期貨與股票(1)ppt課件

1、Lecture #9: Black-Scholes option pricing formula . Brownian MotionThe 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 th

2、e random walk. The discrete-time random walkPk = Pk-1 + k, k = (-) with probability (1-), P0 is fixed. Consider the following continuous time process Pn(t), t 0, T, which is constructed from the discrete time process Pk, k=1,.n as follows: Let h=T/n and define the process Pn(t) = Pt/h = Pnt/T , t 0,

3、 T, where x denotes the greatest integer less than or equal to x. Pn(t) is a left continuous step function.We need to adjust , such that Pn(t) will converge when n goes to infinity. Consider the mean and variance of Pn(T):E(Pn(T) = n(2-1) Var (Pn(T) = 4n(-1) 2.We wish to obtain a continuous time ver

4、sion 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) T4n(-1) 2 T This can be accomplished by setting = *(1+h /), =h. The continuous time limitIt cab be shown that the process P(t) has the following three proper

5、ties:1. For any t1 and t2 such that 0 t1 t2 T: P(t1)-P(t2) (t2-t1), 2(t2-t1)2. For any t1, t 2 , t3, and t4 such that 0 t1 t2 t1 t2 t3 t4 T, the increment P(t2)- P(t1) is statistically independent of the increment P(t4)- P(t3).3. The sample paths of P(t) are continuous. P(t) is called arithmetic Bro

6、wnian 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(t0) = P(t0) +(t-t0)VarP(t) | P(t0) = 2(t-t0)Cov(P(t1),P(t2) = 2 min(t1,t2)Since Var (B(t+h)-B(t)/h = 2/h, therefore, t

7、he derivative of Brownian motion, B(t) does not exist in the ordinary sense, they are nowhere differentiable. Stochastic differential equationsDespite 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

8、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 se

9、nse because dB(t)/dt is a not well defined(althrough dB(t) is).The moments of dB(t):EdB(t) =0VardB(t) = dtE dB dB = dtVardB dB = o(dt)EdB dt = 0VardB 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 U

10、sing th above rule we can calculate (dP)2 = 2dt. It is not a random variable!. Geometric Brownian motionIf 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

11、geometric Brownian motion or lognormal diffusion. . Itos LemmaAlthough 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

12、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 ge

13、neral 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 + 2f/P2 (dP)2 . Applications in FinanceA lognormal distribution for stock price returns is the stan

14、dard 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 ti

15、me 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 p

16、eriod 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)=2h.Technically, these assumptions ensure that as the time

17、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

18、h. This implies that S(t) is lognormally distributed. Lognormal distribution At time t t+hlnSt+h lnSt+(-2/2)h,h0.5where (m,s) denotes a normal distribution with mean m and standard deviation s. Continuously compounded returnln(St+h/St) (-2/2)h,h0.5. Expected returnsEtln(St+h/St) = (-2/2)hEtSt+h/St =

19、 exp(h) Variance of returns Vartln(St+h/St) = 2h VartSt+h/St = exp(2h)(exp(2h)-1). Estimation of n+1: number of stock observationsSj: stock price at the end of jth interval, j=1,nh: length of time intervals in yearsLet uj = lnSj+Dj)/Sj-1u = (u1+un)/n is an estimator for (-2/2)h, s= (u1-u)2+(un-u)2/(

20、n-1)1/2 is an estimator for h1/2.Example: Daily returnsDayClosing priceDividendDaily Return07/0408/0409/0410/0411/0414/0415/0416/0417/0418/0421/0422/0469.4068.5067.2068.7069.5068.3070.1566.9070.7071.0070.5070.00 000000000000-0.01305-0.01916 0.02208 0.01158-0.01742 0.02673-0.04744 0.05525 0.00423-0.0

21、0707-0.00712.DayClosing priceDividendDaily Return23/0424/0425/0428/0429/0430/0401/0502/0505/05MeanS.d.AnnualizedAnnualized 71.1070.8068.7070.2071.1070.5069.8070.4070.20Mean(250 d)s.d. (250 d) 0000000000.01559-0.00423-0.01208 0.02160 0.01274-0.00847-0.00998 0.00856-0.00284 0.00147 0.02157 36.87% 34.1

22、1% . Fundamental equation for derivative securitiesStock price follows Ito process: dS = (S,t)dt + (S,t)dzAt this point, we assume (S,t) =S, and (S,t)= SLet C(S,t) be a derivative security, according to Itos lemma, the process followed by C is dC = C/S (S,t) + C/t + 2C/S2 2(S,t)dt + C/S (S,t)dz .Con

23、sider a portfolio P, combination of S and C to eliminate uncertainty: P = - C + C/S S , the dynamics of P isdP = -dC + C/S dS,dP = - C/S (S,t) + C/t + 2C/S2 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 + 2C/S2 2(S,t) -C/S (S

24、,t)dtdt -C/S (S,t) - C/S (S,t)dzor dP = - C/t + 2C/S2 2(S,t)dt .The portfolio is a riskfree portfolio, hence it should earn risk free return, i.e. dP/P = - C/t + 2C/S2 2(S,t)dt / - C + C/S S = r dt, rearranging terms leads to the well known BS partial differential equation: C/t + r SC/S + 2S22C/S2 r

25、C = 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: cT = max(0,ST-K). . Risk neutral pricingThe drift term does not appear

26、 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,ST-K) where E* denotes the expectation under risk neutral pr

27、obability. The Black-Scholes Formula for European Options (with dividend yield q) c = exp-r(T-t) 0, max(0,ST-K)g(ST)dSTwhere g(ST) is the probability density function of the terminal asset price. By using Itos lemma, we can showln(ST) N(lnS + (r- 2)(T-t), (T-t)1/2) c=Se-qTN(d1)-Xe-rTN(d2)p=Xe-rTN(-d

28、2)-Se-qTN(-d1).whered1=ln(S/X)+(r-q+2/2)T/(T1/2), d2=d1-T1/2Example: X=$70, Maturity date = June 27 (Evaluate on May 5: T=53/365 = 0.1452)_S=70.2000X=70.0000T=0.1452r=0.1000s.d. = 0.3411q = 0.000European option prices: Call = 4.4292 Put = 3.0338 . Implied volatilityThe volatility that makes the mode

29、l 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: 0.45Put implied volatility: 0.42 .Stock Price 93.625 Part Amaturity 22 days, r=5.12%Type of optionsStrike priceMean option priceImplied volati

30、lity(%) CallCallCallPutPut909510090954 3/41 3/81/21/22 7/825.4620.2225.4720.3424.90.Part Bmaturity 50 days, r=5.15%Type of optionsStrike priceMean option priceImplied volatility(%) CallCallCallPutPut909510090955 3/82 3/41.001.503 3/420.1822.2421.2423.6324.45.The prices are mid-point prices. The impl

31、ied 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 GreeksDelta: With respect to an increase in stock

32、 price c=e-qTN(d1) p=e-qTN(d1)-1Gamma: Deltas change with respect to an increase in sock price c=p=N(d1)e-qT/(ST1/2)Theta- with respect to a decrease in maturityc=-SN(d1)e-qT/(2T1/2) + qSN(d1)e-qT-rXe-rTN(d2)p=-SN(d1)e-qT/(2T1/2) - qSN(-d1)e-qT+rXe-rTN(-d2).Vega: with respect to an increase in volat

33、ilityc=p=ST1/2N(d1)e-qTRho: with respect to an increase in interest ratec=XTe-rTN(d2)p=XTe-rTN(-d2).Example1: X=$70, T=0.1452S=70.2000X=70.0000T=0.1452r=0.1000s.d. = 0.4500q = 0.000.European Option Pricesd1=0.1871 N(d1)=0.5742d2=0.0156 N(d2)=0.5062Call=5.3840 Put=4.1749Delta=0.5742 -0.4258Gamma=0.03

34、37 0.0037Theta = -20.3206 -13.4215Vega=10.8602 10.8602Rho= 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 t

35、he 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 optionDay Closing price Daily Return Maturity Delta Stock Position Overall Cash07/0408/0409/0410/0411/0414/0415/0416/0417/0418/0421/0422/0469.4068.5067.2068.7069.5068.3070.1566.9070.7071.0070.5070.00 -0.01305-0.01916 0.02208 0.01158-0.01742 0.02673-0.0