BS公式的推导.ppt

Financial Engineering The Black-ScholesModel The Stock Price Assumption Consider a stock whose price is S In a short period of time of length Dt the change in the stock price is assumed to be normal with mean mSdt and standard deviation m is expected return and s is volatility The Lognormal Property It follows from this assumption that Since the logarithm of ST is normal, ST is lognormally distributed The Lognormal Distribution Continuously Compounded Return, h Estimating Volatility from Historical Data 1. Take observations S0, S1, . . . , Sn at intervals of t years 2. Define: 3. Calculate the standard deviation, s , of the ui ′s 4. The historical volatility estimate is: The Concepts Underlying Black-Scholes The option price the stock price depend on the same underlying source of uncertainty We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate This leads to the Black-Scholes differential equation Assumptions of BS Formula The short-term interest rate is known and is constant through time. The stock price follows a random walk in continuous time with a variance rate proportional to the square of the stock price.Thus the distribution of stock prices is lognormal. The variance rate of the return on the stock is constant. The sock pays no dividends. The option is “European”. There are no transaction costs. It’s possible to borrow money to buy stocks. There are no penalties to short selling. 1 of 3: The Derivation of theBlack-Scholes Differential Equation 2 of 3: The Derivation of theBlack-Scholes Differential Equation 3 of 3: The Derivation of theBlack-Scholes Differential Equation The return on the portfolio must be the risk-free rate. Hence We substitute for and in these equations to get the Black-Scholes differential equation:  The Differential Eq