The laws of Probability, Bayes’ Theorem, Decision Trees. Binomial, Poisson and Normal distributions and their applications; the relationship between these distributions. Different types of sample; sampling distributions of means, standard deviations and proportions. Confidence intervals and hypothesis testing; types of error, significance levels and P values. Power. Quality control: Acceptance sampling and Shewhart charts.

这是一份Bath巴斯大学学院MA10214作业代写的成功案

Let the exercise price be $E$ and the barrier be at $B$ where $B<E$.
Since the Black-Scholes partial differential equation governs the price of the option we can, as before, look for solutions of the form:
$$
c(S, E){d o}=A{1} S^{m_{1}}+A_{2} S^{m_{2}}
$$
subject to the boundary conditions: (i) $c_{d b}(B, E)=0$ and (ii) $c(\infty, E){d o}=S$, see the previous section. From (i) we have: $$ c{d b}(B, E)=A_{1} B^{m_{1}}+A_{2} B^{m_{2}}=0, \text { so } \quad A_{1}=-A_{2} B^{m_{2}-m_{n}}
$$
Therefore
$$
c_{d m}(S, E)=-A_{2} B^{m_{2}-m_{n}} S^{m_{1}}+A_{2} S^{m_{2}}
$$
From (ii), as $S \rightarrow \infty$ :
$$
c_{d m}(S, E)=-A_{2} B^{m_{2}-m_{n}} S^{m_{1}}+A_{2} S^{m_{2}}=S
$$
However, since $m_{2}<0$, we have $A_{2} S^{m_{2}} \rightarrow 0$, as $S \rightarrow \infty$, giving
$$
c_{d o}(S, E)=-A_{2} B^{m_{2}-m_{1}} S^{m_{n}}=S
$$

MA10214 COURSE NOTES ：

The Black approximation, $C_{B L}$, can be expressed more concisely in terms of our previously defined notation as:
$$
C_{B L}(S, E, \tau)=\max \left(v_{1}, v_{2}\right)
$$
where $v_{1}$ and $v_{2}$ are the following European calls
$$
v_{1}=c\left(S_{D}, E, \tau\right) \quad \text { and } \quad v_{2}=c\left(S_{D}^{+}, E, \tau_{1}\right), \quad \tau=T-t \quad \tau_{1}=T-t_{n}
$$
and
$$
S_{D}=S-\sum_{i=1}^{n} D_{i} \quad \text { and } \quad S_{D}^{+}=S-\sum_{i=1}^{n-1} D_{i}
$$