Option Pricing Models Preparation Notes Overview Option pricing models are mathematical frameworks used to determine the fair value or theoretical price of options These models help traders and investors make informed decisions by estimating the expected payoff of options based on various market factors Key Concepts 1 Option Types Call Option Gives the holder the right to buy an asset at a specified price strike price before a specified date expiration date Put Option Gives the holder the right to sell an asset at a specified price before the expiration date 2 Key Variables in Option Pricing Spot Price S Current price of the underlying asset Strike Price K Price at which the option can be exercised Time to Maturity T Time remaining until the option s expiration Volatility Measure of the underlying asset s price fluctuations Risk Free Rate r Theoretical return on a risk free investment often approximated by government bonds Dividends D Payments made to shareholders affecting the pricing of options on dividend paying stocks Major Option Pricing Models 1 Black Scholes Model Assumptions Markets are efficient No dividends are paid out European style options can only be exercised at expiration No transaction costs The risk free rate and volatility are constant over the option s life Formula Call Option C S 0 cdot N d 1 K cdot e rT cdot N d 2 Put Option P K cdot e rT cdot N d 2 S 0 cdot N d 1 Where d 1 frac ln frac S 0 K r frac sigma 2 2 T sigma sqrt T d 2 d 1 sigma sqrt T N is the cumulative distribution function of the standard normal distribution 2 Binomial Options Pricing Model BOPM Assumptions Discrete time model with a step by step approach Price of the underlying asset can move to one of two possible values up or down in each time step Process Construct a binomial tree representing possible future prices of the underlying asset Work backward from the expiration to determine the option s value at each node Formula Up factor u e sigma sqrt Delta t Down factor d e sigma sqrt Delta t Risk neutral probability p frac e r Delta t d u d Option value at each node C e r Delta t p cdot C up 1 p cdot C down 3 Monte Carlo Simulation Assumptions Uses statistical techniques to model and value options Can handle complex features like path dependency e g Asian options and American style options exercisable anytime Process Simulate numerous possible price paths for the underlying asset using random sampling Calculate the option payoff for each path Average the discounted payoffs to estimate the option s fair value 4 Black Scholes Merton Model for Dividends Extension of the Black Scholes model to account for dividends Formula Call Option C S 0 cdot e qT cdot N d 1 K cdot e rT cdot N d 2 Put Option P K cdot e rT cdot N d 2 S 0 cdot e qT cdot N d 1 Where q is the continuous dividend yield Practical Considerations Implied Volatility IV The market s forecast of a likely movement in the underlying asset s price IV is a critical input in option pricing models and can be derived from market prices of options using models like Black Scholes Greeks Sensitivity measures that indicate how the price of an option changes with respect to changes in various parameters e g Delta Gamma Theta Vega Rho Market Conditions Economic events interest rate changes and market sentiment can significantly impact option pricing Applications Trading Strategies Hedging speculation and arbitrage Risk Management Using options to mitigate potential losses in portfolios Portfolio Optimization Enhancing returns through strategic use of options Conclusion Understanding and utilizing option pricing models are essential for making informed trading and investment decisions The choice of model depends on the specific characteristics of the option being valued and the market conditions Proficiency in these models requires not only theoretical knowledge but also practical application and continuous learning