### 金融代写|利率建模代写Interest Rate Modeling代考|MTH5520

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## 金融代写|利率建模代写Interest Rate Modeling代考|Multi-Factor Extensions

In derivatives pricing, we often need to model simultaneously the dynamics of multiple risky securities, using multiple risk factors. Because of that, we must extend several major results established so far to the setting of multiple risk sources or assets. These results include the CMG theorem, the martingale representation theorem, and the option pricing formula, as in Equation 2.52. The proofs are parallel to those for the one-dimensional case and thus are omitted for brevity. Hereafter, we use a superscript “T” to denote the transposition of a matrix.

Theorem 2.6.1 (The CMG Theorem). Let $\mathbf{W}{t}=\left(W{1}(t), W_{2}(t), \ldots\right.$,

$\left.W_{n}(t)\right)^{\mathrm{T}}$ be an $n$-dimensional $\mathbb{P}$-Brownian motion, and let $\gamma_{t}=\left(\gamma_{1}(t)\right.$, $\left.\gamma_{2}(t), \ldots, \gamma_{n}(t)\right)^{\mathrm{T}}$ be an $n$-dimensional $\mathcal{F}{t}$-adaptive process, such that $$E^{P}\left[\exp \left(\frac{1}{2} \int{0}^{T}\left|\gamma_{t}\right|_{2}^{2} d t\right)\right]<\infty$$
Define a new measure, Q1, with a Radon-Nikodym derivative
$$\left.\frac{d \mathbb{Q}}{d \mathbb{P}}\right|{\mathcal{F}{t}}=\exp \left(\int_{0}^{t}-\gamma_{s}^{\mathrm{T}} d \mathbf{W}{s}-\frac{1}{2} \int{0}^{t}\left|\gamma_{s}\right|_{2}^{2} d s\right) \text {. }$$
Then $\mathbb{Q}$ is equivalent to $\mathbb{P}$, and
$$\tilde{\mathbf{W}}{t}=\mathbf{W}{t}+\int_{0}^{t} \gamma_{s} d s$$
is an n-dimensional (1-Brownian motion.
Theorem 2.6.2 (The Martingale Representation Theorem). Let $\mathbf{W}{t}$ be an $n$-dimensional Brownian motion and suppose that $\mathbf{M}{t}$ is an $n$-dimensional Q-martingale process, $\mathbf{M}{t}=\left(M{1}(t), M_{2}(t), \ldots, M_{n}(t)\right)^{\mathrm{T}}$, such that
$$d M_{i}(t)=\sum_{j=1}^{n} a_{i j}(t) d W_{j}(t) .$$
Let $\mathbf{A}=\left(a_{i j}\right)$ be a non-singular matrix. If $N_{t}$ is any one-dimensional $\mathbb{Q}$ martingale with $E^{Q}\left[N_{t}^{2}\right]<\infty$, there exists an $n$-dimensional $\mathcal{F}{t}$-adaptive process, $\Phi{t}=\left(\varphi_{1}(t), \varphi_{2}(t), \ldots, \varphi_{n}(t)\right)^{\mathrm{T}}$, such that
$$E^{Q}\left[\int_{0}^{t}\left(\sum_{j} a_{i j}^{2}(s) \varphi_{j}^{2}(s) d s\right)\right]<\infty, \quad \forall i,$$
and
\begin{aligned} N_{t} &=N_{0}+\sum_{j=1}^{n} \int_{0}^{t} \varphi_{j}(s) d M_{j}(s) \ & \triangleq N_{0}+\int_{0}^{t} \Phi^{\mathrm{T}}(s) d \mathbf{M}(s) . \end{aligned}

## 金融代写|利率建模代写Interest Rate Modeling代考|Existence of a Martingale Measure

We consider a standard model of a complete financial market with a money market account and $n$ risky securities. Let the time $t$ prices be $B_{t}$ and $S_{t}^{i}$, $1 \leq i \leq n$, respectively. We assume lognormal price processes for all assets:
\begin{aligned} \mathrm{d} B_{t} &=r_{t} B_{t} \mathrm{~d} t \ \mathrm{~d} S_{t}^{i} &=S_{t}^{i}\left(\mu_{t}^{i} \mathrm{~d} t+\sum_{j=1}^{n} \sigma_{i j} \mathrm{~d} W_{j}(t)\right) \ &=S_{t}^{i}\left(\mu_{t}^{i} \mathrm{~d} t+\sigma_{i}^{\mathrm{T}}(t) \mathrm{d} \mathbf{W}{t}\right), \quad i=1,2, \ldots, n . \end{aligned} Here, $$\sigma{i}^{\mathrm{T}}(t)=\left(\sigma_{i, 1}, \sigma_{i, 2}, \ldots, \sigma_{i, n}\right) .$$
Let $Z_{t}^{i}=B_{t}^{-1} S_{t}^{i}$ denote the discounted asset price of the $i$ th asset. It then follows that
$$\mathrm{d} Z_{t}^{i}=Z_{t}^{i}\left[\boldsymbol{\sigma}{i}^{\mathrm{T}}(t) \mathrm{d} \mathbf{W}{t}+\left(\mu_{t}^{i}-r_{t}\right) \mathrm{d} t\right], \quad i=1,2, \ldots, n$$
To construct a martingale measure for $Z_{t}^{i}, \forall i$, we must “absorb” the drift terms in Equation $2.62$ into the Brownian motion. For that reason, we define an $\mathcal{F}{t}$-adaptive function, $\gamma{t}$, via the following equations:
$$\boldsymbol{\sigma}{i}^{\mathrm{T}}(t) \gamma{t}=\mu_{t}^{i}-r_{t}, \quad i=1,2, \ldots, n .$$
Suppose that $\gamma_{t}$, the solution to Equation 2.63, exists and satisfies
$$E^{2}\left[\exp \left(\int_{0}^{T}\left|\gamma_{t}\right|^{2} \mathrm{~d} t\right)\right]<\infty$$ for some $T>0$. We then can define a new measure, $\mathbb{Q}$, according to Equation 2.59. Under this newly defined $\mathbb{Q}$,
$$\overline{\mathbf{W}}{t}=\mathbf{W}{t}+\int_{0}^{t} \gamma_{s} \mathrm{~d} s$$
is a multi-dimensional Brownian motion, with which we can rewrite the price processes for the discounted assets into
$$\mathrm{d} Z_{t}^{i}=Z_{t}^{i} \sigma_{i}^{\mathrm{T}}(t) \mathrm{d} \hat{\mathbf{W}}{t}, \quad i=1,2, \ldots, n,$$ and $Z{t}^{i}, i=1,2, \ldots, n$ are lognormal $\mathbb{Q}$-martingales.
We now study the existence of $\gamma_{t}$ and condition 2.64. In matrix form, Equation $2.63$ can be recast into
$$\boldsymbol{\Sigma} \gamma_{t}=\boldsymbol{\mu}{t}-r{t} \mathbf{I}$$

## 金融代写|利率建模代写Interest Rate Modeling代考|Pricing Contingent Claims

Now we are ready to address the pricing of a contingent claim depending on the prices of multiple underlying securities. Having found the martingale measure, $\mathbb{Q}$, for the underlying securities, we define a $\mathbb{Q}$-martingale as
$$N_{t}=E^{Q}\left(B_{T}^{-1} X_{T} \mid \mathcal{F}{t}\right),$$ using the discounted value of $X{T}$, the payoff function of the claim at time $T$. Without loss of generality, we assume that the volatility matrix of the underlying risky securities, $\Sigma$, is non-singular. ${ }^{2}$ According to the

martingale representation theorem, there exists an $\mathcal{F}{t}$-adaptive function, $\Phi{t}=\left(\varphi_{1}(t), \ldots, \varphi_{n}(t)\right)^{\mathrm{T}}$, such that
$$\mathrm{d} N_{t}=\Phi_{t}^{\mathrm{T}} \mathrm{d} \mathbf{Z}{t}$$ where $\mathbf{Z}{t}$ is the vector of the discounted prices. We now define another process,
$$\psi_{t}=N_{t}-\boldsymbol{\Phi}{t}^{\mathrm{T}} \mathbf{Z}{t}$$
and form a portfolio with $\psi_{t}$ units of the money market account and $\phi_{i}(t)$ units of the $i$ th risky security, $i=1, \ldots, n$. The discounted value of the portfolio is
$$\tilde{V}{t}=\Phi{t}^{\mathrm{T}} \mathbf{Z}{t}+\psi{t}=N_{t}$$
The last equation implies replication of the payoff of the contingent portfolio. Furthermore, from Equation 2.70, we can derive
$$\mathrm{d} V_{t}=\Phi_{t}^{\mathrm{T}} \mathrm{d} \mathbf{S}{t}+\psi{t} \mathrm{~d} B_{t}$$
which implies that the replication strategy is a self-financing one. So, we conclude that the value of the contingent claim equals that of the portfolio and thus is given by
$$V_{t}=B_{t} E^{\mathbb{Q}}\left[B_{T}^{-1} X_{T} \mid \mathcal{F}{t}\right]=E^{Q}\left[\mathrm{e}^{-\int{t}^{T} r_{x} \mathrm{~d}{s}} X{T} \mid \mathcal{F}_{t}\right]$$
Formally, Equation $2.71$ is identical to Equation $2.52$, the formula for options on a single underlying security.

## 金融代写|利率建模代写Interest Rate Modeling代考|Multi-Factor Extensions

d问d磷|F吨=经验⁡(∫0吨−Cs吨d在s−12∫0吨|Cs|22ds).

d米一世(吨)=∑j=1n一个一世j(吨)d在j(吨).

ñ吨=ñ0+∑j=1n∫0吨披j(s)d米j(s) ≜ñ0+∫0吨披吨(s)d米(s).

## 金融代写|利率建模代写Interest Rate Modeling代考|Existence of a Martingale Measure

d乙吨=r吨乙吨 d吨  d小号吨一世=小号吨一世(μ吨一世 d吨+∑j=1nσ一世j d在j(吨)) =小号吨一世(μ吨一世 d吨+σ一世吨(吨)d在吨),一世=1,2,…,n.这里，

σ一世吨(吨)=(σ一世,1,σ一世,2,…,σ一世,n).

d从吨一世=从吨一世[σ一世吨(吨)d在吨+(μ吨一世−r吨)d吨],一世=1,2,…,n

σ一世吨(吨)C吨=μ吨一世−r吨,一世=1,2,…,n.

d从吨一世=从吨一世σ一世吨(吨)d在^吨,一世=1,2,…,n,和从吨一世,一世=1,2,…,n是对数正态的问- 鞅。

ΣC吨=μ吨−r吨我

## 金融代写|利率建模代写Interest Rate Modeling代考|Pricing Contingent Claims

ñ吨=和问(乙吨−1X吨∣F吨),使用贴现值X吨, 索赔在时间的支付函数吨. 不失一般性，我们假设基础风险证券的波动率矩阵，Σ, 是非奇异的。2根据

dñ吨=披吨吨d从吨在哪里从吨是折扣价格的向量。我们现在定义另一个过程，

ψ吨=ñ吨−披吨吨从吨

d在吨=披吨吨d小号吨+ψ吨 d乙吨

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