14.3 Option pricing and hedging

Option pricing and hedging are crucial in finance, blending math and market dynamics. These concepts help traders value options accurately and manage risk effectively, forming the backbone of modern derivatives trading.

Black-Scholes and binomial models provide frameworks for pricing options, while Greeks measure option sensitivities. Understanding these tools allows investors to make informed decisions and implement sophisticated hedging strategies in volatile markets.

Option Pricing Models

Black-Scholes and Binomial Models

• Black-Scholes model revolutionized option pricing provides a closed-form solution for European options
• Assumes underlying asset follows geometric Brownian motion with constant volatility
• Key inputs include current stock price, strike price, time to expiration, risk-free rate, and volatility
• Formula calculates theoretical option price based on these inputs
• Binomial option pricing model uses discrete-time framework to value options
• Constructs a binomial tree representing possible price paths of the underlying asset
• Assumes asset price can move up or down by a specific factor at each time step
• Calculates option value at each node working backwards from expiration to present

Advanced Models and Numerical Methods

• Stochastic volatility models address limitations of Black-Scholes by allowing volatility to vary over time
• Popular stochastic volatility models include Heston model and SABR model
• Heston model assumes volatility follows mean-reverting process correlated with asset price
• SABR model combines stochastic volatility with CEV (constant elasticity of variance) dynamics
• Numerical methods for option pricing solve complex option valuation problems
• Monte Carlo simulation generates numerous price paths to estimate option value
• Finite difference methods solve partial differential equations governing option prices
• Lattice methods (binomial and trinomial trees) discretize time and price to approximate option values

Option Characteristics

American vs. European Options and Put-Call Parity

• American options allow exercise at any time before expiration
• European options can only be exercised at expiration
• American options generally more valuable due to additional flexibility
• Put-call parity establishes relationship between prices of European put and call options
• Formula: $C + PV(K) = P + S$
• C represents call price, P represents put price, K represents strike price
• S represents current stock price, PV(K) represents present value of strike price
• Put-call parity enables calculation of one option price given the other and underlying asset price

Implied Volatility and Option Pricing

• Implied volatility represents market's expectation of future volatility
• Derived by solving Black-Scholes equation backwards using observed option prices
• Higher implied volatility indicates greater expected price fluctuations
• Implied volatility smile describes pattern of varying implied volatilities across strike prices
• Volatility skew refers to difference in implied volatility between out-of-the-money puts and calls
• Traders use implied volatility to assess whether options appear overpriced or underpriced

Hedging and Risk Management

Greeks and Option Sensitivity

• Greeks measure sensitivity of option prices to various factors
• Delta (Δ) represents rate of change of option price with respect to underlying asset price
• Gamma (Γ) measures rate of change of delta with respect to underlying asset price
• Theta (Θ) represents rate of change of option price with respect to time
• Vega (ν) measures sensitivity of option price to changes in volatility
• Rho (ρ) represents sensitivity of option price to changes in risk-free interest rate
• Traders use Greeks to assess risk exposure and design hedging strategies

Delta Hedging and Dynamic Hedging

• Delta hedging aims to create a position neutral to small price changes in underlying asset
• Involves taking offsetting position in underlying asset proportional to option's delta
• For call option with delta 0.6, hedge requires short selling 60 shares of underlying stock
• Dynamic hedging continuously adjusts hedge ratio as market conditions change
• Requires frequent rebalancing to maintain delta-neutral position
• Gamma hedging involves neutralizing both delta and gamma to protect against larger price moves
• Vega hedging protects against changes in implied volatility by trading other options