# 金融代写|期权定价理论代写Option Pricing Theory代考|MATH150

## 金融代写|期权定价理论代写Option Pricing Theory代考|Modeling the dynamics of implied volatility

Here we follow [180]. We take $\Phi(x)=(x-K)^{+}$, a call with fixed strike $K$, and we parameterize its market value by its implied volatility $\sigma_t^{T, K}$ defined by
$$\Phi_t^T=\operatorname{BS}\left(\left(\sigma_t^{T, K}\right)^2(T-t), K \mid X_t\right)$$
BS $\left(\sigma^2 T, K \mid X\right)$ denotes the Black-Scholes formula with variance $\sigma^2 T$, strike $K$, and $\operatorname{spot} X$ :
$$\operatorname{BS}\left(\sigma^2 T, K \mid X\right)=X N\left(d_{+}\right)-K N\left(d_{-}\right)$$
with
$$d_{\pm}=\frac{\ln \frac{X}{K}}{\sigma \sqrt{T}} \pm \frac{\sigma \sqrt{T}}{2}$$
$N(x)=\frac{1}{\sqrt{2 n}} \int_{-\infty}^x e^{-\frac{y^2}{2}} d y$ is the standard normal cumulative distribution function.
A straightforward application of Itô’s formula shows that $\Phi_t^T$ is a local martingale if and only if the implied volatility dynamics is given by
$$d \sigma_t^{T, K}=u_t^{T, K} d t+v_t^{T, K} \cdot d W_t$$
with the drift $u_t^{T, K}$ linked to the volatilities $\sigma_t^{T, K}$ and $v_t^{T, K}$ by
\begin{aligned} \sigma_t^{T, K} u_t^{T, K}(T-t)=\frac{1}{2}\left(\left(\sigma_t^{T, K}\right)^2-\right. & \left.\sigma_t^2\right)-\frac{1}{2}\left(v_t^{T, K}\right)^2 \frac{\left(\ln \frac{X_t}{K}\right)^2-\frac{1}{4}\left(\sigma_t^{T, K}\right)^4(T-t)}{\left(\sigma_t^{T, K}\right)^2} \ • & \sigma_t v_t^{T, K, 0} \frac{\ln \frac{X_t}{K}-\frac{1}{2}\left(\sigma_t^{T, K}\right)^2(T-t)}{\sigma_t^{T, K}} \end{aligned}

## 金融代写|期权定价理论代写Option Pricing Theory代考|Modeling the dynamics of log-contract prices

Let us take $\Phi(x)=\ln x$ a log-contract payoff. In the Black-Scholes model with time-dependent volatility $\sigma(t)$, we have (left as an exercise to the reader)
$$\Phi_t^T=\ln X_t-\frac{1}{2} \int_t^T \sigma(s)^2 d s$$
Let us set
$$\Phi_t^T=\ln X_t-\frac{1}{2} \int_t^T \xi_t^s d s$$
for a family of processes $\left(\xi_t^u, u \geq t\right) . \Phi_t^T$ is a local martingale under the risk-neutral measure if and only if
$$\sigma_t^2=\xi_t^t$$
and the $\xi_t^s$ are local martingales:
\begin{aligned} d X_t & =X_t \sqrt{\xi_t^t} d W_t^0 \ d \xi_t^s & =\beta_t^s \cdot d W_t \end{aligned}
$\xi_t^s=\mathbb{E}^{\mathbb{Q}}\left[\xi_s^s \mid \mathcal{F}_t\right]=\mathbb{E}^{\mathbb{Q}}\left[\sigma_s^2 \mid \mathcal{F}_t\right] \geq 0$ represents the forward instantaneous variance at time $s$, as seen at time $t \leq s$. If $\xi_t^s$ behaves nicely when $s \rightarrow T$, condition (3.3) is automatically satisfied. We can trace back the creation of the notion of forward variance, forward instantaneous variance, and variance swaps to Dupire [94]. Bergomi [57] uses this framework to build a variance swap model that admits a low-dimensional Markovian representation, and which is now widely used in the industry:
$$d \xi_t^T=\sigma\left(t, T, \xi_t^T\right) d W_t$$

# 期权理论代考

## 金融代写|期权定价理论代写Option Pricing Theory代考|Modeling the dynamics of implied volatility

$$\Phi_t^T=\operatorname{BS}\left(\left(\sigma_t^{T, K}\right)^2(T-t), K \mid X_t\right)$$

$$\operatorname{BS}\left(\sigma^2 T, K \mid X\right)=X N\left(d_{+}\right)-K N\left(d_{-}\right)$$

$$d_{\pm}=\frac{\ln \frac{X}{K}}{\sigma \sqrt{T}} \pm \frac{\sigma \sqrt{T}}{2}$$
$N(x)=\frac{1}{\sqrt{2 n}} \int_{-\infty}^x e^{-\frac{y^2}{2}} d y$ 是标准正态累积分布函数。
Itô 公式的直接应用表明 $\Phi_t^T$ 是局部鞅当且仅当隐含波动率动态由下式给出
$$d \sigma_t^{T, K}=u_t^{T, K} d t+v_t^{T, K} \cdot d W_t$$

## 金融代写|期权定价理论代写Option Pricing Theory代考|Modeling the dynamics of log-contract prices

$$\Phi_t^T=\ln X_t-\frac{1}{2} \int_t^T \sigma(s)^2 d s$$

$$\Phi_t^T=\ln X_t-\frac{1}{2} \int_t^T \xi_t^s d s$$

$$\sigma_t^2=\xi_t^t$$

$$d X_t=X_t \sqrt{\xi_t^t} d W_t^0 d \xi_t^s \quad=\beta_t^s \cdot d W_t$$
$\xi_t^s=\mathbb{E}^Q\left[\xi_s^s \mid \mathcal{F}_t\right]=\mathbb{E}^Q\left[\sigma_s^2 \mid \mathcal{F}_t\right] \geq 0$ 表示时间的前向瞬时方差 $s$, 如当时所见 $t \leq s$. 如果 $\xi_t^s$ 表 现很好的时候 $s \rightarrow T$ ，条件 (3.3) 自动满足。我们可以追湖到前向方差、前向瞬时方差和方差交换 概念的创建到 Dupire [94]。Bergomi [57] 使用这个框架构建了一个方差交换模型，该模型承认低 维马尔可夫表示，现在在业界广泛使用:
$$d \xi_t^T=\sigma\left(t, T, \xi_t^T\right) d W_t$$

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