对数正态动态的股票价格模型

Stochastic Stock Price Model with Log-Normal Dynamics

专题: Finance / 金融
难度: L6
来源: QuantQuestion

题目详情

一只股票在 $t=0$ 时的初始价格为 $S_0=s_0>0$ 。股票的平均收益率为 $\alpha$ ，波动率为 $\sigma$ 。其价格演化满足随机微分方程

dS_t = S_t (\alpha \, dt + \sigma \, dB_t).

(a) 求 $d(\log S_t)$ 并给出 $S_t$ 的显式解。

(b) 给出条件，使得对任意 $t>0$ 都有 $\mathbb{E}[S_t]>s_0$ 。

A stock has an initial price $S_0$ (where $S_0 = s_0 > 0$ ) at time $t = 0$ . The stock’s mean return rate is denoted by $\alpha$ and its volatility by $\sigma$ . The price evolution of this stock can be described using the following stochastic differential equation:

dS_t = S_t (\alpha \, dt + \sigma \, dB_t)

(a) Determine $d(\log S_t)$ and find an explicit solution for $S_t$ .

(b) Establish the condition under which the expected value of the stock, $\mathbb{E}[S_t]$ , is always greater than its initial value $s_0$ for any $t > 0$ .

(c) Identify the condition that ensures the probability of the stock price being above $s_0$ at any time $t > 0$ , denoted as $\mathbb{P}(S_t > s_0)$ , is more than 50%.

解析

(a) 对 $\log S_t$ 用伊藤公式。对 $f(s)=\log s$ 有 $f'(s)=1/s,\ f''(s)=-1/s^2$ ，因此

d(\log S_t)=\frac{1}{S_t}dS_t-\frac{1}{2}\frac{1}{S_t^2}(dS_t)^2 =(\alpha-\tfrac{1}{2}\sigma^2)dt+\sigma\, dB_t.

积分得到

\log S_t=\log S_0+(\alpha-\tfrac{1}{2}\sigma^2)t+\sigma B_t \Rightarrow S_t=S_0\exp\Big((\alpha-\tfrac{1}{2}\sigma^2)t+\sigma B_t\Big).

(b) 由几何布朗运动性质

\mathbb{E}[S_t]=S_0 e^{\alpha t}.

对任意 $t>0$ 都满足 $\mathbb{E}[S_t]>S_0$ 当且仅当 $\alpha>0$ 。

(c) 由 (a)

\log\frac{S_t}{S_0}=(\alpha-\tfrac{1}{2}\sigma^2)t+\sigma B_t.

其中 $B_t\sim N(0,t)$ 。因此 $\log(S_t/S_0)$ 为正态，均值 $(\alpha-\tfrac{1}{2}\sigma^2)t$ ，标准差 $\sigma\sqrt{t}$ 。

当 $\sigma>0$ 时，正态变量大于 0 的概率超过 $1/2$ 当且仅当其均值为正，即

\alpha-\tfrac{1}{2}\sigma^2>0.

所以条件为 $\boxed{\alpha>\tfrac{1}{2}\sigma^2}$ （若 $\sigma=0$ 则退化为确定性过程，条件为 $\alpha>0$ ）。