Options Pricing Models

Options pricing models are mathematical frameworks used to estimate the fair value of options, incorporating various factors such as the underlying asset price, volatility, time to expiration, interest rates, and dividends. Below are the most commonly used models:

1. Black-Scholes Model (BSM)

• Use Case: European options (no early exercise).
• Key Assumptions:
  • The underlying asset follows a geometric Brownian motion with constant volatility.
  • The risk-free interest rate is constant.
  • The market is frictionless (no transaction costs or taxes).
• Formula:
  • For a European call option:
    $C = S_0 N(d_1) - Xe^{-rT} N(d_2)$
  • Where:
    • $d_1 = \frac{\ln(S_0/X) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$
    • $d_2 = d_1 - \sigma\sqrt{T}$
  • $S_0$: Current stock price
    $X$: Strike price
    $T$: Time to expiration
    $\sigma$: Volatility
    $r$: Risk-free interest rate
    $N(d)$: Cumulative distribution function of the standard normal distribution.

2. Binomial Options Pricing Model (BOPM)

• Use Case: Both European and American options.
• Key Features:
  • Uses a discrete-time approach to model the price evolution of the underlying asset.
  • Captures the possibility of early exercise for American options.
• Process:
  • Break the time to expiration into $n$ intervals.
  • In each interval, the stock price can move up ($u$) or down ($d$).
  • Risk-neutral valuation is applied to determine the option value at each node.

3. Monte Carlo Simulation

• Use Case: Complex options like Asian or path-dependent options.
• Key Features:
  • Simulates numerous possible price paths for the underlying asset.
  • Computes the payoff for each path and averages them, discounting for time value.
• Advantages:
  • Flexible and can handle exotic options.
  • Not limited by analytical constraints.

4. Heston Model

• Use Case: Options where volatility is not constant.
• Key Features:
  • A stochastic volatility model where volatility itself follows a mean-reverting process.
  • Often used in markets where volatility skew or smile is observed.
• Equations: Involves solving complex stochastic differential equations numerically.

5. Bachelier Model

• Use Case: Used for low-volatility environments like interest rate options.
• Key Assumption: Underlying price changes follow normal (not lognormal) distribution.
• Key Drawback: Can result in negative prices.

6. Local Volatility Models (e.g., Dupire Model)

• Use Case: Addressing implied volatility inconsistencies.
• Key Features:
  • Volatility is a function of the underlying asset price and time.
  • Helps explain volatility smiles and skews.

7. SABR Model

• Use Case: Fitting implied volatility surfaces.
• Key Features:
  • Combines stochastic volatility with the correlation between asset price and volatility.
  • Widely used in interest rate and forex derivatives.

Applications in Quantitative Strategies

• Delta Hedging: Used to create risk-neutral portfolios.
• Volatility Trading: Pricing models help in analyzing volatility spreads.
• Exotic Derivatives: Monte Carlo and stochastic models handle path-dependent payoffs.