Sub-topic 3: Options Contracts: Pricing and Valuation
This module delves into the critical aspects of options pricing and valuation, a cornerstone for understanding derivatives and their application in competitive financial exams like the CFA. We will explore the fundamental factors influencing option prices, introduce key valuation models, and discuss practical considerations for their application.
Factors Influencing Option Prices
The price of an option, known as the premium, is determined by several key factors. Understanding these drivers is crucial for both traders and investors. These factors interact dynamically to influence the value of a call or put option.
The primary factors are: underlying asset price, strike price, time to expiration, volatility of the underlying asset, interest rates, and dividends (for stock options).
Options Valuation Models
Several models are used to price options, with the Black-Scholes-Merton model being the most foundational. These models provide a theoretical framework for determining fair option values.
The Black-Scholes-Merton model calculates the price of a European call option (C) and put option (P) using the following formulas:
For a Call Option:
C = S_0 * N(d1) - K * e^(-rT) * N(d2)
For a Put Option:
P = K * e^(-rT) * N(-d2) - S_0 * N(-d1)
Where:
d1 = [ln(S_0/K) + (r + sigma^2/2) * T] / (sigma * sqrt(T))
d2 = d1 - sigma * sqrt(T)
And:
S_0 = Current price of the underlying asset
K = Strike price of the option
r = Risk-free interest rate
sigma = Volatility of the underlying asset
T = Time to expiration (in years)
N(x) = Cumulative standard normal distribution function
e = Base of the natural logarithm
N(d1) and N(d2) represent probabilities related to the option finishing in-the-money. The terms in the formulas represent the expected value of the underlying asset at expiration, adjusted for the probability of exercise, and the present value of the strike price, also adjusted for probability.
Text-based content
Library pages focus on text content
The 'Greeks' in Option Pricing
The 'Greeks' are a set of risk measures that quantify the sensitivity of an option's price to changes in its underlying parameters. They are essential for managing option portfolios.
| Greek | Measures Sensitivity To | Impact on Option Price | Call Option | Put Option |
|---|---|---|---|---|
| Delta | Underlying Asset Price | Higher Delta means price moves more with underlying | +ve (0 to 1) | -ve (-1 to 0) |
| Gamma | Change in Delta with Underlying Price | Measures curvature of option price | +ve | +ve |
| Theta | Time to Expiration | Rate of time decay | -ve (usually) | -ve (usually) |
| Vega | Implied Volatility | Sensitivity to changes in volatility | +ve | +ve |
| Rho | Risk-Free Interest Rate | Sensitivity to interest rate changes | +ve (for calls) | -ve (for puts) |
Understanding the Greeks is crucial for hedging and risk management. For example, a portfolio manager might use Delta hedging to create a position that is insensitive to small movements in the underlying asset's price.
Practical Considerations and Applications
In real-world trading and investment, option pricing and valuation are applied in various strategies, from hedging to speculation. Understanding the limitations of models and market dynamics is key.
Historical volatility measures past price fluctuations, while implied volatility is the market's expectation of future volatility derived from option prices.
Mastering options pricing and valuation is essential for success in competitive financial exams and for making informed investment decisions. The interplay of various factors and the application of valuation models provide a robust framework for understanding these complex instruments.
Learning Resources
Official curriculum material from the CFA Institute covering options and other derivatives, providing foundational knowledge for exam preparation.
A comprehensive guide to the basics of options, including definitions, types, and how they are traded, suitable for beginners.
A series of video lessons explaining options trading, including calls, puts, and basic strategies, presented in an accessible format.
An in-depth explanation of the Black-Scholes model, its formula, assumptions, and limitations in option pricing.
A clear breakdown of the 'Greeks' (Delta, Gamma, Theta, Vega, Rho) and how they measure an option's risk sensitivity.
A PDF document detailing the principles and application of the binomial option pricing model, useful for understanding its mechanics.
An overview of various option pricing models, including Black-Scholes and binomial models, with practical insights.
A detailed encyclopedia entry covering the history, types, pricing, and trading of financial options.
Explains the concept of implied volatility, its importance in option pricing, and how traders use it.
A sample lecture from a Coursera course on financial markets, focusing on the valuation of options and related concepts.