### 金融代写|金融数学作业代写Financial Mathematics代考|VALUATION BASED ON TWO-STATE INTENSITY MODEL

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## 金融代写|金融数学作业代写Financial Mathematics代考|Assumptions

The main assumptions and specifications of the framework of Jarrow and Yu (2001) are as follows:

• Default is modeled using Cox-processes from which it follows that the default is triggered if the default process $H_{t}^{j}$ jumps. The stochastic intensity of the Coxprocesses is considered as being dependent on the $d$-dimensional state variable $Y_{t}$.
• Jarrow and Yu (2001) assume equivalent recovery after the default. $\ddagger$ This recovery rate expressed as $\delta^{j} \in[0 ; 1]$ depends on the obligor and is constant over time.
• Process of the short rate is not specified but every arbitrage free (riskless) term structure model can be drawn on.

Under the assumption that no default of the obligor has been occurred so far, the price of a default-free zero bond with maturity $T$ can be expressed at time $t$ as follows:
$$D^{j}(t, T)=\delta^{j} B(t, T)+\left(1-\delta^{j}\right) E_{Q^{}}\left(\mathrm{e}^{-\int_{t}^{T}\left(r_{s}+\lambda_{s}^{j}\right) d s}\right), \quad T \geq t$$ Equation $2.5$ can only be used for debtors if their default probability is solely influenced by the macroeconomic state variables $Y_{t^{}}$ In cases in which the likelihood of the default of a debtor depends on his counterparties or another third party, Equation $2.5$ is not valid. To model firm-specific dependencies, two distinct methodologies are being discussed in the literature: unilateral or bilateral. In the latter case, $A$ influences the default probability of $B$, which itself influences the default probability of $A$ (i.e., looping default). One explanation for this would be if an interlocking participation exists between two counterparties. This kind of dependency in terms of modeling is rather difficult to handle but for the primary objective namely the modeling of a default of financial institutions not necessary. ${ }^{*}$ Given this, Jarrow and Yu (2001) focus in their analysis on a one-sided dependency, which they label primary-secondary framework.

For the above-described purpose, the set of firms $\mathbb{J}$ is being separated into two different subsets $S_{1}$ and $S_{2}$ :

• $\mathcal{S}{1} \subset \mathbb{J}$ contains all primary firms. The default probabilities of those companies depend solely on state variables $Y{t}$.
• $\mathcal{S}{2} \subset \mathbb{J}$ contains all secondary firms. The default probabilities of these companies are affected by $Y{t}$ as well as the default process of the above-described primary firms.
For the intensity of secondary firms, the following structure is assumed:
$$\lambda_{t}^{j}=a_{0, t}^{j}+\sum_{k=1}^{m^{j}} a_{k, t}^{j} \mathbf{1}{\left[\tau{k} \leq t\right\rangle}, \quad \forall j \in \mathcal{S}{2}$$ where $i$ represents the index of primary firms $j$ stands for the index of secondary firms $m^{j}$ is the number of primary firms that influence the default probability of $j$ Thereby, it is assumed that the secondary firm $j$ influences the primary firms $i{1}, i_{2}, \ldots$, $i_{m}$, and therefore, for example, the secondary firm holds equity of debt stakes of the primary firm. The impact of the defaults of primary firms depends on the sign of $a_{k, t}^{j}$. In cases in which $a_{k, t}^{j}>0$, the default of the primary firm will increase the default probability of the secondary firms. On the other side in cases in which $a_{k, t}^{j}<0$, the probability of default is going to decrease given the dependency between both groups.

## 金融代写|金融数学作业代写Financial Mathematics代考|Valuation of Credit Default Swaps

Following Jarrow and Yu (2001), we consider for our analysis a CDS inheriting the following properties:

• Zero bond of the reference party $R$ with maturity $T^{\prime}$ and face value 1 is considered as reference bond of the swap. From $R \in S_{1}$ follows that the reference party is a primary firm.
• Maturity date of the swap is expressed as $T$, with $T<T^{\prime}$.
• Risk seller A pays a periodical fixed swap-rate $\dot{\kappa}^{\prime \prime}$ until the maturity of the swap. This payment has to be paid even if a default of the reference party or the risk buyer $B$ has occurred.
• If the reference party defaults till $T, B$ sets off the loss $A$. This payment is due at the maturity of the swap. Furthermore, zero recovery is assumed for $R$ as recovery rate. From this it follows that $R$ faces a loss of $100 \%$ of his claims in the event of default.
• If during the duration, the risk buyer defaults before or at the same time as $R, B$ does not pay in the event of the default of $R$, which is equivalent to a zero recovery rate of $B$. From $\mathrm{B} \in S_{2}$ follows that the risk buyer is a secondary firm.
• If the risk seller defaults during the duration, the payment of the swap rates is going to be suspended (i.e., zero recovery rate). For simplicity it is assumed that at the default of $A$, the swap is not going to be terminated. From this it follows that in cases in which $R$ defaults during the duration of the swap at a later stage, $B$ has to pay the agreed compensation.

Each participating party $(A, B$, and $R$ ) is assumed to be inheriting a likelihood of default. The expected value of the premium payments can be calculated as follows:

$$E_{Q^{}}\left(\int_{0}^{T} \dot{\kappa}^{\prime \prime} \mathbf{1}{[\tau \mathcal{\tau}>s}} \mathrm{e}^{-\int{0}^{\mu} r_{u} \mathrm{~d} u} \mathrm{~d} s\right)=\dot{\kappa}^{\prime \prime} \int_{0}^{T} D^{\mathrm{A}}(0, s) \mathrm{d} s$$
From the risk seller’s perspective (i.e., $A$ ), the price of the swap is equivalent to the expected rate of return of the discounted payment stream received by $A$ at $t=0$ :
$$\ddot{\boldsymbol{w}}{0}^{\prime \prime}=E{Q^{}}\left(\mathbf{1}{\left[\tau^{\tilde{n}} \leq T\right}} \mathbf{1}{\left{\tau^{\mathrm{B}}>T\right}} \mathrm{e}^{-\int_{0}^{\mathrm{T}} r_{u} \mathrm{~d} u}\right)$$
This leads to the following solution for the expected rate of return:*
$$\ddot{\varpi}{0}^{\prime \prime}=D^{\mathrm{B}}(0, T)-\mathrm{e}^{-\int{0}^{T}\left(r_{s}+\lambda_{s}^{\mathrm{B}}+\lambda_{s}^{\mathrm{B}}\right) \mathrm{ds}}$$

## 金融代写|金融数学作业代写Financial Mathematics代考|Assumptions and Valuation

The rating-based model, which is described further in this section, is based on Huge and Lando (1999) as well as Lando (1998a,b). For the joint development of the rating of two obligors, a two-dimensional stochastic process described by $A_{t}=\left(A_{t}^{\mathrm{B}}, A_{t}^{R}\right)$ is assumed. The state space with a continuous-time process is given by
$$\mathbb{K}^{K}={1, \ldots, K} \times{1, \ldots, K}$$

Denote $K$ as default state. The intensity of the joint rating-transition from pair $\left(i^{\mathscr{B}}, i^{\mathbb{R}}\right) \in$ $\mathbb{K}$ to pair $\left(j^{\mathbb{B}}, j^{\mathscr{R}}\right) \in \mathbb{K}$ is denoted by $\lambda\left[\left(i^{\mathcal{B}}, i^{\mathbb{R}}\right),\left(j^{\mathbb{B}}, j^{\mathscr{R}}\right)\right]$. These intensities are combined in the $K^{2} \times K^{2}$-intensity matrix $\Lambda$. Furthermore, the subset of the state space is denoted as $\mathbb{D}=\mathbb{K}^{K} / \mathbb{K}^{K-1}$, in which at least one of the two firms is subject to default.

The price of a financial security is denoted by $S\left(t, r_{b}, A_{t}\right)$. This financial security consists of a stream of payments, which are due to at fixed dates $T_{n}$ or at those points of times where transitions in the Markov-chain take place. The maturity of the security is $T_{N}=T$. It is imperative that $A_{t} \in \mathbb{D}: S\left(t, r_{t}, A_{t}\right)=0$ because in the case of default, a final payment ensues and the contract will be ceased prematurely.

Assume a point of time just before maturity with $t \in\left[T_{N-1} ; T\right)$ but after the last fixed payment. The $K^{2}$-dimensional vector $S\left(t, r_{t}\right)$ consists of all $S\left(t, r_{t}, i\right), i \in \mathbb{K}^{K}$ and can be given by the solution of the subsequent stochastic differential equation:
$$\frac{\partial \mathbf{S}\left(t, r_{t}\right)}{\partial t}+\mu_{r} \frac{\partial \mathbf{S}\left(t, r_{t}\right)}{\partial r}+\frac{1}{2} \sigma_{r} \frac{\partial^{2} \mathbf{S}\left(t, r_{t}\right)}{\partial r^{2}}+\boldsymbol{\Lambda} \mathbf{S}\left(t, r_{t}\right)+\operatorname{diag}\left(\boldsymbol{\Lambda} \Xi_{t}^{\mathrm{T}}\right)-r_{t} \mathbf{S}\left(t, r_{t}\right)=0$$
where
diag(•) represents the vector of the diagonal elements
$\Lambda$ is an intensity matrix
$\Xi$ stands for a $K^{2} \times K^{2}$ matrix
The side condition of the stochastic differential equation (Equation 2.8) is $S\left(T, r_{t}, i\right)=$ $d\left(T, r_{T}, i\right), i \in \mathbb{K}^{K}$. The elements of the matrix $\Xi$ correspond to the payments $\hat{\boldsymbol{\omega}}\left(t, r_{t}, \mathrm{~A}{t-}, \mathrm{A}{t}\right)$ which are going to be paid at the time of rating-transitions from $\mathrm{A}{t-}$ to $\mathrm{A}{t}$. The valuation at time $t<T_{N-1}$ is based on Equation $2.8$ and was done by Lando (2000) in a recursive way. Equation $2.8$ is the basis for a numerical valuation of defaultable CDSs.

## 金融代写|金融数学作业代写Financial Mathematics代考|Assumptions

Jarrow and Yu (2001) 框架的主要假设和规范如下：

• 默认使用 Cox-processes 建模，从中可以得出，如果默认过程触发默认H吨j跳跃。Coxprocesses 的随机强度被认为取决于d维状态变量是吨.
• Jarrow 和 Yu (2001) 假设违约后的等效恢复。‡该回收率表示为dj∈[0;1]取决于债务人，并且随着时间的推移是恒定的。
• 短期利率的过程没有具体说明，但可以借鉴每个无套利（无风险）期限结构模型。

Dj(吨,吨)=dj乙(吨,吨)+(1−dj)和问(和−∫吨吨(rs+λsj)ds),吨≥吨方程2.5只能用于债务人的违约概率仅受宏观经济状态变量影响的情况是吨在债务人违约的可能性取决于其交易对手或其他第三方的情况下，等式2.5无效。为了模拟公司特定的依赖关系，文献中讨论了两种不同的方法：单边或双边。在后一种情况下，一种影响违约概率乙，它本身会影响违约概率一种（即，循环默认）。对此的一种解释是，如果两个交易对手之间存在连锁参与。这种在建模方面的依赖是相当难以处理的，但对于主要目标，即金融机构违约的建模是不必要的。∗鉴于此，Jarrow 和 Yu (2001) 将他们的分析重点放在单向依赖上，他们将其标记为小学-中学框架。

• 小号1⊂Ĵ包含所有初级公司。这些公司的默认概率仅取决于状态变量是吨.
• 小号2⊂Ĵ包含所有二级公司。这些公司的违约概率受以下因素影响是吨以及上述初级公司的默认流程。
对于二级企业的强度，假设如下结构：
λ吨j=一种0,吨j+∑ķ=1米j一种ķ,吨j1[τķ≤吨⟩,∀j∈小号2在哪里一世代表初级公司的指数j代表二级公司的指数米j是影响违约概率的初级公司的数量j因此，假设二级企业j影响初级公司一世1,一世2,…, 一世米，因此，例如，二级公司持有一级公司的债务股权。初级公司违约的影响取决于一种ķ,吨j. 在这种情况下一种ķ,吨j>0, 一级企业的违约会增加二级企业的违约概率。另一方面，在这种情况下一种ķ,吨j<0, 考虑到两组之间的依赖关系，违约概率将会降低。

## 金融代写|金融数学作业代写Financial Mathematics代考|Valuation of Credit Default Swaps

• 参考方零保证金R成熟的吨′面值 1 被视为互换的参考债券。从R∈小号1由此可见，参考方是一家主要公司。
• 互换的到期日表示为吨， 和吨<吨′.
• 风险卖方 A 支付定期固定掉期利率ķ˙′′直到互换到期。即使参考方或风险买方违约，也必须支付这笔款项乙已经发生了。
• 如果参考方违约直到吨,乙弥补损失一种. 这笔款项应在掉期到期时支付。此外，假设零恢复R作为恢复率。由此可知R面临损失100%在违约的情况下他的索赔。
• 如果在存续期间，风险买方在违约之前或同时违约R,乙在违约的情况下不支付R，这相当于零回收率乙. 从乙∈小号2由此可见，风险买方是一家二级公司。
• 如果风险卖方在存续期内违约，将暂停支付掉期利率（即零回收率）。为简单起见，假设在默认情况下一种，交换不会终止。由此可以得出，在以下情况下R在稍后阶段的掉期期间违约，乙必须支付约定的赔偿金。

\ddot{\boldsymbol{w}}{0}^{\prime \prime}=E{Q^{}}\left(\mathbf{1}{\left[\tau^{\tilde{n}} \ leq T\right}} \mathbf{1}{\left{\tau^{\mathrm{B}}>T\right}} \mathrm{e}^{-\int_{0}^{\mathrm{T }} r_{u} \mathrm{~d} u}\right)\ddot{\boldsymbol{w}}{0}^{\prime \prime}=E{Q^{}}\left(\mathbf{1}{\left[\tau^{\tilde{n}} \ leq T\right}} \mathbf{1}{\left{\tau^{\mathrm{B}}>T\right}} \mathrm{e}^{-\int_{0}^{\mathrm{T }} r_{u} \mathrm{~d} u}\right)

ϖ¨0′′=D乙(0,吨)−和−∫0吨(rs+λs乙+λs乙)ds

## 金融代写|金融数学作业代写Financial Mathematics代考|Assumptions and Valuation

ķķ=1,…,ķ×1,…,ķ

\frac{\partial \mathbf{S} \left(t, r_{t}\right)}{\partial t}+\mu_{r} \frac{\partial \mathbf{S} \left(t, r_{t}\right)}{\partial r}+\frac{1}{2} \sigma_{r} \frac{\partial^{2} \mathbf{S} \left(t, r_{t} \right)}{\partial r^{2}}+\boldsymbol{\Lambda} \mathbf{S} \left(t, r_{t}\right)+ \operatorname{diag} \left(\boldsymbol{\ Lambda} \Xi_{t}^{\mathrm{T}}\right)-r_{t} \mathbf{S} \left(t, r_{t}\right)=0


diag(•) 表示对角线元素的向量
Λ是一个强度矩阵
X代表一个ķ2×ķ2矩阵

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