Ch11_TheBlack-ScholesModel(金融工程学,华东师大).ppt

11.1,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,The Black-ScholesModelChapter 11,11.2,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,The Stock Price Assumption,Consider a stock whose price is SIn a short period of time of length Dt the change in then stock price S is assumed to be normal with mean mSdt and standard deviation,that is,S follows geometric Brownian motion ds=Sdt+Sdz.Thenm is expected return and s is volatility,11.3,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,The Lognormal Property,It follows from this assumption that Since the logarithm of ST is normal,ST is lognormally distributed,11.4,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,Modeling Stock Prices in Finance,In finance,frequently we model the evolution of stock prices as a generalized Wiener Process Also,assume prices are distributed lognormal and returns are distributed normal,11.5,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,The Lognormal Distribution,11.6,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,Continuously Compounded Rate of Return,h(Equation(11.7),11.7,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,The Expected Return,The expected value of the stock price is E(ST)=S0eTThe expected continuously compounded return on the stock is E()=s2/2(the geometric average)is the the arithmetic average of the returns Note that Eln(ST)is not equal to lnE(ST)lnE(ST)=ln S0+T,Eln(ST)=ln S0+(-2/2)T,11.8,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,The Expected Return Example,Take the following 5 annual returns:10%,12%,8%,9%,and 11%The arithmetic average is However,the geometric average is Thus,the arithmetic average overstates the geometric average.The geometric is the actual return that one would have earned.The approximation for the geometric return is This differs from g as the returns are not normally distributed.,11.9,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,Is Normality Realistic?,If returns are normal and thus prices are lognormal and assuming that volatility is at 20%(about the historical average)On 10/19/87,the 2 month S&P 500 Futures dropped 29%This was a-27 sigma event with a probability of occurring of once in every 10160 daysOn 10/13/89,the S&P 500 index lost about 6%This was a-5 sigma event with a probability of 0.00000027 or once every 14,756 years,11.10,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,The Concepts Underlying Black-Scholes,The option price&the stock price depend on the same underlying source of uncertaintyWe can form a portfolio consisting of the stock and the option which eliminates this source of uncertaintyThe portfolio is instantaneously riskless and must instantaneously earn the risk-free rateThis leads to the Black-Scholes differential equation,11.11,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,The Assumptions UnderlyingBlack-Scholes,1.The stock follows a Brownian motion with constant and 2.Short selling of securities with full use of proceeds is permitted3.No transaction cost or taxes4.Securities are perfectly divisible5.No dividends paid during the life of the option6.There are no arbitrage opportunities7.Security trading is continuous8.The risk-free rate of interest,r,is constant and is the same for all maturities,11.12,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,1 of 3:The Derivation of theBlack-Scholes Differential Equation,11.13,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,2 of 3:The Derivation of theBlack-Scholes Differential Equation,11.14,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,3 of 3:The Derivation of theBlack-Scholes Differential Equation,11.15,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,The Algebra of the Differential Equation I(=),notice that all of the s cancel out,11.16,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,The Algebra of the Differential Equation II,Again,this is the Black-Scholes partial differential equation,rf,S,S,f,S,S,f,r,t,f,S,S,f,r,rf,S,S,f,t,f,S,S,f,f,r,S,S,f,t,f,t,S,S,f,f,r,t,S,S,f,t,f,=,+,+,-,=,+,-,+,=,+,+,-,=,-,-,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,1,2,1,2,1,s,s,s,d,d,s,11.17,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,The Differential Equation,Any security whose price is dependent on the stock price satisfies the differential equationThe particular security being valued is determined by the boundary conditions of the differential equationIn a forward contract the boundary condition is=S K when t=T The solution to the equation is=S K er(T t),11.18,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,Risk-Neutral Valuation,The variable m does not appearin the Black-Scholes equationThe equation is independent of all variables affected by risk preferenceThe solution to the differential equation is therefore the same in a risk-free world as it is in the real worldThis leads to the principle of risk-neutral valuation,11.19,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,Applying Risk-Neutral Valuation,1.Assume that the expected return from the stock price is the risk-free rate2.Calculate the expected payoff from the option3.Discount at the risk-free rate,11.20,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,Application to Forward contracts on a Stock,Long forward contract with maturity time T and delivery price KThe value of the contract at T=ST-KIn the risk-neutral world,f,the value of the forward contract at tT is f=e-r(T-t)(ST-K)where K is constand and(ST)=Se(T-t)Take=rf=S-K e-r(T-t)(same as eq.3.9),11.21,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,The Black-Scholes Formulas,and N(x)is the CDF for standard normal distribution.Section 11.8 gives a polynomial approximation with 6-decial-place of accuracy,11.22,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,Example using Black-Scholes,Consider a European call with S0=105,K=100,r=10%,D=0,T=0.25,and=30%.Calculate c.First,calculate d1 and d2,11.23,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,Example using Black-Scholes(continued),Hence,N(d1)=0.7123+0.73*(0.7157-0.7123)=0.7148 N(d2)=0.6618So,11.24,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,Intuition Resulting from the Black-Scholes Formulas,NPV,Expected value of a variable that is ST if ST K and 0 otherwise,Strike Price*Pr of exercise,11.25,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,Implied Volatility,Implied Volatility is simply the volatility implied by the market price of the optionIn other words,given the market price of theoption and the B-S inputs for S0,K,r,D,T,what is necessary to equate the B-S pricewith the market priceThe is a one-to-one correspondence between prices and implied volatilitiesNote:the B-S formula CAN NOT be inverted tosolve for,hence one must use an iterativetechnique,11.26,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,Example with Implied Volatility,This example uses trial and error in EXCEL.However,I suggest that you use EXCELs solver if at all possibleYou have a European call with c=2.49S0=90,K=100,r=0.08,D=0,and T=0.25 c1.03 20%1.72 25%2.49 30%3.31 35%4.15 40%,11.27,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,Causes of Volatility,Random arrival of new informationHowever,volatility is larger when the exchanges are open than when they are closedMost information arrives while the exchanges are closedTradingHence,for annualizing daily volatility use 252 trading days per year,11.28,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,The Volatility,The volatility is the STD of the continuously compounded rate of return in 1 year(eq.11.7)As an approximation it is the STD of the proportional change S/S in the stock price in 1 year(eq.10.9)It is also the STD of ln S at the end of 1 year(eq.11.2),11.29,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,Estimating Volatility from Historical Data,1.Take observations S0,S1,.,Sn at intervals of t years2.Define the continuously compounded return as:3.Calculate the standard deviation,s,of the ui s4.The historical volatility estimate is:,11.30,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,Estimating Volatility from Historical Data Example,11.31,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,Estimating Volatility from Historical Data Example(continued),The standard deviation of the uis isNow,this is the weekly volatility.To annualize,simply multiply by the square root of the periodicity.,11.32,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,Warrants&Dilution(稀释),When a regular call option is exercised the stock that is delivered must be purchased in the open marketWhen a warrant is exercised new Treasury stock is issued by the companyThis will dilute the value of the existing stockOne valuation approach is to assume that all equity(warrants+stock)follows geometric Brownian motion,11.33,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,Valuation of European Warrant(Page 254),A Company has N outstanding shares and issues M European warrants NOW.The holder has the right to buy shares at T at a price of X per share.If the warrant is exercised at T,the Companys equity increased from VT to VT+MXShare price after exercise=(VT+MX)/(N+M)The payoff per warrant to the holder is,11.34,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,Valuation of European Warrant(continued),The value of the warrant is where V is the value of the companys equity.At time 0,V0=NS0+MW(W=?),11.35,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,Valuation of European Warrant(continued),The procedure to give the warrant price W by modifying the B-S formula The stock price S0 us replaced by S0+(M/N)W is the volatility of the companys equity(including both the shares and the warrants)The B-S formula is multiplied by the coefficient,11.36,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,Dividends,European options on dividend-paying stocks are valued by substituting the stock price less the present value of dividends into Black-ScholesOnly dividends with ex-dividend dates during life of option should be included The“dividend”should be the expected reduction in the stock price expectedExample 11.7(page 258),11.37,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,American Calls,An American call on a NON-dividend-paying stock shouldNEVER be exercised earlyAn American call on a dividend-paying stock should ONLYever be exercised immediately prior to an ex-dividend date,11.38,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,Blacks Approach to Dealing withDividends in American Call Options,Set the American price equal to the maximum of two European prices:1.The 1st European price is for an option maturing at the same time as the American option2.The 2nd European price is for an option maturing just before the final ex-dividend dateExample 11.8(page 260),11.39,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,Early Exercise on American Calls for Dividend Paying Stocks,For early exercise to be optimal,the dividend must outweigh the interest savings on the strike priceOnly need to look at days that dividends are paidLooking at the last dividend if then no early exercise at that timeSimilarly,for any i n if then no early exercise at those times,11.40,Options,Futures,and Other Derivatives,4th edition 2000 by John C.Hull Tang Yincai,2003,Shanghai Normal University,Assignment,11.4+11.5,11.8,11.11,11.13,11.15,11.16,11.17,11.18,11.19,11.23Assignment Questions,