\[y_t=\mu y_t dt+ \sigma_y y_t dw_t \]
The solution is:


\[ y_t=y_0 e^{(\mu-\frac{1}{2}\sigma_y^2)t+\sigma_y \sqrt{t} z}\] where z is a standard normal.


\[ E((\int_{0}^{\infty} e^{-(r+h)t}h y_t dt) -I ) \]


\[ E((\int_{0}^{\infty} e^{-(r+h)t}h y_0 e^{(\mu-\frac{1}{2}\sigma_y^2)t+\sigma_y \sqrt{t} z} dt) -I ) \]


\[ h y_0 E(( \int_{0}^{\infty} e^{(\mu-r-h-\frac{1}{2}\sigma_y^2) t+\sigma_y \sqrt{t} z} dt) )-I \]


change the order of integral:


\[ h y_0(\int_{0}^{\infty} E(e^{(\mu-r-h-\frac{1}{2}\sigma_y^2) t+\sigma_y\sqrt{t}z} dt))-I \]


\[e^{(\mu-r-h-\frac{1}{2}\sigma_y^2) t+\sigma_y\sqrt{t}z} \] this is a lognormal random variable. The mean and variance of the corresponding normal distribution is \[ N((\mu-r-h-\frac{1}{2}\sigma_y^2) t, \sigma_y^2t)\]


if \[ z \sim N(\mu,\sigma^2)\] then \[ E(e^z)=e^{\mu+\frac{1}{2}\sigma^2}\]
so:

\[ E(e^{(\mu-r-h-\frac{1}{2}\sigma_y^2) t+\sigma_y\sqrt{t}z})\]
\[=e^{(\mu-r-h-\frac{1}{2}\sigma_y^2) t+\frac{1}{2}\sigma_y^2t}\]
\[=e^{(\mu-r-h) t}\]


the original pricing formula is then equal to:


\[ h y_0( \int_{0}^{\infty} e^{(\mu-r-h) t})dt-I \]

let \[ \alpha=\mu-r-h \]


\[=\frac{hy_0}{\alpha}( \int_{0}^{\infty} e^{\alpha t})d\alpha t-I \]


\[=\frac{hy_0}{\alpha}( e^{\alpha \infty}-1)-I \]


assume \[\alpha <0->\mu<r+h\]


\[=-\frac{hy_0}{\alpha}-I \]


\[=\frac{hy_0}{r+h-\mu}-I \]

牛人！请问您这是看得那本书的啊？能推荐一下不？最近再研究实物期权，推导太难了。

I derived myself. This is just basic financial math

墙裂安利楼主shreve那本书的第二册，那个讲的比较简单而且也讲的非常明白

though basic it is, you don't have to calculate the expectation by normal, since there is a exp-martingale which is used in Girsanov frequently. No offence :)