### 金融代写|金融工程作业代写Financial Engineering代考|Multivariate Black-Scholes Model

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## 金融代写|金融工程作业代写Financial Engineering代考|Black-Scholes Model for Several Assets

In many cases, an option’s payoff may depend on several assets, e.g., swap options, quantos, basket options, etc. Therefore one has to model the dynamical behavior of several assets.

Definition 2.1.1 $W=\left(W_{1}, \ldots, W_{d}\right)$ is a d-dimensional Brownian motion with correlation matrix $R$ if it is a continuous random vector starting at 0 , with independent increments, and such that $W(t)-W(s) \sim N_{d}(0, R(t-s))$, whenever $0 \leq s \leq t$.

Note that it follows that the components $W_{1}, \ldots, W_{d}$ are correlated Brownian motions with
$$\operatorname{Cov}\left{W_{j}(s), W_{k}(t)\right}=R_{j k} \min (s, t), \quad s, t, \geq 0, \quad j, k \in{1, \ldots, d}$$
One can now state the extension of the Black-Scholes model to several assets.

Definition 2.1.2 The values $S_{1}, \ldots, S_{d}$ of assets are modeled by geometric Brownian motions if
$$S_{j}(t)=S_{j}(0) e^{\left(\mu_{j}-\frac{\sigma_{j}^{2}}{2}\right) t+\sigma_{j} W_{j}(t)}, \quad t \geq 0, \quad j \in{1, \ldots, d},$$
where the Brownian motions $W_{i}$ are correlated, with correlation matrix $R$. One can also say that $S=\left(S_{1}, \ldots, S_{d}\right)$ are correlated geometric Brownian

motions with parameters $\mu$ and $\Sigma$, where the covariance matrix $\Sigma$ is defined $b y$
$$\Sigma_{j k}=\sigma_{j} \sigma_{k} R_{j k}, \quad j, k \in{1, \ldots, d}$$
Remark 2.1.1 As in the one-dimensional case, the geometric Brownian motions satisfy the following system of stochastic differential equations:
$$d S_{j}(t)=\mu_{j} S_{j}(t) d t+\sigma_{j} S_{j}(t) d W_{j}(t), \quad j \in{1, \ldots, d} .$$

## 金融代写|金融工程作业代写Financial Engineering代考|Representation of a Multivariate Brownian Motion

Using the properties of Gaussian random vectors (Section A.6.18), a $d-$ dimensional Brownian motion $W$ with correlation matrix $R$ can be constructed as a linear combination of $d$ independent univariate Brownian motions $Z=$ $\left(Z_{1}, \ldots, Z_{d}\right)^{\top}$ by setting $W(t)=b^{\top} Z(t)$, where $b^{\top} b=R$. We can then rewrite (2.1) as
$$\frac{S_{j}(t)}{S_{j}(0)}=e^{\left(\mu_{j}-\frac{\sigma_{1}^{2}}{2}\right) t+\sigma_{j} \sum_{k=1}^{d} b_{k j} Z_{k}(t)}, \quad t \geq 0, \quad j \in{1, \ldots, d}$$
An interesting decomposition of $R=b^{\top} b$ is when $b$ is an upper triangular matrix, as in Cholesky decomposition. In this case,
$$\frac{S_{j}(t)}{S_{j}(0)}=e^{\left(\mu_{j}-\frac{\Sigma_{j j}}{2}\right) t+\sum_{k=1}^{j} a_{k j} Z_{k}(t)}, \quad t \geq 0, \quad j \in{1, \ldots, d},$$
where the Brownian motions $Z_{1}, \ldots, Z_{d}$ are independent, and $a_{j k}=\sigma_{k} b_{j k}$. The matrix $a$ is upper triangular, contains all the information on the dependence, and is uniquely determined by the Cholesky decomposition $a^{\top} a=\Sigma$, i.e..
$$\Sigma_{j k}=\left(a^{\top} a\right){j k}=\sum{l=1}^{\min (j, k)} a_{l j} a l k=\sigma_{j} \sigma_{k} R_{j k}, \quad j, k \in{1, \ldots, d}$$
under the constraints $a_{j j}>0, j \in{1, \ldots, d}$. In the bivariate case, one can check that $a=\left(\begin{array}{cc}\sigma_{1} & R_{12} \sigma_{2} \ 0 & \sigma_{2} \sqrt{1-R_{12}^{2}}\end{array}\right)$.

## 金融代写|金融工程作业代写Financial Engineering代考|Numerical Method

In general, we cannot compute explicitly the maximum likelihood estimates of a model, so we have to rely on numerical methods for optimization. One of the main problems encountered with numerical methods for optimization is the problem of constraints on the parameters.

For example, for model (2.1), the matrix $R$ must be positive definite and symmetric. In two dimensions, this condition is simple since the correlation matrix $R$ is determined by the unique number $\rho=R_{12}$. In this case, one needs to assume that the correlation coefficient $\rho$ is in the interval $(-1,1)$. To get rid of this constraint, we can set $\rho=\tanh (\alpha)$. Its inverse, called the Fisher transformation, is $\alpha=\frac{1}{2} \ln \left(\frac{1+\rho}{1-\rho}\right)$. See Figure $2.1$.

For higher dimensions, it is impossibly difficult to find explicit constraints on a matrix coefficients to ensure that it is positive definite. This is where representation (2.4) becomes interesting. The only constraint on the upper triangular matrix $a$ is that its diagonal is positive. As parameters are expressed by vectors, it is therefore natural to work with the volatility vector $v$.

In addition, as we will see later, it is relatively easy to compute the estimator error on option prices in terms of the error on $v$.

## 金融代写|金融工程作业代写Financial Engineering代考|Black-Scholes Model for Several Assets

\operatorname{Cov}\left{W_{j}(s), W_{k}(t)\right}=R_{j k} \min (s, t), \quad s, t, \geq 0, \四边形 j, k \in{1, \ldots, d}\operatorname{Cov}\left{W_{j}(s), W_{k}(t)\right}=R_{j k} \min (s, t), \quad s, t, \geq 0, \四边形 j, k \in{1, \ldots, d}

Σjķ=σjσķRjķ,j,ķ∈1,…,d

d小号j(吨)=μj小号j(吨)d吨+σj小号j(吨)d在j(吨),j∈1,…,d.

## 金融代写|金融工程作业代写Financial Engineering代考|Representation of a Multivariate Brownian Motion

Σjķ=(一种⊤一种)jķ=∑l=1分钟(j,ķ)一种lj一种lķ=σjσķRjķ,j,ķ∈1,…,d

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