对数正态动态的股票价格模型

(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 $\alpha>0$）。

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英文解析

• • (a) * * Use the Ito formula for $\log S_t$. $f'(s)=1/s,\f''(s)=-1/s^2$for $f(s)=\log s$, so

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

Points earned

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

• • (b) * * Nature by Geometric Brownian Motion

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

Any $t>0$ would qualify for $\mathbb{E}[S_t]>S_0$ if and only if $\alpha>0$.

• • (c) * * By (a)

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

$B_t\sim N(0,t)$, so$\log(S_t/S_0)$ is normal, with a mean of$(\alpha-\tfrac{1}{2}\sigma^2)t$ and a standard deviation of$\sigma\sqrt{t}$.

When $\sigma>0$, the probability that the normal variable is greater than 0 exceeds $1/2$if and only if its mean is positive, i.e.

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

So the condition is $\boxed{\alpha>\tfrac{1}{2}\sigma^2}$ (if $\sigma=0$ it degenerates into a deterministic process, the condition is $\alpha>0$).