这是一份ed.ac爱丁堡格大学MATH11157作业代写的成功案

In the local volatility the risk-neutral dynamics of the underlying index or stock value is described by the SDE
$$
d S_{t}=\left(r_{t}-\delta_{t}\right) S_{t} d t+\sigma\left(t, S_{t}\right) S_{t} d W_{t},
$$
where $r_{t}$ is the deterministic instantaneous risk-free interest rate, $\delta_{t}$ is the deterministic instantaneous dividend yield and the volatility $\sigma\left(t, S_{t}\right)$ is a deterministic function dependent on the current index value $S_{t}$. Discretizing the SDE (1.3), yields the following difference equation
$$
S_{t_{i}}-S_{t_{i-1}}=\left(r_{t_{i-1}}-\delta_{t_{i-1}}\right) S_{t_{i-1}} \Delta_{t}+\sigma\left(t_{i-1}, S_{t_{i-1}}\right) S_{t_{i-1}} \Delta W_{t_{i-1}},
$$
where $\Delta W_{t_{i-1}}=W_{t_{i}}-W_{t_{i-1}} \sim N\left(\Delta_{t}\right)$. Hence we have the following conditional distribution of $S_{t_{i}}$
$$
S_{t_{i}} \mid S_{t_{i-1}} \sim N\left(S_{t_{i-1}}+\left(r_{t_{i-1}}-\delta_{t_{i-1}}\right) S_{t_{i-1}} \Delta_{t}, \sigma\left(t_{i-1}, S_{t_{i-1}}\right) S_{t_{i-1}} \Delta_{t}\right) .
$$

MATH11157 COURSE NOTES ：

A portfolio is a vector $\theta=\left(\theta_{0}, \theta_{1}, \ldots, \theta_{m}\right) \in \mathbb{R}^{m+1}$ $\theta_{j}$ is the number of units held in asset $A_{j}$ for all $j=0,1, \ldots, r$ Spot Value $($ at $t=0)$ of portfolio $\theta$ denoted $V_{\theta}^{(0)}$ is:
$$
V_{\theta}^{(0)}=\sum_{j=0}^{m} \theta_{j} \cdot S_{j}^{(0)}
$$
Value of portfolio $\theta$ in state $\omega_{i}($ at $t=1)$ denoted $V_{\theta}^{(i)}$ is:
$$
V_{\theta}^{(i)}=\sum_{j=0}^{m} \theta_{j} \cdot S_{j}^{(i)} \text { for all } i=1, \ldots, n
$$