### 金融代写|金融数学作业代写Financial Mathematics代考|VALUATION BASED ON MERTON’S APPROACH

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

Hull and White (2001) focus on CDSs with periodical risk-free premium payments in arrear. Moreover, the protection buyer is regarded as a risk-free agent and therefore the payments of the buyer are risk-free as well.
The following swap payment streams are considered for our analysis:

1. Risk seller $A$ pays the swap rate till maturity $T$. If the first default occurs before $T$ then A terminates his regular payments and
a. pays a final accrual payment ${ }^{*}$ of $e$, if the defaulted party is the reference obligator $R$, otherwise
b. pays no final accrual payment, if the risk buyer $B$ defaults.
2. If $R$ defaults during the duration of the swap and before $B$, then $B$ pays the difference between the face value of the debt claim of $A$ against $R$ and the recovery rate. The recovery rate is calculated as the face value of the bond including accrued interests.
3. If $B$ defaults before $R$ during the duration, then no payment will be made and the swap is going to be terminated.
At $t=0$ the present value of the expected premium paid by $A$ is calculated as follows:
$$\dddot{\kappa}{\rho v}^{\prime}=\dddot{\kappa}^{\prime}\left(\int{0}^{T}\left{f_{t}^{R B}\left[u_{t}+m_{t}\left(e w^{-1}\right)\right]+f_{t}^{\mathrm{BR}} u_{t}\right} \mathrm{d} t+G_{T}^{\mathrm{BR}} u_{T}\right)$$
where, $w$ represents the annual payments of the premium calculated as a proportion of the face value of the swaps, $f_{t}^{R B}$ is the default density of $R$ under the condition that $B$ did not default until $t=0,{ }^{\dagger} f_{t}^{\mathrm{B} R}$ stands for the default density function of $B$ under the assumption that $R$ do not default until $t=0, \mp u_{t}$ represents the present value of the annuity factor for annuity payments of 1 within the period $[0 ; t], m_{t}\left(e^{-1}\right)$ describes the present value factor of an one-off final payment of $e w^{-1}$ in $t$, and $G_{T}^{\mathrm{BR}}$ is the probability that the counterparty $B$ and the reference entity $R$ do not default until $T$.

The value of the swap for $A$ is calculated as the present value of the expected payments generated by the swap as follows:
$$\ddot{\varpi}{0}^{\prime}=\int{0}^{T}\left[1-E\left(\delta_{t}^{R}\right)\left(1+c_{t}\right)\right] f_{t}^{R B} m_{t}(1) \mathrm{d} t$$

where $\delta_{t}^{R}$ is the expected recovery rate at the default of $R$ and $c_{t}$ represents the interest accruals until $t$ is calculated as a percentage of the face value on the debt claims.

Hull and White (2001) calculate the fair swap-rate $\dddot{\kappa}{\rho \nu}^{\prime}$ drawing on Equations $2.2$ and $2.3$ through using a Monte Carlo simulation. They use different sets of parameters first for the quality of the risk buyer and second for the correlation between the risk buyer and the reference entity. This fair swap rate equals $w$, from which it follows that $\ddot{\varpi}{0}^{\prime}=\dddot{\kappa}_{\rho v^{\prime}}$ High correlations between the risk buyer and the reference entity as well as low ratings of the swap seller show a significant impact on the swap rate.

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

To evaluate vulnerable puts, different methods (e.g., Johnson and Stulz 1987, Klein 1996, Klein and Inglis 1999) based on Merton’s (1974) credit risk model have been proposed. ${ }^{\dagger}$ Looking at defaultable CDSs, both credit risks have to be considered explicitly: First, the credit risk of the reference asset and second the credit risk of the risk buyer have to be evaluated. It should be noted that the valuation of these credit risks does not have to be based on an identical methodology. For the purpose of this review chapter, we assume a standard Merton (1974) firm value model:

• Firm value of the risk buyer $V_{t}^{\mathrm{B}}$ follows a lognormal distribution. The process of $V_{t}^{\mathrm{B}}$ can be expressed as follows: $\mathrm{d} V_{t}^{R}=V_{t}^{\mathrm{B}} \mu_{V^{\mathrm{B}}} \mathrm{d} t+V_{t}^{\mathrm{B}} \sigma_{V^{\mathrm{B}}} \mathrm{d} W_{t}^{V^{\mathrm{B}}}$.
• Firm value of the reference party $V_{t}^{R}$ follows a lognormal distribution. We are therefore able to use the following expression:
$$\mathrm{d} V_{t}^{R}=V_{t}^{R} \mu_{V^{R}} \mathrm{~d} t+V_{t}^{R} \sigma_{V^{R}} \mathrm{~d} W_{t}^{V^{\mathbb{R}}}$$
The correlation coefficient between the firm values $B$ and $R$, both of which follow a Brownian motion, is denoted as $\rho_{\mathrm{B} R}$.Risk buyer $B$ defaults only at $T$. The default occurs, if the firm’s value $V_{t}^{\mathrm{B}}$ drops below an assessed fixed default barrier. This threshold level differs, as shown below, between the various models used to evaluate vulnerable puts:
• a. Option (a): The claim of the swap can be used as the default barrier. This is equivalent to the assumption that the default swap is the sole liability of the option writer and therefore this follows Johnson and Stulz (1987).
• b. Option (b): Alternatively, the total amount of all liabilities of the risk buyer can be drawn on as the default barrier. If the liabilities are assumed to be constant over time, then this is equivalent to the assumption that the claim of the swap is negligibly small. This is based on the model of Klein (1996) and Klein and Inglis $(1999) ^{}$
• c. Option (c): Moreover, the threshold level can be set equivalent to the sum of all other liabilities of the option writer additional to the claim on the swap. This is based on the approach of Klein and Inglis (2001).
• Reference party $R$ can only default at $T$. The default occurs, if the firm value $V_{T}^{R}$ drops below the value of the liabilities.
• At default, the recovery rate $\delta_{T}^{\mathrm{B}}\left(V_{T}^{\mathrm{B}}\right)$ of the risk taker $B$ is calculated as the ratio of the firm value $V_{T}^{\mathrm{B}}$ and the total sum of all liabilities multiplied with the factor $(1-\alpha)$. The latter factor represents the dead-weighted costs associated with the default.
• At default, the recovery rate of $R$ corresponds to the ratio of $V_{T}^{R}$ to the overall liabilities.

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

The value of the CDS, $\ddot{\varpi}{0}$, can be derived from the discounted expected stream of payments. The expected values each depending on the default barrier of $\mathrm{B}$ can be expressed as follows: \begin{aligned} \ddot{\varpi}{0}^{\prime}=& E_{p^{}}\left{B_{T}^{-1}\left[\left(F-V_{T}^{R}\right)^{+} \mathbf{1}{\left{V{T}^{\mathrm{B}} \geq F-V_{T}^{\mathbb{\pi}}\right}}+V_{T}^{\mathrm{B}} \mathbf{1}{\left{V{T}^{\mathrm{B}}{0}^{\prime \prime}=& E{p^{}}\left{B_{T}^{-1}\left[\left(F-V_{T}^{R}\right)^{+} \mathbf{1}{\left{V{T}^{\mathrm{B}} \geq D^{\mathrm{B}}\right}}+(1-\alpha) \frac{V_{T}^{\mathrm{B}}}{D^{}}\left(F-V_{T}^{R}\right)+\mathbf{1}{\left{V{T}^{\mathrm{B}}{0}^{\prime \prime \prime}=& E{p^{}}\left[B_{T}^{-1}\left(\left(F-V_{T}^{R}\right)^{+} \mathbf{1}{\left{V{T}^{\mathrm{B}} \geq D^{\mathrm{B}}+F-V_{T}^{\mathrm{R}}\right.}\right}\right.\ &\left.\left.+(1-\alpha) \frac{V_{T}^{\mathrm{B}}}{D^{}+\left(F-V_{T}^{R}\right)}\left(F-V_{T}^{R}\right)^{+} \mathbf{1}{\left{V{T}^{\mathrm{B}}{0}^{\prime} to \ddot{\varpi}{0}^{\prime \prime \prime} represent the above described options (a) to (c), respectively. The CDS equals considering its payout-structure a defaultable put option, which is expressed as \tilde{P}{0}\left(F, V{t}^{R}\right). Owing to this, the equations for European put options derived by Johnson and Stulz (1987), Klein (1996), and Klein and Inglis (2001) can be used to determine the expected rate of return in Equation 2.4. We use \ddot{\varpi} 0_{0}^{\prime \prime} below as illustrative example: \begin{aligned} \ddot{\omega}{t}^{\prime \prime}=&-V{t}^{R} \Phi_{2}\left(-d_{1}, b_{1},-p\right)+F \mathrm{e}^{-r(T-t)} \Phi_{2}\left(-d_{2}, b_{2},-p\right) \ &-(1-\alpha) \frac{V_{t}^{\mathrm{B}}}{D^{\star}} V_{t}^{R} \mathrm{e}^{p \sigma} v^{\mathrm{B}^{\alpha}} v^{R^{(T-t)}} \mathrm{e}^{r(T-t)} \Phi_{2}\left(-\tilde{d}{1}, \tilde{b}{1}, p\right) \ &+(1-\alpha) \frac{V_{t}^{\mathrm{B}}}{D^{\star}} F \Phi_{2}\left(-\tilde{d}{2}, \tilde{b}{2}, p\right) \end{aligned} $$where \Phi_{2}(\cdot) is the function with a standard bivariate normal distribution and d_{1}, d_{2}, b_{1}, b_{2}, t^{}, and p are give by: d_{1}=\frac{\ln \left(V_{t}^{R} / F\right)+\left(r+\frac{1}{2} \sigma_{V^{R}}^{2}\right) t^{}}{\sigma_{V^{\mathbb{R}} \sqrt{t^{}}}}=d_{1}\left(t^{}, V_{t}^{R}\right), \quad \tilde{d}{1}=d{1}+p \sigma_{V^{\mathrm{B}}} \sqrt{t^{}} d_{2}=d_{1}-\sigma_{V^{\mathbb{R}} \sqrt{t^{}}} \quad \tilde{d}{2}=d{2}+p \sigma_{V^{\mathrm{B}}} \sqrt{t^{}} b_{1}=\frac{\ln \left(V_{t}^{\mathrm{B}} / D^{}\right)+\left(r-\frac{1}{2} \sigma_{V^{\mathrm{B}}}^{2}+p \sigma_{V^{\mathrm{B}}} \sigma_{V^{\mathrm{R}}}\right) t^{}}{\sigma_{V^{\mathrm{B}}} \sqrt{t^{}}}=b_{1}\left(t^{\star}, V_{t}^{\mathrm{B}}\right), \quad \tilde{b}{1}=-b{1}-\sigma_{V^{\mathrm{B}}} \sqrt{t^{\star}} b_{2}=b_{1}-p \sigma_{V^{\mathbb{n}}} \sqrt{t^{}}, \tilde{b}{2}=-b{2}-\sigma_{V^{\mathrm{B}}} \sqrt{t^{\star}} t^{*}=T-t \rho=\rho_{\mathrm{BR}} This equation follows directly from Klein’s (1996) equation for European put options with a defaultable option writer and a constant interest rate. Within this framework, the firm value V of the option writer, the value of the underlying U, and the strike price K are specified as follows:*$$
\begin{gathered}
V_{t}=V_{t}^{\mathrm{B}} \
U_{t}=V_{t}^{R} \
K=F
\end{gathered}
$$In the case of stochastic interest rates, the equation framework of Klein and Inglis (1999) has to be used instead of Klein’s framework. { }^{\dagger} ## 金融数学代写 ## 金融代写|金融数学作业代写Financial Mathematics代考|Valuation of Credit Default Swaps Hull 和 White (2001) 专注于定期无风险支付拖欠保费的 CDS。此外，保护买方被视为无风险代理人，因此买方的付款也是无风险的。 我们的分析考虑了以下掉期支付流： 1. 风险卖方一种支付掉期利率直到到期吨. 如果第一个默认值发生在之前吨然后 A 终止他的定期付款和 a。支付最终应计付款∗的和, 如果违约方是参考义务人R, 否则 b. 如果风险买方不支付最终应计付款乙默认值。 2. 如果R在交换期间和之前的默认值乙， 然后乙支付债权面值的差额一种反对R和恢复率。回收率按包括应计利息在内的债券面值计算。 3. 如果乙之前的默认值R在此期间，将不支付任何款项，并且将终止掉期。 在吨=0支付的预期保费的现值一种计算如下： \dddot{\kappa}{\rho v}^{\prime}=\dddot{\kappa}^{\prime}\left(\int{0}^{T}\left{f_{t}^{R B }\left[u_{t}+m_{t}\left(e w^{-1}\right)\right]+f_{t}^{\mathrm{BR}} u_{t}\right} \mathrm {d} t+G_{T}^{\mathrm{BR}} u_{T}\right)\dddot{\kappa}{\rho v}^{\prime}=\dddot{\kappa}^{\prime}\left(\int{0}^{T}\left{f_{t}^{R B }\left[u_{t}+m_{t}\left(e w^{-1}\right)\right]+f_{t}^{\mathrm{BR}} u_{t}\right} \mathrm {d} t+G_{T}^{\mathrm{BR}} u_{T}\right) 在哪里，在表示按掉期面值的比例计算的每年支付的保费，F吨R乙是默认密度R条件下乙直到没有违约吨=0,†F吨乙R代表默认的密度函数乙在假设R不要默认，直到吨=0,∓在吨表示该期间内年金支付为 1 的年金因子的现值[0;吨],米吨(和−1)描述一次性最终付款的现值因子和在−1在吨， 和G吨乙R是交易对手的概率乙和参考实体R不要默认，直到吨. 掉期的价值一种计算为掉期产生的预期支付的现值，如下所示： ϖ¨0′=∫0吨[1−和(d吨R)(1+C吨)]F吨R乙米吨(1)d吨 在哪里d吨R是默认情况下的预期回收率R和C吨表示应计利息，直到吨按债权面值的百分比计算。 Hull and White (2001) 计算公平掉期利率ķ⃛ρν′绘制方程2.2和2.3通过使用蒙特卡罗模拟。他们首先对风险购买者的质量使用不同的参数集，其次对风险购买者和参考实体之间的相关性使用不同的参数集。这个公平的掉期利率等于在，由此得出ϖ¨0′=ķ⃛ρ在′风险买方和参考实体之间的高相关性以及掉期卖方的低评级表明对掉期利率有重大影响。 ## 金融代写|金融数学作业代写Financial Mathematics代考|Assumptions 为了评估易受攻击的看跌期权，基于 Merton (1974) 信用风险模型提出了不同的方法（例如 Johnson 和 Stulz 1987、Klein 1996、Klein 和 Inglis 1999）。†对于可违约 CDS，必须明确考虑两种信用风险：首先，必须评估参考资产的信用风险，其次必须评估风险买方的信用风险。应该注意的是，这些信用风险的估值不必基于相同的方法。出于本章回顾的目的，我们假设一个标准的 Merton (1974) 公司价值模型： • 风险买方的公司价值在吨乙服从对数正态分布。的过程在吨乙可以表示如下：d在吨R=在吨乙μ在乙d吨+在吨乙σ在乙d在吨在乙. • 参考方的公司价值在吨R服从对数正态分布。因此，我们可以使用以下表达式： d在吨R=在吨Rμ在R d吨+在吨Rσ在R d在吨在R 公司价值之间的相关系数乙和R，两者都遵循布朗运动，记为ρ乙R.风险买家乙默认仅在吨. 默认发生，如果公司的价值在吨乙跌至评估的固定默认障碍以下。如下所示，用于评估易受攻击看跌期权的各种模型之间的阈值水平不同： • 一种。选项（a）：可以将交换的声明用作默认障碍。这相当于假设违约掉期是期权卖方的唯一责任，因此遵循 Johnson 和 Stulz (1987)。 • 湾。选项（b）：或者，可以将风险买方的所有负债总额作为违约障碍。如果假设负债随着时间的推移是恒定的，那么这相当于假设掉期的债权可以忽略不计。这是基于 Klein (1996) 和 Klein and Inglis (1999) ^{ }的模型 • C。选项（c）：此外，阈值水平可以设置为等于期权卖方除掉期债权之外的所有其他负债的总和。这是基于 Klein 和 Inglis (2001) 的方法。 • 参考方R只能默认为吨. 默认发生，如果坚定的价值在吨R低于负债的价值。 • 默认情况下，恢复率d吨乙(在吨乙)冒险者的乙计算为公司价值的比率在吨乙和所有负债的总和乘以因子(1−一种). 后一个因素代表与违约相关的无谓加权成本。 • 默认情况下，恢复率为R对应的比例在吨R对总负债。 ## 金融代写|金融数学作业代写Financial Mathematics代考|Valuation of Credit Default Swaps CDS的价值，ϖ¨0, 可以从贴现的预期支付流中推导出来。每个期望值取决于乙可以表示如下：$$ \begin{aligned} \ddot{\varpi}{0}^{\prime}=& E_{p^{}}\left{B_{T}^{-1}\left [\left(F-V_{T}^{R}\right)^{+} \mathbf{1}{\left{V{T}^{\mathrm{B}} \geq F-V_{T} ^{\mathbb{\pi}}\right}}+V_{T}^{\mathrm{B}} \mathbf{1}{\left{V{T}^{\mathrm{B}}{0} ^{\prime \prime}=& E{p^{}}\left{B_{T}^{-1}\left[\left(F-V_{T}^{R}\right)^{+ } \mathbf{1}{\left{V{T}^{\mathrm{B}} \geq D^{\mathrm{B}}\right}}+(1-\alpha) \frac{V_{T }^{\mathrm{B}}}{D^{}}\left(F-V_{T}^{R}\right)+\mathbf{1}{\left{V{T}^{\mathrm {B}}{0}^{\prime \prime \prime}=& E{p^{}}\left[B_{T}^{-1}\left(\left(F-V_{T}^ {R}\right)^{+} \mathbf{1}{\left{V{T}^{\mathrm{B}} \geq D^{\mathrm{B}}+F-V_{T}^ {\mathrm{R}}\right.}\right}\right.\
&\left.\left.+(1-\alpha) \frac{V_{T}^{\mathrm{B}}}{D^{}+\left(F-V_{T}^{R}\右)}\left(F-V_{T}^{R}\right)^{+} \mathbf{1}{\left{V{T}^{\mathrm{B}}{0}^{\主要}吨这\ddot {\varpi} {0} ^ {\素数\素数\素数}r和pr和s和n吨吨H和一种b这在和d和sCr一世b和d这p吨一世这ns(一种)吨这(C),r和sp和C吨一世在和l是.吨H和CD小号和q在一种lsC这ns一世d和r一世nG一世吨sp一种是这在吨−s吨r在C吨在r和一种d和F一种在l吨一种bl和p在吨这p吨一世这n,在H一世CH一世s和Xpr和ss和d一种s\波浪号{P}{0}\left(F, V{t}^{R}\right).这在一世nG吨这吨H一世s,吨H和和q在一种吨一世这nsF这r和在r这p和一种np在吨这p吨一世这nsd和r一世在和db是Ĵ这Hns这n一种nd小号吨在l和(1987),ķl和一世n(1996),一种ndķl和一世n一种nd一世nGl一世s(2001)C一种nb和在s和d吨这d和吨和r米一世n和吨H和和Xp和C吨和dr一种吨和这Fr和吨在rn一世n和q在一种吨一世这n2.4.在和在s和\ ddot {\ varpi} 0_ {0} ^ {\素数\素数}b和l这在一种s一世ll在s吨r一种吨一世在和和X一种米pl和:ω¨吨′′=−在吨R披2(−d1,b1,−p)+F和−r(吨−吨)披2(−d2,b2,−p) −(1−一种)在吨乙D⋆在吨R和pσ在乙一种在R(吨−吨)和r(吨−吨)披2(−d~1,b~1,p) +(1−一种)在吨乙D⋆F披2(−d~2,b~2,p)在H和r和\Phi_{2}(\cdot)一世s吨H和F在nC吨一世这n在一世吨H一种s吨一种nd一种rdb一世在一种r一世一种吨和n这r米一种ld一世s吨r一世b在吨一世这n一种ndd_{1}、d_{2}、b_{1}、b_{2},t^{},一种ndp一种r和G一世在和b是:d_{1}=\frac{\ln \left(V_{t}^{R} / F\right)+\left(r+\frac{1}{2} \sigma_{V^{R}}^{ 2}\right) t^{}}{\sigma_{V^{\mathbb{R}} \sqrt{t^{}}}}=d_{1}\left(t^{}, V_{t} ^{R}\right), \quad \tilde{d}{1}=d{1}+p \sigma_{V^{\mathrm{B}}} \sqrt{t^{}}d_{2}=d_{1}-\sigma_{V^{\mathbb{R}} \sqrt{t^{}}} \quad \tilde{d}{2}=d{2}+p \sigma_ {V^{\mathrm{B}}} \sqrt{t^{}}b_{1}=\frac{\ln \left(V_{t}^{\mathrm{B}} / D^{}\right)+\left(r-\frac{1}{2} \sigma_{ V^{\mathrm{B}}}^{2}+p \sigma_{V^{\mathrm{B}}} \sigma_{V^{\mathrm{R}}}\right) t^{}} {\sigma_{V^{\mathrm{B}}} \sqrt{t^{}}}=b_{1}\left(t^{\star}, V_{t}^{\mathrm{B}} \right), \quad \tilde{b}{1}=-b{1}-\sigma_{V^{\mathrm{B}}} \sqrt{t^{\star}}b_{2}=b_{1}-p \sigma_{V^{\mathbb{n}}} \sqrt{t^{}},\波浪号{b}{2}=-b{2}-\sigma_{V^{\mathrm{B}}} \sqrt{t^{\star}}t^{*}=Tt\rho=\rho_{\mathrm{BR}}\$

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