这是一份nottingham诺丁汉大学ECON1047作业代写的成功案例

Up to this point we have been dealing with the standard Black-Scholes equation, which is
$$
\frac{\partial f}{\partial t}+(r-q) S \frac{\partial f}{\partial S}+\frac{\sigma^{2} S^{2}}{2} \frac{\partial^{2} f}{\partial S^{2}}=r f
$$
However, if we introduce the change of variable $Z=\log S$, we obtain the following equation:
$$
\frac{\partial f}{\partial t}+b \frac{\partial f}{\partial Z}+\frac{\sigma^{2}}{2} \frac{\partial^{2} f}{\partial Z^{2}}=r f
$$
where $b=r-q-\left(\sigma^{2} / 2\right)$. This has beneficial numerical properties since it does not contain the original Black-Scholes terms in $S$ and $S^{2}$.

ECON1047 COURSE NOTES ：

$$
S_{1}^{1}=115, \quad S_{1}^{2}=60, \quad \text { and } \quad S_{1}^{3}=114
$$
and for the second time step
$$
S_{2}^{1}=116, \quad S_{2}^{2}=90, \quad S_{2}^{3}=149, \ldots, \quad S_{2}^{7}=102, \quad S_{2}^{8}=88, \quad S_{2}^{9}=80
$$
The $k$ th asset price at the $i$ th time step, $S_{i}^{k}$ then generates the following asset prices at the $(i+1)$ th time step:
$$
\frac{S_{i+1}^{(k-1) b+j}}{S_{i}^{k}}=d S^{j}, \quad j=1, \ldots, b, \quad k=1, \ldots, b^{i}
$$
where $d S^{j}$ is, as before, a random variate from a given distribution. When $S_{i}$ follows GBM we therefore have