Change in Spot Price).
- Properties:
- Represents the slope of the curve relating option price to underlying stock price.
- Delta of the underlying stock itself is always 1.0.
- Call Option Delta is always positive (0 to +1).
- Put Option Delta is always negative (-1 to 0).
- Formulae for Dividend-Paying Stock (Dividend Yield q):
- European Call Delta = N(d1) * e^(-q * T)
- European Put Delta = e^(-q * T) * [N(d1) - 1]
- Delta Neutrality / Delta Hedging: Maintaining a portfolio Delta at zero by buying or selling a specific quantity of underlying stock to neutralize directional risk.
6.3.2 Gamma (Γ)
- Definition: The rate of change of the option's Delta with respect to the price of the underlying asset.
- Mathematical Property: Second derivative of the option price with respect to the price of the underlying asset.
6.3.3 Theta (Θ)
- Definition: The rate of change of the portfolio value with respect to the passage of time, holding all other factors constant (time decay).
- Time Measurement Conversions:
- Per Calendar Day: Theta_calendar = Theta_formula / 365
- Per Trading Day: Theta_trading = Theta_formula / 250
6.3.4 Vega (ν)
- Definition: The rate of change in portfolio value with respect to changes in the volatility of the underlying asset.
- Interpretation: High absolute Vega indicates significant sensitivity to volatility changes, whereas low absolute Vega means volatility shifts have minimal impact.
6.3.5 Rho (ρ)
- Definition: The rate of change of portfolio value with respect to changes in the risk-free interest rate.
Summary: Option Greeks Overview
| Greek | Symbol | Measures Sensitivity To | Key Notes |
|---|---|---|---|
| Delta | Δ | Spot Price Change | Call is positive, Put is negative; basis for Delta Hedging |
| Gamma | Γ | Delta Change / Spot Price Change | Second derivative; measures curvature of option price |
| Theta | Θ | Passage of Time | Represents Time Decay; divided by 365 (calendar) or 250 (trading) |
| Vega | ν | Volatility Change | High Vega indicates sensitivity to volatility movements |
| Rho | ρ | Interest Rate Change | Measures portfolio risk relative to interest rate movements |
Important Terms & Key Takeaways
- Option Premium Components: Option Premium consists of Intrinsic Value (Max(0, S - X) for Call; Max(0, X - S) for Put) and Time Value (Option Premium - Intrinsic Value).
- Call Upper Limit: Spot price S.
- Put Upper Limit: Present value of strike price X * e^(-r * T).
- Delta Neutrality: Rebalancing underlying asset positions to maintain zero portfolio Delta.
- Annual Volatility Rule: Multiply daily standard deviation by sqrt(250).