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
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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 + P V ( K ) = P + S C + PV(K) = P + S C + P V ( 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