# 金融代写|期权理论代写Mathematical Introduction to Options代考|MATH150

## 金融代写|期权理论代写Mathematical Introduction to Options代考|BARONE-ADESI AND WHALEY APPROXIMATION

(i) This method can be applied to continuous dividend puts and calls. We will restrict our analysis to put options where price divergence between European and American options is greater, but the analysis for calls is exactly analogous (Barone-Adesi and Whaley, 1987).

The price of an American put option can be written $P_{\mathrm{A}}=P_{\mathrm{E}}+\varphi$, where $P_{\mathrm{E}}$ is the price of the European option and $\varphi$ is a premium for the possibility of early exercise. This method seeks a way of calculating $\varphi$; the Black Scholes model is used to calculate the values of $P_{\mathrm{E}}$.
$P_{\mathrm{A}}$ is a solution to the Black Scholes equation in the region $S_0^*<S_0$, i.e. above the exercise boundary. Therefore in this region, $\varphi$ is also a solution of the Black Scholes equation. Rearranging from the normal order a little gives
$$\frac{1}{2} S_0^2 \sigma^2 \frac{\partial^2 \varphi}{\partial S_0^2}+(r-q) S_0 \frac{\partial \varphi}{\partial S_0}-r \varphi-\frac{\partial \varphi}{\partial T}=0$$ (ii) Consider the evolution of $\varphi$ over time which is illustrated in Figure 6.3. The key properties of this graph are as follows:

(A) The quantity $\varphi$ is defined only in the region $S_0^$ is a function of $r, q, T$ and $\sigma$; it decreases as $T$ increases.
(C) As $S_0 \rightarrow \infty$ we expect $\varphi \rightarrow 0$ since it is unlikely that the stock price will reach the $S_0^$ where early exercise occurs. (D) If $S_0$ is small (but nevertheless above $\left.S_0^\right), \varphi$ will approach its asymptotic value $\left(X-S_0\right)-$ $\left(X \mathrm{e}^{-r T}-S_0 \mathrm{e}^{-q T}\right) \approx X\left(1-\mathrm{e}^{-r T}\right)$ for small dividend yield $q$.
(E) If $T \rightarrow 0$ we must have $\varphi \rightarrow 0$ since the early exercise possibility ceases to have any meaning.

## 金融代写|期权理论代写Mathematical Introduction to Options代考|PERPETUAL PUTS

(i) The concept of a perpetual option seems bizarre. For European options it is meaningless: a call option may go further and further in-the-money indefinitely, but it cannot be exercised until maturity which is never reached!

For American options, which can be exercised at any time, the concept makes more sense, and we examine the case of a perpetual put. If $r>q$, the expected value of the stock price $S$ (the forward rate) drifts upwards over time so that the option gets indefinitely further out-of-the-money; but in the early stages of this infinitely long process, there is some probability that it will pay to exercise the option. Later, the probability recedes to zero.
(ii) An exact solution to this problem can be obtained using the techniques of Section 6.2. The Black Scholes equation in the region above the exercise boundary can be written
$$\frac{1}{2} S^2 \sigma^2 \frac{\partial^2 P_{\infty}}{\partial S_0^2}+(r-q) S \frac{\partial P_{\infty}}{\partial S_0}-r P_{\infty}-\frac{\partial P_{\infty}}{\partial T}=0$$
Since the put is perpetual, it cannot be a function of time, so the last term is zero and we are left with Cauchy’s equation:
$S^2 \frac{\mathrm{d}^2 P_{\infty}}{\mathrm{d} S_0^2}+b S \frac{\mathrm{d} P_{\infty}}{\mathrm{d} S_0}+c P_{\infty}=0 \quad$ where $\quad b=(r-q) / \frac{1}{2} \sigma^2 ; \quad c=-r / \frac{1}{2} \sigma^2$
We have already come across the solution to this equation in the last section:
$$\begin{gathered} P_{\infty}=A S^{\gamma_1}+B S^{\gamma_2} \ \gamma_1=\frac{1}{2}\left{-(b-1)+\sqrt{(b-1)^2-4 c}\right} ; \quad \gamma_2=\frac{1}{2}\left{-(b-1)-\sqrt{(b-1)^2-4 c}\right} \end{gathered}$$
Again, $\gamma_1$ must be positive and $\gamma_2$ must be negative, so that the boundary condition $\lim {S_0 \rightarrow \infty} P{\mathrm{A}} \rightarrow 0$ gives $A=0$. We are left with the two-part solution
$$P_{\infty}= \begin{cases}X-S_0 & S_0<S_0^* \ B S_0^{\gamma_2} & S_0^ which leads to the conditions$$
\left.\begin{array}{l}
X-S_0^=B S_0^{ \gamma \gamma_2} \
-1=B \gamma_2 S_0^{* \gamma_2-1}
S_0^*=\frac{\gamma_2 X}{\gamma_2-1} \
B=-\frac{1}{\gamma_2}\left(\frac{\gamma_2-1}{\gamma_2 X}\right)^{\gamma_2-1}
\end{array}\right.
$$# 期权理论代考 ## 金融代写|期权理论代写Mathematical Introduction to Options代考|BARONE-ADESI AND WHALEY APPROXIMATION (i) 此方法可应用于连续股息看跌期权和看涨期权。我们将分析限制在欧式和美式期权价格差异较大的看跌 期权，但看涨期权的分析完全类似 (Barone-Adesi 和 Whaley，1987)。 美式看跌期权的价格可以写成 P_{\mathrm{A}}=P_{\mathrm{E}}+\varphi ，在哪里 P_{\mathrm{E}} 是欧式期权的价格，并且 \varphi 是早期钣炼的可能性 的溢价。该方法寻求一种计算方法 \varphi; Black Scholes 模型用于计算 P_{\mathrm{E}}. P_{\mathrm{A}} 是区域中 Black Scholes 方程的解 S_0^*<S_0 ，即高于运动边界。所以在这个地区， \varphi 也是 Black Scholes 方程的解。从正常顺序重新排列一点$$
\frac{1}{2} S_0^2 \sigma^2 \frac{\partial^2 \varphi}{\partial S_0^2}+(r-q) S_0 \frac{\partial \varphi}{\partial S_0}-r \varphi-\frac{\partial \varphi}{\partial T}=0
$$(ii) 考虑 \varphi 随看时间的推移，如图 6.3 所示。该图的主要属性如下: (一)数量 \varphi 仅在区域中定义 S_{-} 0^{\wedge} 是一个函数 r, q, T 和 \sigma; 它减少为 T 增加。 (C) 作为 S_0 \rightarrow \infty 我们期待 \varphi \rightarrow 0 因为股价不太可能达到 S_{-} 0^{\wedge} 早期运动发生的地方。(D) 如果 S_0 很小 (但 利 q. (E) 如果 T \rightarrow 0 我们必须有 \varphi \rightarrow 0 因为提前行使的可能性不再具有任何意义。 ## 金融代写|期权理论代写Mathematical Introduction to Options代考|PERPETUAL PUTS (i) 永续期权的概念似乎很奇怪。对于欧式期权来说，这是没有意义的：看涨期权可能会无限期地在价内走 得更远，但直到永远无法达到的到期日才能行使! 对于可以随时行使的美式期权，这个概念更有意义，我们研究了永续看跌期权的情况。如果 r>q ，股票价 格的期望值 S (远期利率) 随看时间的推移向上漂移，使得期权无限期地进一步虚值; 但在这个无限长过 程的早期阶段，行使期权可能会有所回报。后来，概率下降到零。 (ii) 使用第 6.2 节的技术可以得到这个问题的精确解。运动边界以上区域的 Black Scholes 方程可以写成$$
\frac{1}{2} S^2 \sigma^2 \frac{\partial^2 P_{\infty}}{\partial S_0^2}+(r-q) S \frac{\partial P_{\infty}}{\partial S_0}-r P_{\infty}-\frac{\partial P_{\infty}}{\partial T}=0
$$由于看跌期权是永久的，它不能是时间的函数，所以最后一项为零，我们得到柯西方程: S^2 \frac{\mathrm{d}^2 P_{\infty}}{\mathrm{d} S_0^2}+b S \frac{\mathrm{d} P_{\infty}}{\mathrm{d} S_0}+c P_{\infty}=0 在哪里 \quad b=(r-q) / \frac{1}{2} \sigma^2 ; \quad c=-r / \frac{1}{2} \sigma^2 我们已经在上一节中遇到了这个方程的解: 再次， \gamma_1 必须是积极的 \gamma_2 必须为负，因此边界条件 \lim S_0 \rightarrow \infty P A \rightarrow 0 给 A=0. 我们只剩下两部分的 解决方案 剩下。$$
X-S_0^{=} B S_0^{r \gamma_2}-1=B \gamma_2 S_0^{* \gamma_2-1}

S_0^*=\frac{\gamma_2 X}{\gamma_2-1} B=-\frac{1}{\gamma_2}\left(\frac{\gamma_2-1}{\gamma_2 X}\right)^{\gamma_2-1}

【正确的。

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