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二叉树下的平值看跌期权

The price of a stock

专题
Finance / 金融
难度
L4

题目详情

股票现价为 5050。三个月后股价要么为 4747，要么为 5252。假设无分红且利率为 0，一份平值看跌期权应值多少？

英文原题

The price of a stock is $50. In three months, it will either be $47 or $52, each with 50% probability. How much would you pay for an at-the-money put? Assume for simplicity that the stock pays no dividends and that interest rates are zero.

解析

真实概率不直接进入无套利定价，应使用风险中性概率。

上下行因子为

u=5250=1.04,d=4750=0.94.u=\frac{52}{50}=1.04,\quad d=\frac{47}{50}=0.94.

利率为 0 时，风险中性上涨概率

p=1−du−d=0.060.10=0.6,p=\frac{1-d}{u-d}=\frac{0.06}{0.10}=0.6,

所以下跌概率为 0.40.4。平值 put 在上涨状态收益为 00，在下跌状态收益为 50−47=350-47=3。因此

P0=0.6⋅0+0.4⋅3=1.2.P_0=0.6\cdot 0+0.4\cdot 3=\boxed{1.2}.

英文解析

The stated real-world probabilities are irrelevant for no-arbitrage pricing. Use the risk-neutral probability.

The up and down factors are

u=5250=1.04,d=4750=0.94.u=\frac{52}{50}=1.04,\quad d=\frac{47}{50}=0.94.

With zero interest rate, the risk-neutral up probability is

p=1−du−d=0.060.10=0.6,p=\frac{1-d}{u-d}=\frac{0.06}{0.10}=0.6,

so the down probability is 0.40.4. The ATM put pays 00 in the up state and 50−47=350-47=3 in the down state. Therefore

P0=0.6⋅0+0.4⋅3=1.2.P_0=0.6\cdot 0+0.4\cdot 3=\boxed{1.2}.