### 金融代写|金融工程作业代写Financial Engineering代考| Joint Distribution of Returns

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## 金融代写|金融工程作业代写Financial Engineering代考|Joint Distribution of Returns

Proposition 1.2.1 Under the Black-Scholes model, for $h>0$ given, the returns $X_{i}=\ln {S(i h)}-\ln [S{(i-1) h}], i \in{1, \ldots, n}$, are independent and $X_{i} \sim N\left(\mu h-\frac{\sigma^{2}}{2} h, \sigma^{2} h\right)$, i.e., $X_{i}$ has a Gaussian distribution with mean $\left(\mu-\frac{\sigma^{2}}{2}\right) h$ and variance $\sigma^{2} h$.
PROOF. For $i \in{1, \ldots, n}$, one has
\begin{aligned} S(i h) / S{(i-1) h} &=\frac{s e^{\mu i h-\frac{i h \sigma^{2}}{2}}+\sigma W(i h)}{s e^{\mu(i-1) h-\frac{(i-1) h \sigma^{2}}{2}+\sigma W{(i-1) h}}} \ &=e^{\mu h-\frac{h \sigma^{2}}{2}+\sigma{W(i h)-W{(i-1) h}}} \end{aligned}
so
\begin{aligned} X_{i} &=\ln [S(i h) / S{(i-1) h}] \ &=\mu h-\frac{\sigma^{2}}{2} h+\sigma{W(i h)-W{(i-1) h}} \ & \sim N\left(\mu h-\frac{\sigma^{2}}{2} h, \sigma^{2} h\right) \end{aligned}
The increments $W(i h)-W{(i-1) h}, i \in{1 \ldots n}$, being independent, by definition of Brownian motion, it follows that the returns $X_{i}, i \in{1 \ldots n}$, are also independent.

Remark 1.2.2 Since $\sigma$ appears to be a measure of variability, it is also called volatility in financial applications and it is often reported in percentage. For example, a volatility of $20 \%$ per annum means that $\sigma=0.2$, on an annual time scale.

## 金融代写|金融工程作业代写Financial Engineering代考|Joint Law of Prices

One of the most efficient method for estimating parameters is the maximum likelihood principle, described in Appendix B.1. As shown in Remark B.1.2, the maximum likelihood principle can be used with prices or returns. Since the returns are independent and identically distributed in the BlackScholes model, we will estimate the parameters using the returns instead of the prices.

However, for sake of completeness, we also give the conditional law of the prices, together with their joint law.

Proposition 1.2.2 The conditional distribution of $S{(i+1) h}$, given $S(0)=$ $s, \ldots, S(i h)=s_{i}$, depends only on $S(i h)$, and its density is
$$f_{S{(i+1) h} \mid S(i h)}\left(x \mid s_{i}\right)=\frac{e^{-\frac{1}{2 \sigma^{2} h}\left{\ln \left(x / s_{i}\right)-\left(\mu-\sigma^{2} / 2\right) h\right}^{2}}}{x \sigma \sqrt{2 \pi h}} \mathbb{I}(x>0), \quad x \in \mathbb{R} .$$
PROOF. It follows from Proposition $1.2 .1$ that for all $i \geq 0, S{(i+1) h}=$ $S(i h) e^{X_{i+1}}$, and $S(i h)=s e^{X_{1}+\cdots+X_{i}}$ is independent of $e^{X_{i+1}}$. Therefore, the conditional distribution of $S{(i+1) h}$ given $S(0)=s_{0}, \ldots, S(i h)=s_{i}$, is a log-normal distribution (see A.6.8), being the exponential of a Gaussian variate with mean $\ln \left(s_{i}\right)+\left(\mu-\sigma^{2} / 2\right) h$ and variance $\sigma^{2} h$.

Remark 1.2.3 As a by-product of Proposition 1.2.2 and the multiplication formula (A.19), the joint law of $S(h), \ldots, S(n h)$, given $S(0)=s$, is
\begin{aligned} f_{S(h), \ldots, S(n h) \mid S(0)}\left(s_{1}, \ldots, s_{n} \mid s\right) &=\prod_{i=1}^{n} f_{S((i h) \mid S(0), \ldots, S((i-1) h)}\left(s_{i} \mid s, \ldots, s_{i-1}\right) \ &=\prod_{i=1}^{n} f_{S(i h) \mid S((i-1) h)}\left(s_{i} \mid s_{i-1}\right) \ &=\prod_{i=1}^{n} \frac{e^{-\frac{\left(\ln \left(x_{i}\right)-\ln \left(x_{i-1}\right)-\left(\mu-\sigma^{2} / 2\right)^{h}\right)^{2}}{2 \sigma^{2} h}}}{s_{i} \sqrt{2 \pi \sigma^{2} h}} \end{aligned}

## 金融代写|金融工程作业代写Financial Engineering代考|Estimation of Parameters

The maximum likelihood principle described in Appendix B.1 will now be used to estimate parameters $\mu$ and $\sigma$, using the returns as the observations. Usually this method of estimation is more precise than any other one. Since the data are Gaussian, the method of moments, described in Appendix B.8, could also be used. However these two methods yield different results in general.

Recall that the original data set are the prices $S(0)=s_{0}, S(h)=$ $s_{1}, \ldots, S(n h)=s_{n}$, from which we compute the returns $x_{i}=\ln \left(s_{i}\right)-\ln \left(s_{i-1}\right)$, $i \in{1, \ldots, n} .$

Proposition 1.3.1 The estimations of $\mu$ and $\sigma$, obtained by the maximum likelihood principle, are given by
\begin{aligned} \hat{\mu}{n} &=\frac{\bar{x}}{h}+\frac{s{x}^{2}}{2 h}, \ \hat{\sigma}{n} &=\frac{s{x}}{\sqrt{h}} \end{aligned}
where $\bar{x}=\frac{1}{n} \sum_{i=1}^{n} x_{i}$ and $s_{x}^{2}=\frac{1}{n} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}$.
The proof is given in Appendix 1.C.1.
Remark 1.3.1 In practice, $\sigma^{2}$ is estimated by
$$\frac{1}{(n-1) h} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}$$
the reason being that this estimator is unbiased, while being also quite close to $\frac{s^{2}}{h}$. For example, with MATLAB, the function std $(x)$ returns
$$\sqrt{\frac{1}{n-1} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}}$$
$$\sqrt{\frac{1}{n} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}}$$
Remark 1.3.2 When parameters are estimated using the maximum likelihood principle and the observations are not Gaussian, one rarely finds explicit expressions for the estimators. Therefore, one has to use numerical algorithms to maximize the likelihood. In this case, domain constraints on the parameters must be taken into account. For example, in the Black-Scholes model, $\sigma>0$. One can easily replace this sign constraint by setting $\sigma=e^{\alpha}$, with $\alpha \in \mathbb{R}$.

## 金融代写|金融工程作业代写Financial Engineering代考|Joint Distribution of Returns

X一世=ln⁡[小号(一世H)/小号(一世−1)H] =μH−σ22H+σ在(一世H)−在(一世−1)H ∼ñ(μH−σ22H,σ2H)

## 金融代写|金融工程作业代写Financial Engineering代考|Joint Law of Prices

f_{S{(i+1) h} \mid S(i h)}\left(x \mid s_{i}\right)=\frac{e^{-\frac{1}{2 \sigma^{ 2} h}\left{\ln \left(x / s_{i}\right)-\left(\mu-\sigma^{2} / 2\right) h\right}^{2}}}{ x \sigma \sqrt{2 \pi h}} \mathbb{I}(x>0), \quad x \in \mathbb{R} 。f_{S{(i+1) h} \mid S(i h)}\left(x \mid s_{i}\right)=\frac{e^{-\frac{1}{2 \sigma^{ 2} h}\left{\ln \left(x / s_{i}\right)-\left(\mu-\sigma^{2} / 2\right) h\right}^{2}}}{ x \sigma \sqrt{2 \pi h}} \mathbb{I}(x>0), \quad x \in \mathbb{R} 。

F小号(H),…,小号(nH)∣小号(0)(s1,…,sn∣s)=∏一世=1nF小号((一世H)∣小号(0),…,小号((一世−1)H)(s一世∣s,…,s一世−1) =∏一世=1nF小号(一世H)∣小号((一世−1)H)(s一世∣s一世−1) =∏一世=1n和−(ln⁡(X一世)−ln⁡(X一世−1)−(μ−σ2/2)H)22σ2Hs一世2圆周率σ2H

## 金融代写|金融工程作业代写Financial Engineering代考|Estimation of Parameters

μ^n=X¯H+sX22H, σ^n=sXH

1(n−1)H∑一世=1n(X一世−X¯)2

1n−1∑一世=1n(X一世−X¯)2

1n∑一世=1n(X一世−X¯)2

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