问题如下：
6.The value of the American-style put option on Beta Company shares is closest to:
选项：
A. 4.53.
B. 5.15.
C. 9.32.
解释：
B is correct.
Using the expectations approach, the risk-neutral probability of an up move is
π= [FV(1) - d]/(u - d) = (1.03 - 0.800)/(1.300 - 0.800) = 0.46.
An American-style put can be exercised early. At Time Step 1, for the up move, p+ is 0.2517 and the put is out of the money and should not be exercised early (X < S, 40 < 49.4). However, at Time Step 1, p- is 8.4350 and the put is in the money by 9.60 (X - S = 40 - 30.40). So, the put is exercised early, and the value of early exercise (9.60) replaces the value of not exercising early (8.4350) in the binomial tree. The value of the put at Time Step 0 is now
p = PV[πp+ + (1 - π)p-] = [1/(1.03)][0.46(0.2517) + 0.54(9.60)] = 5.1454.
Following is a supplementary note regarding Exhibit 1.
The values in Exhibit 1 are calculated as follows.
At Time Step 2:
p++ = Max(0,X - u2S) = Max[0,40 - 1.3002(38)] = Max(0,40 - 64.22) = 0.
p-+ = Max(0,X - udS) = Max[0,40 - 1.300(0.800)(38)] = Max(0,40 - 39.52) = 0.48.
p- - = Max(0,X - d2S) = Max[0,40 - 0.8002(38)] = Max(0,40 - 24.32)= 15.68.
At Time Step 1:
p+ = PV[πp++ + (1 - π)p-+] = [1/(1.03)][0.46(0) + 0.54(0.48)] = 0.2517.
p- = PV[πp-+ + (1 - π)p- -] = [1/(1.03)][0.46(0.48) + 0.54(15.68)]=8.4350.
At Time Step 0:
p = PV[πp+ + (1 - π)p-] = [1/(1.03)][0.46(0.2517) + 0.54(8.4350)] = 4.5346.
为什么结果不是p= [1/(1.03)][0.46(0)+ 0.54(9.60)] = 5.03呢，因为在p1+时，49.4>40，p1+=0