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单击此处编辑母版标题样式,*,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,Lecture #9: Black-Scholes option pricing formula,Brownian Motion,The first formal mathematical model of financial asset prices, developed by Bachelier (1900), was the continuous-time random walk, or Brownian motion. This continuous-time process is closely related to the discrete-time versions of the random walk.,The discrete-time random walk,P,k,= P,k-1,+,k,k,=,(-,) with probability,(1-,), P,0,is fixed. Consider the following continuous time process P,n,(t), t,0, T, which is constructed from the discrete time process P,k, k=1,.n as follows: Let h=T/n and define the process,P,n,(t) = P,t/h,= P,nt/T, t,0, T, where x denotes the greatest integer less than or equal to x. P,n,(t) is a left continuous step function.,We need to adjust,such that P,n,(t) will converge when n goes to infinity. Consider the mean and variance of P,n,(T):,E(P,n,(T) = n(2,-1),Var (P,n,(T) = 4n,(,-1),2,We wish to obtain a continuous time version of the random walk, we should expect the mean and variance of the limiting process P(T) to be linear in T. Therefore, we must have,n(2,-1),T,4n,(,-1),2,T,This can be accomplished by setting,= *(1+,h /,),=,h,The continuous time limit,It cab be shown that the process P(t) has the following three properties:,1. For any t,1,and t,2,such that 0,t,1, t,2,T:,P(t,1,)-P(t,2,),(,(t,2,-t,1,),2,(t,2,-t,1,),2. For any t,1, t,2 ,t,3, and t,4,such that 0,t,1, t,2,t,1, t,2,t,3, t,4,T, the increment,P(t,2,)- P(t,1,) is statistically independent of the increment P(t,4,)- P(t,3,).,3. The sample paths of P(t) are continuous.,P(t) is called arithmetic Brownian motion or Winner process.,If we set,=0,=1, we obtain standard Brownian Motion which is denoted as B(t). Accordingly, P(t) =,t +,B(t),Consider the following moments:,EP(t) | P(t,0,) = P(t,0,) +,(t-t,0,),VarP(t) | P(t,0,) =,2,(t-t,0,),Cov(P(t,1,),P(t,2,) =,2,min(t,1,t,2,),Since Var (B(t+h)-B(t)/h =,2,/h, therefore, the derivative of Brownian motion, B(t) does not exist in the ordinary sense, they are nowhere differentiable.,Stochastic differential equations,Despite the fact, the infinitesimal increment of Brownian motion, the limit of B(t+h) = B(t) as h approaches to an infinitesimal of time (dt) has earned the notation dB(t) and it has become a fundamental building block for constructing other continuous time process. It is called white noise. For P(t) define earlier we have dP(t) =,dt +,dB(t). This is called stochastic differential equation. The natural transformation dP(t)/dt =,+,dB(t)/dt doesnt male sense because dB(t)/dt is a not well defined,(althrough dB(t) is).,The moments of dB(t):,EdB(t) =0,VardB(t) = dt,E dB dB = dt,VardB dB = o(dt),EdB dt = 0,VardB dt = o(dt),If we treat terms of order of o(dt) as essentially zero, the (dB),2,and dBdt are both non-stochastic variables.,| dB dt,dB | dt 0,dt | 0 0,Using th above rule we can calculate (dP),2,=,2,dt. It is not a random variable!,Geometric Brownian motion,If the arithmetic Brownian motion P(t) is taken to be the price of some asset, the price may be negative. The price process p(t)= exp(P(t), where P(t) is the arithmetic Brownian motion, is called geometric Brownian motion or lognormal diffusion.,Itos Lemma,Although the first complete mathematical theory of Brownian motion is due to Wiener(1923), it is the seminal contribution of Ito (1951) that is largely responsible for the enormous number of applications of Brownian motion to problems in mathematics, statistics, physics, chemistry, biology, engineering, and of course, financial economics. In particular, Ito constructs a broad class of continuous time stochastic process based on Brownian motion now known as Ito process or Ito stochastic differential equations which is closed under general non-linear transformation.,Ito (1951) provides a formula Itos lemma for calculating explicitly the stochastic differential equation that governs the dynamics of f(P,t):,df(P,t) =,f/,P dP +,f/,t dt + ,2,f/,P,2,(dP),2,Applications in Finance,A lognormal distribution for stock price returns is the standard model used in financial economics. Given some reasonable assumptions about the random behavior of stock returns, a lognormal distribution is implied. These assumptions will characterize lognomal distribution in a very intuitive manner.,Let S(t) be the stocks price at date t. We subdivided the time horizon 0 T into n equally spaced subintervals of length h. We write S(ih) as S(i), i=0,1,n. Let z(i) be the continuous compounded rate of return over (i-1)h ih, ie S(i)=S(i-1)exp(z(i), i=1,2,.,n. It is clear that S(i)=S(0)expz(1)+z(2)+z(i).,The continuous compounded return on the stock over the period 0 T is the sum of the continuously compounded returns over the n subintervals.,Assumption A1. The returns z(j) are i.i.d.,Assumption A2. Ez(t)=,h, where,is the expected continuously compounded return per unit time.,Assumption A3. varz(t)=,2,h.,Technically, these assumptions ensure that as the time decrease proportionally, the behavior of the distribution for S(t) dose not explode nor degenerate to a fixed point.,Assumption 1-3 implies that for any infinitesimal time subintervals, the distribution for the continuously compounded return z(t) has a normal distribution with mean,h, and variance,2,h. This implies that S(t) is lognormally distributed.,Lognormal distribution,At time t t+h,lnS,t+h,lnS,t,+(,-,2,/2)h,h,0.5,where,(m,s) denotes a normal distribution with mean m and standard deviation s.,Continuously compounded return,ln(S,t+h,/S,t,) ,(,-,2,/2)h,h,0.5,Expected returns,E,t,ln(S,t+h,/S,t,) = (,-,2,/2)h,E,t,S,t+h,/S,t, = exp(,h),Variance of returns,Var,t,ln(S,t+h,/S,t,) =,2,h,Var,t,S,t+h,/S,t, = exp(2,h)(exp(,2,h)-1),Estimation of,n+1: number of stock observations,S,j,: stock price at the end of jth interval, j=1,n,h: length of time intervals in years,Let,u,j,= lnS,j,+D,j,)/S,j-1,u = (u,1,+u,n,)/n is an estimator for (,-,2,/2)h, s= (u,1,-u),2,+(u,n,-u),2,/(n-1),1/2,is an estimator for,h,1/2,.,Example: Daily returns,Day,Closing price,Dividend,Daily Return,07/04,08/04,09/04,10/04,11/04,14/04,15/04,16/04,17/04,18/04,21/04,22/04,69.40,68.50,67.20,68.70,69.50,68.30,70.15,66.90,70.70,71.00,70.50,70.00,0,0,0,0,0,0,0,0,0,0,0,0,-0.01305,-0.01916,0.02208,0.01158,-0.01742,0.02673,-0.04744,0.05525,0.00423,-0.00707,-0.00712,Day,Closing price,Dividend,Daily Return,23/04,24/04,25/04,28/04,29/04,30/04,01/05,02/05,05/05,Mean,S.d.,Annualized,Annualized,71.10,70.80,68.70,70.20,71.10,70.50,69.80,70.40,70.20,Mean(250 d),s.d. (250 d),0,0,0,0,0,0,0,0,0,0.01559,-0.00423,-0.01208,0.02160,0.01274,-0.00847,-0.00998,0.00856,-0.00284,0.00147,0.02157,36.87%,34.11%,Fundamental equation for derivative securities,Stock price follows Ito process:,dS =,(S,t)dt +,(S,t)dz,At this point, we assume,(S,t) =,S, and,(S,t)=,S,Let C(S,t) be a derivative security, according to Itos lemma, the process followed by C is,dC = ,C/,S,(S,t) +,C/,t + ,2,C/,S,2,2,(S,t)dt + ,C/,S,(S,t)dz,Consider a portfolio P, combination of S and C to eliminate uncertainty: P = - C +,C/,S S , the dynamics of P is,dP = -dC +,C/,S dS,dP = - ,C/,S,(S,t) +,C/,t + ,2,C/,S,2,2,(S,t)dt -,C/,S,(S,t)dz +,C/,S,(S,t)dt +,(S,t)dz,Collecting terms involving dt and dz together we get,dP = - ,C/,S,(S,t) +,C/,t + ,2,C/,S,2,2,(S,t) -,C/,S,(S,t)dtdt -,C/,S,(S,t) -,C/,S,(S,t)dz,or dP = - ,C/,t + ,2,C/,S,2,2,(S,t)dt,The portfolio is a riskfree portfolio, hence it should earn risk free return, i.e.,dP/P = - ,C/,t + ,2,C/,S,2,2,(S,t)dt / - C +,C/,S S = r dt, rearranging terms leads to the well known BS partial differential equation:,C/,t + r S,C/,S + ,2,S,2,2,C/,S,2, rC = 0,This is the fundamental partial differential equation for derivatives. The solution for an specific derivative is determined by boundary conditions. For example, the European call option is determined by boundary condition: c,T,= max(0,S,T,-K).,Risk neutral pricing,The drift term,does not appear in the fundamental equation. Rather, the reiskfree rate r is there. Under risk neutral measure, the stock price dynamics is dS = rSdt +,Sdz.,If interest rate is constant as in BS, the European option can be priced as,c = exp-r(T-t) E,*,max(0,S,T,-K),where E,*,denotes the expectation under risk neutral probability.,The Black-Scholes Formula for European Options (with dividend yield q),c = exp-r(T-t),0,max(0,S,T,-K)g(S,T,)dS,T,where g(S,T,) is the probability density function of the terminal asset price. By using Itos lemma, we can show,ln(S,T,) N(lnS + (r- ,2,)(T-t),(T-t),1/2,),c=Se,-qT,N(d,1,)-Xe,-rT,N(d,2,),p=Xe,-rT,N(-d,2,)-Se,-qT,N(-d,1,),where,d,1,=ln(S/X)+(r-q+,2,/2)T/(,T,1/2,), d,2,=d,1,-,T,1/2,Example: X=$70, Maturity date = June 27 (Evaluate on May 5: T=53/365 = 0.1452),_,S=70.2000,X=70.0000,T=0.1452,r=0.1000,s.d. = 0.3411,q = 0.000,European option prices: Call = 4.4292 Put = 3.0338,Implied volatility,The volatility that makes the model price equal its market price.,Assume that the call and put options in the above example are traded at 5.383 and 3.860, respectively.,Call implied volatility: 0.45,Put implied volatility: 0.42,Stock Price 93.625,Part A,maturity 22 days, r=5.12%,Type of options,Strike price,Mean option price,Implied volatility(%),Call,Call,Call,Put,Put,90,95,100,90,95,4 3/4,1 3/8,1/2,1/2,2 7/8,25.46,20.22,25.47,20.34,24.90,Part B,maturity 50 days, r=5.15%,Type of options,Strike price,Mean option price,Implied volatility(%),Call,Call,Call,Put,Put,90,95,100,90,95,5 3/8,2 3/4,1.00,1.50,3 3/4,20.18,22.24,21.24,23.63,24.45,The prices are mid-point prices. The implied volatility seems to depend upon whether the option is in/out or at-the-money. The implied volatility for calls seems to differ from the implied volatility for puts.,There are many reasons why the implied volatility estimates differ. (why?),Option Greeks,Delta: With respect to an increase in stock price,c,=e,-qT,N(d,1,),p,=e,-qT,N(d,1,)-1,Gamma: Deltas change with respect to an increase in sock price,c,=,p,=N(d,1,)e,-qT,/(S,T,1/2,),Theta- with respect to a decrease in maturity,c,=-SN(d,1,),e,-qT,/(2T,1/2,) + qSN(d,1,)e,-qT,-rXe,-rT,N(d,2,),p,=-SN(d,1,),e,-qT,/(2T,1/2,) - qSN(-d,1,)e,-qT,+rXe,-rT,N(-d,2,),Vega: with respect to an increase in volatility,c,=,p,=ST,1/2,N(d,1,)e,-qT,Rho: with respect to an increase in interest rate,c,=XTe,-rT,N(d,2,),p,=XTe,-rT,N(-d,2,),Example1: X=$70, T=0.1452,S=70.2000,X=70.0000,T=0.1452,r=0.1000,s.d. = 0.4500,q = 0.000,European Option Prices,d,1,=0.1871 N(d,1,)=0.5742,d,2,=0.0156 N(d,2,)=0.5062,Call=5.3840 Put=4.1749,Delta=0.5742 -0.4258,Gamma=0.0337 0.0037,Theta = -20.3206 -13.4215,Vega=10.8602 10.8602,Rho= 5.0712 -4.9467,Synthetic option,Set aside cash in the amount equal to the model value.,Maintain the stock position equal to the delta of the target option.,Cash balance is invested in risk-free assets to earn interests.,Close the position at the desired matuirity.,If the model is good, the terminal payoff of this dynamic strategy should be close to the payoff of the target option at the maturity.,Example: Synthetic put option,Day,Closing price,Daily Return,Maturity,Delta,Stock Position,Overall Cash,07/04,08/04,09/04,10/04,11/04,14/04,15/04,16/04,17/04,18/04,21/04,22/04,69.40,68.50,67.20,68.70,69.50,68.30,70.15,66.90,70.70,71.00,70.50,70.00,-0.01305,-0.01916,0.02208,0.01158,-0.01742,0.02673,-0.04744,0.05525,0.00423,-0.00707,-0.00712,0.11200,0.10800,0.10400,0.10000,0.09600,0.08400,0.08000,0.07600,0.07200,0.06800,0.05600,0.05200,-0.6067,-0.6528,-0.7166,-0.6532,-0.6191,-0.6956,-0.6047,-0.7770,-0.5839,-0.5712,-0.6198,-0.6599,0.00,-42.11,-44.71,-48.16,-44.87,-43.03,-47.51,-42.42,-51.98,-41.28,-40.56,-43.69,-46.19,4.94,47.04,50.22,54.53,50.19,47.48,53.13,46.77,58.32,44.69,43.81,47.28,50.11,23/04,24/04,25/04,28/04,29/04,30/04,01/05,02/05,05/05,71.10,70.80,68.70,70.20,71.10,70.50,69.80,70.40,70.20,0.01559,-0.00423,-0.03011,0.02160,0.01274,-0.00847,-0.00998,0.00856,-0.00284,0.04800,0.04400,0.04000,0.02800,0.02400,0.02000,0.01600,0.01200,0.00000,-0.5938,-0.6230,-0.7765,-0.7189,-0.6580,-0.7347,-0.8256,-0.8140,0.0000,-42.22,-44.14,-53.35,-50.46,-46.78,-51.79,-57.63,-57.05,0.00,45.43,47.52,58.08,54.10,49.79,55.22,61.59,60.54,3.73,Mean,s.d. a. m.,250 d,a. s.d.,250 d,0.00057,0.02252,14.33%,35.62%,X=73.0,T=0.112,r=0.1000,P=4.93939,Duration of an option,An options,is its partial derivative with respect to a change in the continuously compounded interest rate. Specifically, the call option pricing formula (Black and Scholes) is,c=SN(d,1,)-Xe,-rT,N(d,2,),where,d,1,=ln(S/X)+(r+,2,/2)T/(,T,1/2,), d,2,=d,1,-,T,1/2,It follows that,c/,r = XTe,-rT,N(d,2,),and,-(,c/,r)/c = -(X/c)Te,-rT,N(d,2,)0.13, then D,c,is positive.,