Preparation Notes on Black-Scholes Model

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Preparation Notes on Black Scholes Model The Black Scholes model also known as the Black Scholes Merton model is a mathematical model for pricing an options contract The model was developed by economists Fischer Black Myron Scholes and Robert Merton in the early 1970s It provides a theoretical estimate of the price of European style options and has become a foundational tool in financial markets Key Concepts 1 European Option An option that can only be exercised at the expiration date 2 Call Option The right to buy an asset at a specified price strike price on the expiration date 3 Put Option The right to sell an asset at a specified price strike price on the expiration date 4 Underlying Asset The asset on which the option is based such as a stock 5 Strike Price K The price at which the option can be exercised 6 Expiration Date T The date on which the option expires 7 Volatility A measure of the asset s price fluctuations over time 8 Risk Free Rate r The return on an investment with zero risk typically taken as the yield on government bonds 9 Current Stock Price S The current price of the underlying asset 10 d1 and d2 Intermediate calculations used in the Black Scholes formula Assumptions of the Model 1 The option is European and can only be exercised at expiration 2 Markets are efficient i e market movements cannot be predicted 3 No dividends are paid out during the life of the option 4 There are no transaction costs or taxes 5 The risk free rate and volatility of the underlying asset are known and constant 6 The returns on the underlying asset are normally distributed Black Scholes Formula For a European call option the Black Scholes formula is C S 0 N d 1 K e rT N d 2 For a European put option the formula is P K e rT N d 2 S 0 N d 1 where d 1 frac ln S 0 K r frac sigma 2 2 T sigma sqrt T d 2 d 1 sigma sqrt T Here C is the call option price P is the put option price S 0 is the current stock price K is the strike price r is the risk free interest rate sigma is the volatility of the stock T is the time to expiration N d is the cumulative distribution function of the standard normal distribution Applications Limitations 1 Pricing Options The primary use of the Black Scholes model is to price European call and put options 2 Risk Management Traders use the model to understand the sensitivities of the option price to various factors the Greeks Delta Gamma Theta Vega Rho 3 Hedging The model helps in creating delta neutral portfolios to hedge risk 1 Assumption of Constant Volatility Real market conditions exhibit changing volatility 2 No Dividends The model does not account for dividends paid by the underlying asset 3 Market Conditions Assumes ideal market conditions which may not hold in practice e g liquidity transaction costs 4 European Options The model is strictly for European options and not directly applicable to American options which can be exercised at any time before expiration Conclusion The Black Scholes model is a cornerstone of modern financial theory Despite its assumptions and limitations it provides a robust framework for understanding and pricing options Mastery of this model is essential for professionals in finance particularly those involved in trading risk management and financial engineering

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