金融工程学课件：Chapter 5The Black-Scholes Model

1、Chapter 5 The Black-Scholes Model,The relationship between Binomial Model and the Black-Scholes Model,The former is discrete time models and the latter is continuous time models The Black-Scholes Model is the limit of the Binomial Model,Assumptions,Stock prices behave randomly and evolve according t

2、o a lognormal distribution The risk-free rate and volatility of the log return on the stock are constant throuthout the options life There are no taxes or transaction costs The stock pays no dividends The options are European,Assumption 1Prices behave randomly,In the long run, we can recognize the t

3、rend of price movement In short time, we are not so sure of the price movement. Stock returns are lognormally distributed,Assumption 2,Risk free rate is constant Volatility of logreturn on the stock is constant,Assumption 3,No transaction costs No taxes,Assumption 4,Stock pays no dividends,Assumptio

4、n 5,The options are European,Formula,Example,S=125.9375 X=125 r=0.4466 T=0.0959 =0.83 C=?,Interpretation of the formula,Risk averse investors determine the prices of the primary assets in the financial markets while the forces of arbitrage determine the prices of derivatives. So we can price an opti

5、on as if investors were risk neutral.,The lower bound of a European call,The formula when T=0,The formula when S=0,We get C=0,The formula when =0,The formula when X=0,The call is equivalent to the stock,Delta,The delta is the change in the call price for very small change in the stock price.,Delta a

6、lso changes as the option evolves through its life.,Delta hedge and Delta neutral,Page 129-130,Gamma,The gamma is the change in the delta for a very small change in the stock price.,The exercise price,The risk-free rate,The call price is nearly linear in the risk-free rate and does not change much o

7、ver a very broad range of risk-free rates.,The volatility or stand deviation,The sensitivity of a call price to a very small change in volatility is called its vega.,The time to expiration,The rate of the time value decay is measured by the options theta, which is given as,Known discrete dividends,W

8、e subtract the present value of the dividend from the stock price and use the adjusted stock price in the formula.,Known continuous dividend yield,The adjustment procedure requires substituting the value S for S in the Black-Scholes model.,The BS model and American call options,Page 141-142,Estimating the volatility,Historical volatility Implied volatility,Historical volatility,The Historical volatility estimate is based on the assumptions that the volatility that prevailed over the recent past will continue to hold in the future.