这是一份manchester曼切斯特大学ECON20222T作业代写的成功案例

Since an American option gives more choice its value is always at least that of its European counterpart. This early exercise premium $(\nu(S, E, \tau) \geq 0)$ is now defined more precisely for American puts and calls. If at current time $t$ the asset price is $S$, then the early exercise premium for an American call which expires at time $T$, and therefore has maturity $\tau=T-t$, is:
$$
\nu_{c}(S, E, \tau)=C(S, E, \tau)-c(S, E, \tau) \geq 0
$$
where $C(S, E, \tau)$ denotes the value of the American call and $c(S, E, \tau)$ denotes the value of the corresponding European call. The early exercise premium of an American put option, $\nu_{p}(S, E, \tau)$, is similarly defined as:
$$
\nu_{p}(S, E, \tau)=P(S, E, \tau)-p(S, E, \tau) \geq 0
$$

ECON20222T COURSE NOTES ：

Substituting these results into Equation $10.39$ yields the following transformed Black-Scholes equation:
$$
\frac{S^{2} \sigma^{2} h}{2} \frac{\partial^{2} g}{\partial S^{2}}+(r-q) S h \frac{\partial g}{\partial S}+r g(h-1)+r h(h-1) \frac{\partial g}{\partial h}=r g h
$$
which can be further simplified to give:
$$
S^{2} \sigma^{2} \frac{\partial^{2} g}{\partial S^{2}}+\frac{2(r-q) S}{\sigma^{2}} \frac{\partial g}{\partial S}-\frac{2 r g}{h \sigma^{2}}-\frac{2 r(1-h)}{\sigma^{2}} \frac{\partial g}{\partial h}=r g h
$$
or
$$
S^{2} \frac{\partial^{2} g}{\partial S^{2}}+\beta S \frac{\partial g}{\partial S}-\frac{\alpha}{h} g-(1-h) \alpha \frac{\partial g}{\partial h}=0
$$
where $\alpha=2 r / \sigma^{2}$ and $\beta=(2(r-q)) / \sigma^{2}$.