week7blackscholesmodelsandoptionpricing.pptVIP

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 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 趋于一连线的随机过程 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： 期权的价格 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 jk, W(tk) - W(tj) = ?i=jk-1 e(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=1n e(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 Princ