### 金融代写|波动率模型代写Market Volatility Modelling代考|FM321

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|波动率模型代写Market Volatility Modelling代考|Risk-Neutral Pricing

We mentioned in Section 1.2.1 that, unless $\mu=r$, the expected value under the $s u b$ jective probability $\mathbb{P}$ of the discounted payoff of a derivative (1.23) would lead to an opportunity for arbitrage. This is closely related to the fact that the discounted price $\widetilde{X}_t=e^{-r t} X_t$ is not a martingale since, from (1.18),
$$d \tilde{X}_t=(\mu-r) \tilde{X}_t d t+\sigma \tilde{X}_t d W_t,$$
which contains a nonzero drift term if $\mu \neq r$.
The main result we want to build in this section is that there is a unique probability measure $\mathbb{P}^$ equivalent to $\mathbb{P}$ such that, under this probability, (i) the discounted price $\widetilde{X}_t$ is a martingale and (ii) the expected value under $\mathbb{P}^$ of the discounted payoff of a derivative gives its no-arbitrage price. Such a probability measure describing a risk-neutral world is called an equivalent martingale measure.

In order to find a probability measure under which the discounted price $\tilde{X}t$ is a martingale, we rewrite (1.43) in such a way that the drift term is “absorbed” into the martingale term: We set $$d \widetilde{X}_t=\sigma \widetilde{X}_t\left[d W_t+\left(\frac{\mu-r}{\sigma}\right) d t\right] .$$ $$\theta=\frac{\mu-r}{\sigma},$$ called the market price of asset risk, and define $$W_t^{\star}=W_t+\int_0^t \theta d s=W_t+\theta t,$$ so that $$d \tilde{X}_t=\sigma \tilde{X}_t d W_t^{\star} .$$ Using the characterization (1.3), it is easy to check that the positive random variable $\xi_T^\theta$ defined by $$\xi_T^\theta=\exp \left(-\theta W_T-\frac{1}{2} \theta^2 T\right)$$ has an expected value (with respect to $\mathbb{P}$ ) equal to 1 (the Cameron-Martin formula). More generally, it has a conditional expectation with respect to $\mathcal{F}_t$ given by $$\mathbb{E}\left{\xi_T^\theta \mid \mathcal{F}_t\right}=\exp \left(-\theta W_t-\frac{1}{2} \theta^2 t\right)=\xi_t^\theta \quad \text { for } 0 \leq t \leq T,$$ which defines a martingale denoted by $\left(\xi_t^\theta\right){0 \leq t \leq T}$.
We now introduce the probability measure $\mathbb{P}^{\star}$. It is an equivalent measure to $\mathbb{P}$, meaning that it has the same null sets $\left(\mathbb{P}^*\right.$ and $\mathbb{P}$ agree on which events have zero probability); $\mathbb{P}^{\star}$ has the density $\xi_T^\theta$ with respect to $\mathbb{P}$,
$$d \mathbb{P}^{\star}=\xi_T^\theta d \mathbb{P} .$$

## 金融代写|波动率模型代写Market Volatility Modelling代考|Risk-Neutral Expectations and Partial Differential Equations

In Section 1.4.4 we used the Markov property of the stock price $\left(X_t\right)$ and, in order to compute $X_T$ knowing that $X_t=x$ at time $t \leq T$, we solved the stochastic differential equation (1.2) between $t$ and $T$. This was a particular case of the general situation where $\left(X_t\right)$ is the unique solution of the stochastic differential equation (1.11). We denote by $\left(X_s^{t, x}\right){s \geq t}$ the solution of that equation, starting from $x$ at time $t$ : $$X_s^{t, x}=x+\int_t^s \mu\left(u, X_u^{t, x}\right) d u+\int_t^s \sigma\left(u, X_u^{t, x}\right) d W_u,$$ and we assume enough regularity in the coefficients $\mu$ and $\sigma$ for $\left(X_s^{t, x}\right)$ to be jointly continuous in the three variables $(t, x, s)$. The flow property for deterministic differential equations can be extended to stochastic differential equations like (1.11); it says that, in order to compute the solution at time $s>t$ starting at time 0 from point $x$, one can use $$x \longrightarrow X_t^{0, x} \longrightarrow X_s^{1, X_t^{0, x}}=X_s^{0, x} \quad(\mathbb{P} \text {-a.s.). }$$ In other words, one can solve the equation from 0 to $t$, starting from $x$, to obtain $X_t^{0, x}$. Then we solve the equation from $t$ to $s$, starting from $X_t^{0, x}$. This is the same as solving the equation from 0 to $s$, starting from $x$. The Markov property is a consequence and can be stated as follows: $$\mathbb{E}\left{h\left(X_s\right) \mid \mathcal{F}_1\right}=\left.\mathbb{E}\left{h\left(X_s^{t \cdot x}\right)\right}\right|{x=X_1},$$
which is what we have used with $s=T$ to derive (1.55). Observe that the discounting factor could be pulled out of the conditional expection since the interest rate is constant (not random). In the time-homogeneous case ( $\mu$ and $\sigma$ independent of time) we further have $$\mathbb{E}\left{h\left(X_s^{t, X}\right)\right}=\mathbb{E}\left{h\left(X_{s-t}^{0, X}\right)\right},$$
which could have been used with $s=T$ to derive (1.57) since $W_{T-t}^*$ is $\mathcal{N}(0, T-t)$ distributed.

# 波动率模型代考

## 金融代写|波动率模型代写Market Volatility Modelling代考|Risk-Neutral Pricing

$$d \tilde{X}_t=(\mu-r) \tilde{X}_t d t+\sigma \tilde{X}_t d W_t,$$

$$\begin{gathered} d \widetilde{X}_t=\sigma \widetilde{X}_t\left[d W_t+\left(\frac{\mu-r}{\sigma}\right) d t\right] . \ \theta=\frac{\mu-r}{\sigma}, \end{gathered}$$

$$W_t^{\star}=W_t+\int_0^t \theta d s=W_t+\theta t$$

$$d \tilde{X}_t=\sigma \tilde{X}_t d W_t^{\star} .$$

$$\xi_T^\theta=\exp \left(-\theta W_T-\frac{1}{2} \theta^2 T\right)$$

$$d \mathbb{P}^{\star}=\xi_T^\theta d \mathbb{P} .$$

## 金融代写|波动率模型代写Market Volatility Modelling代考|Risk-Neutral Expectations and Partial Differential Equations

$$X_s^{t, x}=x+\int_t^s \mu\left(u, X_u^{t, x}\right) d u+\int_t^s \sigma\left(u, X_u^{t, x}\right) d W_u,$$

$$x \longrightarrow X_t^{0, x} \longrightarrow X_s^{1, X_t^{0, x}}=X_s^{0, x} \quad(\mathbb{P} \text {-a.s. }) .$$

Imathbb ${E} \backslash$ eft $\left{h \backslash l\right.$ eft $\left(X_{-} s^{\wedge}{t, X} \backslash\right.$ ight $\left.) \backslash r i g h t\right}=\backslash m a t h b b{E} \backslash l$ eft $\left{h \backslash l\right.$ eft $\left(X_{-}{\right.$st $} \wedge{0, X} \backslash$ ight $) \backslash$ 正确的 $}$ ，

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