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Week 7: Black Scholes Models and Option Pricing. Binomial tree. 优点：直观简单，能够很好地解析无套利定价和利用关于风险中性测度的期望值来计算期权价格的思想。 缺点 : 不符合股票市场时时交易的特性 . 如何更好地刻画现实中的期权价格： （ a ）多周期二叉树模型 （ b ）连续时间模型. 多步二叉树. r=0 时，执行期为 t 个周期的期权价格. P t 为时间 t 时刻的股票价格，风险中性概率为 1/2,

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## Week 7: Black Scholes Models and Option Pricing

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**Binomial tree**• 优点：直观简单，能够很好地解析无套利定价和利用关于风险中性测度的期望值来计算期权价格的思想。 • 缺点: 不符合股票市场时时交易的特性. • 如何更好地刻画现实中的期权价格： （a）多周期二叉树模型 （b）连续时间模型**r=0时，执行期为t个周期的期权价格**• Pt为时间t时刻的股票价格，风险中性概率为1/2, Pt=P0+10{2(W1+W2+…..+Wt)-t}, Wi以1/2概率取0或1。 则期权价格为 C=E{(Pt-E)+}**无穷多步期权定价**• [0, 1]分成n个区间， 股票价格以1/2概率上涨或下跌σ/n1/2,t=m/n时刻的价格为 pm:= Pt=P0+ σ/n1/2 {2(W1+W2+…..+Wm)-m}, 则 (1) {pm}为一 鞅（？）; (2) E(Pt|P0)=P0; (3) Var(Pt|P0)=t σ2. 当n充分大时，Pt 趋于一连续的随机过程 P0+ σBt (Bt为BM, BM的性质?)**GBM**• 利用random-walk (BM)来刻画期权价格的缺点：可能使得价格取到负值，这是不合理的。 • 我们利用GBM来克服该缺点。**期权价值**到期日的价值： 现价：**Brownian Motions**• W(tk+1)= W(tk) + e(tk) t, where t = tk+1 – tk, k=0,…,N, t0 = 0, and e(tk) iid N(0,1). • For j<k, W(tk) - W(tj) = i=jk-1e(ti) t. • The right-hand side is normally distributed, so is the left-hand side. • Clearly, E(W(tk) - W(tj) ) = 0. • Var (W(tk) - W(tj)) = E [i=jk-1 e(ti) t]2 = (k-j) t = tk – tj. • For t1 < t2 t3 < t4, W(t4) - W(t3) is uncorrelated with W(t2) - W(t1).**Simulation of Brownian Motion**• Partition [0,1] into n subintervals each with length 1/n. For each t in [0,1], let [nt] denote the greatest integer part of the number nt. For example, if n=10 and t=1/3, then [nt] =[10/3]=3. • For each t in [0,1], define a stochastic process S[nt] = i=1[nt]e(i)/n, e(i) iid N(0,1). • Clearly,S[nt] = S[nt]-1 + e([nt])/ n, a special form of the additive model defined at the beginning with t =1/n and W(t)= S[nt]. • At time t=1, S[nt] = Sn = i=1ne(i)/n, which has a standard normal distribution. Even if e(i)s are not normally distributed, CLT shows that Sn will still be normally distributed as long as n is big. This is the key idea in constructing a Brownian motion.**By letting n goes to infinity (t goes to 0), Donsker**(1953) proved that the stochastic process S[nt] constructed in this way tends to a Brownian motion. This result is known as the Functional Central Limit Theorem or the Invariance Principle. Therefore, the above equation provides a means to simulate the path of a Brownian motion. All we have to do is to iterate the equation S[nt] = S[nt]-1 + e([nt])/ n by taking n bigger and bigger and in the limit, we have a Brownian motion.**When t goes to 0, the discrete-time random walk equation**S[nt] = S[nt]-1+e([nt])/n, can be approximated by the continuous-time equation dW(t) = e(t)dt . • In advanced courses in Probability, it will be shown that this limiting operation is well-defined and the limiting process is the Brownian motion. Formally, we define a Brownian motion as follows.**Wiener Process**Definition. A Wiener process (standard Brownian motion) is a stochastic process which satisfies the following properties: • For s<t, W(t) – W(s) is a N(0, t-s) random variable. • For t1 < t2 t3 < t4, W(t4) - W(t3) is uncorrelated with W(t2) - W(t1). This is known as the independent increment property. • W(t0) = 0 with probability one.**Properties of Wiener Process**• For each t, W(t) is normally distributed. • For t<s, E(W(s)|W(t)) = E(W(s)-W(t)+W(t)|W(t))=W(t), hence a martingale. • With probability 1, W(t) is nowhere differentiable. E[W(s)-W(t)/(s-t)]2 =1/(s-t) which tends to infinity when s tends to t. Consequently, the process (t) = dW(t)/dt is undefined mathematically.**Diffusion Processes**• Consider dX(t) = dt dW(t), integrating this equation, we get • X(t) = X(0) + t W(t). In other words, the process X(t) is the solution to the integral equation dX(t)= dt + dW(t). • X(t) defined in this way is known a diffusion process. It can be expressed explicitly in terms of W(t) and the constants and . To generalize, we have:**Generalized Wiener Process**• Definition. An Ito’s process or a generalized Wiener process is a stochastic process which satisfies the following stochastic differential equation dX(t) = (x,t)dt + (x,t) dW(t), where (x,t) is known as the drift function and (x,t) is known as the diffusion function. • This equation has to be interpreted through integration.**Stock Returns**• Recall the multiplicative model log S(k+1) = log S(k)+w(k), w(k) iid N(2). • The continuous-time approximation is dlog S(t) = dt + dW(t). Integrating this equation, we have log S(t) = log S(0) + t + W(t). The process S(t) is the geometric Brownian motion.**Definition.Let X(t) be a Brownian motion with drift and**diffusion coefficient (variance) 2 , that is, dX(t) = dt + dW(t). • The stochastic process S(t)= exp(X(t)) is said to be a geometric Brownian motion with drift parameter and variance 2, where = +2/2. Or equivalently, the process X(t) =logS(t) is a Brownian motion such that dlogS(t) = (-2/2)dt + dW(t).**The following figure are simulations of S(t)=exp{0.01t +**0.01W(t)}. Notice the linear growth of the mean, =.01.**Let S(0)=z. An equivalent way of defining a geometric**Brownian motion is that the process S(t) satisfies S(t) = z e X(t) = z exp{t + W(t)} =z exp{(-2/2)t + W(t)}. • Consider the successive ratios S(t1)/S(t0), S(t2)/S(t1), …,S(tn)/S(tn-1). Since the Brownian motion has independent increments, these ratios are independent random variables. • log S(t) = log S(0)+X(t) ~ N(logS(0) + t, 2 t).**As S(t)=ze X(t), to find the moments of S(t), consider**E(S(t)) = E(E(S(t)|S(0)=z)) =E(E(z exp{(-2/2)t +W(t)}|S(0)=z)) = z exp{(-2/2)t}E(eW(t)) = z exp{(-2/2)t}E(e t ), ~N(0,1), = z exp{(-2/2)t}exp{2 t/2} =zet=S(0)et.**For >0, E(S(t)) tends to infinity as t gets large. But**for 0< < 2/2, the process X(t) = X(0) + (-2/2)t + W(t) has a negative drift, which means X(t) tends to negative infinity as t gets bigger. Consequently, the price S(t) = S(0)eX(t) tends to zero as t tends to infinity. In other words, the geometric Brownian motion S(t) drifts to zero but at the same time, its mean is drifting to infinity. Using the mean value can be misleading in describing the process. • Similarly, we can show that Var S(t) = S(0)2 e 2t (exp{2 t}-1). (check)**Ito’s Formula**• In the preceding section, we define the geometric Brownain motion in terms of dlogS(t) = dt + dW(t). • From ordinary calculus, dS(t)/S(t)=dlogS(t) Guess: dS(t)/S(t) = dt + dW(t). • Unfortunately, this is not correct. We need an extra correction term. • Rule: Whenever dW(t) is involved, we need to account for a correction term. This is the essence of the Ito’s calculus.**The correct formula is**dS(t)/S(t) = (2/2)dt + dW(t) = dt + dW(t), as =-2/2. • The extra term 2/2 when transforming dlogS(t) to dS(t) is known as the Ito’s lemma. • Remarks: • The above equation describes the dynamics of the instantaneous return process dS(t)/S(t). • When =0, the process is deterministic and the above equation leads back to S(t)=S(0)et .**One can simulate S(t) from the above equation,**S(tk+1) – S(tk) = S(tk)t + S(tk)(tk)t, or S(tk+1) =[1+ t + (tk)t]S(tk), (tk) iid N(0,1). This is a multiplicative model in S(t), but the coefficient is normal, not lognormal • Alternatively, we can discretize dlogS(t) = dt+dW(t) to simulate logS(tk+1)- logS(tk)=t+(tk)t, or S(tk+1) = exp{t+(tk)t}S(tk). This time we have a multiplicative model again, but with lognormal coefficient. In practice, as long as t is small, both expressions can be used to simulate S(t).**Ito’s Lemma**• Theorem. Let x(t) be a diffusion process that satisfies dx(t) = a(x,t)dt + b(x,t)dW(t). Let the process y(t) = F(x,t) for some function F. Then y(t) satisfies the Ito’s equation dy(t) = [(F/x)a + F/t + (1/2)(2F/x2)b2]dt + (F/x)bdW(t).**Proof. It will only be a sketch. Recall from calculus that**for a function of two variables y(t)=F(x,t), dy(t) = (F/x) dx + (F/t) dt = (F/x)(adt + bdW) + (F/t) dt. Comparing this with the Ito’s equation, we see that there is the extra correction term(1/2)(2F/x2)b2 in front of the coefficient dt in the Ito’s equation. The reason for this correction is due to the fact that dW is of order dt, and as dx is expressed in terms of dW, it is also of order dt. To see how this is done, consider the Taylor’s expansion of F up to the order of dt. Specifically, calculate the expression**F(x+x, t+t)**= F(x,t) + (F/x)x + (F/t)t + (1/2)(2F/x2)(x)2 =F(x,t)+(F/x)(at+bW)+(F/t)+(1/2)(2F/x2)(at+bW)2. Consider this quadratic term, expanding (at+bW)2= a2 (t)2 + 2ab (t)(W) + b2(W)2. Note that both the first term and the second term of the right hand side are of higher order than t. Recall the fact that since dW ~ e(t)dt, W is of order t and thus, (W)2 ~ t. Using this fact and remembering that we are expanding y only up to order t (i.e. terms of order higher than t can be ignored), we get (at+bW)2 ~ b2 t. Substituting this into the pink term, F(x+x, t+t) = F(x,t) + [(F/x)a + (F/t) + (1/2)(2F/x2)b2] t + (F/x)bW. Taking limit t goes to zero, the Ito’s formula is obtained.**Example. Consider the geometric Brownian motion S(t) that**satisfies : dS(t) = S(t)dt + S(t)dW(t). Consider the process y(t)=F(S(t))=logS(t). What kind of equation does y satisfy? To find out, use Ito’s lemma. 1. Identify a= S and b= S. 2. We know F/S=1/S and 2F/S2=-1/S2. 3. Plug these into Ito’s formula, dlogS =[a/S – (1/2)(b2/S2]dt + (b/S)dW =(- 2/2)dt + dW, which agrees with the earlier derivation. **Example. sdW(s) =? First a guess,**tW(t) - W(s)ds. To check, • Let X(t)=W(t), dX(t)=dW(t) and identify a=0, b=1. • Let Y(t)=F(W(t))=tW(t). Then F/W=t, 2F/W2=0, and F/t =W(t). • Substitute these into the Ito’s lemma, dY(t)=tdW(t) + W(t)dt. Integrating, • Y(t) =tW(t)= sdW(s) + W(s)ds, that is, sdW(s) = tW(t) - W(s)ds as expected.**Example. W(s)dW(s) = ? To find out, first guess an**answer W2(t)/2. Is this correct? Use Ito’s formula to check. • Let X(t)=W(t), dX(t)=dW(t) and identify a=0, b=1. • Let Y(t)=F(X(t))= W2(t)/2. Then F/W=W, 2F/W2=1, and F/t =0. • Recite Ito’s lemma: dY(t) = [(F/x)a + F/t+(1/2)(2F/x2)b2]dt + (F/x)bdW(t) so that dY(t) = (1/2)dt + W(t)dW(t). Now integrate both sides of this equation, we have • W2(t)/2 = Y(t) = t/2 + W(s)dW(s), that is, W(s)dW(s) = W2(t)/2 - t/2!!! This time, our initial guess wasn’t correct. We need the extra correction term from the Ito’s lemma. **Black-Scholes Equation**• There are two securities: a stock that follows the geometric Brownian motion, dS = Sdt + S dW, and a bond that follows dB=rBdt. • Consider a contingent claim (call option, say) to S. This derivative has a price, which is a function of S and t, f(S,t). • Goal: To find an equation that describes the behavior of f(S,t). This goal is achieved by the Black-Scholes equation.**Theorem. With the notation just defined, then the price of**the derivative satisfies: f/t +(f/S)rS + (1/2)(2f/S2)2S2 = rf.**Proof. (Analytical). The idea of this proof is the same as**in the binomial case. We try to construct a portfolio which replicates the characteristics of the contingent claim at each instant. • Recall from Ito’s lemma that since f is a function of S, df = [(f/S)S + f/t + (1/2)(2f/S2)2S2]dt + (f/S) S dW. This equation shows that f is also a diffusion process with drift […] and diffusion coefficient (…).**Construct a portfolio that replicates f. At each time t,**select xt of stock and yt of bond, giving a total portfolio value G(t) = xtS(t) + ytB(t). • The instantaneous gain in value of this portfolio comes from changes in S and B so that dG = xtdS + ytdB = xt (Sdt + S dW) + ytrBdt = (xtS + ytrB)dt + xtS dW. • We want this to be like df, match the coefficient of dt and dW between the preceding equation and that of df. From the coefficient of dW, we identify xt = f/S.**From G(t) = xtS(t) + ytB(t) and using G=f (our goal),**6. yt = [G(t) -xtS(t)]/B(t) = [f(S,t) – (f/S) S(t)]/B(t).**Substituting this expression into the equation of dG and**matching it with the coefficient of dt in the equation of df, we have xtS + ytrB = (f/S)S + f/t+(1/2)(2f/S2)2S2. That is, (f/S)S +[1/B(t) ] [f(S,t) – (f/S) S(t)] r B(t) = (f/S)S + f/t +(1/2)(2f/S2)2S2. Consequently, f/t + (f/S) S(t)r + (1/2)(2f/S2)2S2 =rf. **If f(S,t) = S, then f/t =0, f/S=1, and**2f/S2=0 so that the Black-Scholes equation becomes rS=rS. Clearly, f=S is a trivial solution to the Black-Scholes (BS) equation. • Similarly, if f(S,t) = ert, it is easily shown that it is also a trivial solution to the BS (check!!). • For a European call option with strike price K and maturity T, f(S,t)=C(S,t) with boundary conditions C(0,t)=0 and C(S,T)= max{S-K,0}. • For a European put option with strike price K and maturity T, f(S,t)=P(S,t) with boundary conditions P(,t)=0 and P(S,T) = max{K-S,0}. • For an American call option that allows early exercise, we have the extra boundary condition max{0,S-K}C(S,t).**Notice that the BS is a partial differential equation.**There is no guarantee that it has a solution. As a matter of fact, except in simple cases such as a European call or put option, one cannot solve the BS analytically. As a result, either simulations or numerical solutions are possible alternatives. • One can also derive the BS through a deltahedgingargument. To see how it works, construct a portfolio that consists of shorting one derivative (call option, say) and longing f/S shares of the underlying stock. Let the value of this portfolio be . Then = f + (f/S)S. The change of the portfolio value in the time interval t is given by = f + (f/S) S.**Since S follows a Geometric BM, S = S t + S**W. Recall from the Ito’s lemma, the discrete version of df is f =[(f/S)S+f/t+(1/2)(2f/S2)2S2]t+(f/S) SW. Substitute these expressions into the equation = f + (f/S) S, we obtain =[(f/S)S+f/t+(1/2)(2f/S2)2S2]t (f/S)SW + (f/S) (S t + S W) =[ f/t (1/2)(2f/S2)2S2]t.**This portfolio has no random element since Whas been**eliminated by construction. In other words, regardless of the outcome of the stock prices, the change in the portfolio value is nonrandom. It must equal to the risk-free rate since if otherwise, there will be arbitrage opportunities (check!!). Consequently, = r t. That is, [ f/t (1/2)(2f/S2)2S2]t = r[f + (f/S)S] t, f/t + (1/2)(2f/S2)2S2 + (f/S)rS = rf, which is just the Black-Scholes equation again. **Example. Let f be the price of a forward contract of a**non-dividend-paying stock with delivery price K and maturity T. Then the price at time t is given by f(S,t) = S Ker(T t). To check the validity of this pricing formula, we need to check if it satisfies the BS equation. Observe that f/S=1, 2f/S2=0, and f/t= rKer(T t). Substitute these quantities into the BS equation, we have f/t+(1/2)(2f/S2)2S2 +(f/S)rS = rKer(T t) + rS =r(S Ker(T t)) =rf. Therefore, the given formula satisfies the BS and thus is the correct formula of the price for the forward contract. **Black-Scholes formula**Lemma. Let S be a lognormally distributed random variable such that log S ~ N(m, 2) and let K>0 be a given constant. Then E(max{SK,0}) = E(S)(d1) K(d2), where () denotes the cdf of a standard normal random variable and d1 = 1(logK+m+2) = 1(log E(S/K) + 2/2), d2 = 1(logK+m) = 1(log E(S/K) 2/2).**Proof. Recall that since log S ~N(m,2),**1. E(S) = exp(m+2/2) so that log E(S) = m+2/2. 2. Let g(s) denote the pdf of S and let Q = 1(logSm). Then Q~N(0,1) with pdf (q)=(2) 1/2 exp(q2/2). 3. Also, g(s) = ((log s m)/)/(s). (check !!) 4. Observe that q= 1(log sm), s=eq+m so that dq=ds/(s). Consider E(max{SK,0}) = 0 max{sK,0}g(s)ds = K(sK)g(s)ds = (log Km)/(eq+m K)g(eq+m)sdq

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