### 金融代写|金融数学作业代写Financial Mathematics代考|Valuation of Credit Derivatives with Counterparty Risk

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

The valuation of credit derivatives has for a long time been based on default-free counterparties (i.e., contractual partners), as this allows a risk-free valuation of the payments made under credit derivatives. Even though financial institutions own subsidiaries, which could act as counterparties in OTC derivatives (over-the-counter derivatives), and reach strong ratings of “AAA”, Ammann (2001) shows that less than half of the market participants have a rating of “A” or above. Moreover, no exchange traded credit derivatives exist up to now. Following these arguments, the consideration of counterparty risk is essential for a correct and consistent valuation of credit derivatives.

Given the stated findings and reasoning, this chapter discusses the valuation of credit derivatives with defaultable counterparties.* Especially, we focus on the modeling of joint default risks of risk buyers and reference parties. The following section concentrates on the valuation of standard credit derivatives, i.e., credit default swaps (CDSs). As described in Hull and White (2000), a CDS is a contract that provides protection against the default of a particular company. This company is referred to as reference entity and the default of the reference entity is the credit event. The buyer of protection has the right to sell to the protection seller at par a specified bond issued by the reference entity when a credit event occurs. That bond is commonly referred to as the reference obligation and the total volume of the bond that can be sold is the notional of the CDS.

The most important aspect of modeling counterparty risk is the treatment of correlations between the credit risk of the underlying asset and the credit risk of the counterparty. If, for example, perfect correlation is assumed, one can easily see the importance of correlation for valuation purposes. If a financial institution with an AAA-rating sells CDSs (risk buyer, protection seller) and the reference asset is its own bond, then the swap with an assumed recovery rate of zero is worthless. This is due to the fact that in the event of default of the reference entity, which triggers the credit event under the CDS, the risk buyer-being identical to the reference entity-defaults as well. If, in contrast, the risk buyer is solvent, i.e., no credit event occurs, the risk buyer will not be drawn on. Tavakoli (2001) summarizes these results as follows: “Counterintuitive as it may seem, it is better to buy credit protection from an uncorrelated lower-rated protection seller than from a protection seller that is highly correlated with the reference asset one is trying to hedge” (Tavakoli 2001, p. 25).

This chapter is organized as follows: Section $2.2$ presents the approach of Hull and White (2001). The authors define variables that are directly or indirectly observable from market data as endogenous and all other variables as exogenous. In Section 2.3, we review how defaultable CDSs can be evaluated based on Merton (1974). Sections $2.4$ and $2.5$ present the models of Jarrow and Yu (2001) and Lando (2000), respectively. Both approaches focus on a joint default process, which explicitly incorporates the reciprocal action between both obligors. Therefore, the latter concepts abandon the conditional independencies of default events. Section $2.6$ concludes the chapter.

## 金融代写|金融数学作业代写Financial Mathematics代考|VALUATION BASED ON OBSERVABLE MARKET DATA

The approach of Hull and White (2001) defines both directly and indirectly observable market data as exogenous. Because of this approach, the model is commonly considered as rather practical and applicable. The valuation of CDSs follows a three-stage procedure:

1. Calculation of risk-neutral default probabilities of the relevant contractual agents (i.e., protection seller and reference entity). To accomplish this, one can either draw on listed bonds issued by the reference entity or bonds from companies with the same risk of default as the reference entity.
2. Calculation of the default correlation between the protection seller and the reference entity. Hull and White (2001) suggest the usage of market data.
3. Calculation of the expected future stream of payments, which are linked with the default swap (i.e., premium payments and loss compensations at default).

It should be noted that the model of Hull and White (2001) cannot be solved analytically. Instead, one has to use a simulation-based approach, e.g., Monte Carlo simulation.

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

Hull and White (2001) base the modeling of the default on a first-passage-time approach. Given this methodology, the default occurs, if the (obligor-specific) default barrier variable $\tilde{G}{t}^{j}$ (where $j$ represents the index of the analysed obligor) reaches the default barrier $\tilde{g}{t}^{j} . \tilde{G}{t}^{j}$ is referred to as the credit index for company $j$ at time $t$. Hull and White (2001) propose two different interpretations of this variable. First, they define this variable as a function of the value of the total firm assets and second as a discrete credit rating. A more specific valuation is not important as the default probability will not be calculated within this model. Instead, it is an exogenous input parameter taken from observable market data. The risk-neutral default probability is used to calibrate the default barrier $\tilde{g}{t}^{j}$.

To determine the correct default barrier $\tilde{g}{t}^{j}$, the following assumptions have to be considered. First, it has to be assumed that the risk-neutral process for $\tilde{G}{t}^{j}$ follows a geometric Brownian motion with zero drift and variance of one per year. Second, the default occurs at discrete points of time referred to as $t_{i}$ (where $1 \leq i \leq n$ ). Therefore, the cumulative default probability of company $j$ till time $t_{i}$ could be expressed as follows, if no default occurs before $t_{i-1}$ :
$$\mathrm{P}^{}\left{\tau_{j}=t_{i} \mid \tau_{j}>t_{i-1}\right}=1-\int_{\bar{g}{i}^{j}}^{\infty} f{i}^{j}(x) \mathrm{d} x$$
where
$\tau_{j}$ is the default time of the company $j$
$f_{i}^{j}$ is the (conditional) density function of the credit index $\tilde{G}{t}^{j}$ with $\tau{j}>t_{i-1}$
If the (unconditional) default probability $p_{i}^{j}=\mathrm{P}^{}\left{\tau_{j}=t_{i}\right}$ is known, the conditional density function, $f_{1}^{j}(x)$, and default barrier, $p_{i}^{j}$, can be calculated as follows:

For $i=1$
\begin{aligned} f_{1}^{j}(x) &=\frac{1}{\sqrt{2 \pi \Delta_{1}}} \mathrm{e}^{-x^{2} / 2 \Delta_{1}} \ p_{1}^{j} &=\Phi\left(\frac{\tilde{g}{1}^{j}}{\sqrt{\Delta{1}}}\right) \end{aligned}
with $\Delta_{i}=t_{i}-t_{i-1}, 1 \leq i \leq n$ and $t_{0}=0 . \Phi$ is the standard normal distribution function.
From Equation $2.1$ and $\tilde{g}{1}^{j}=\sqrt{\Delta{1}} \Phi^{-1}\left(p_{1}^{j}\right)$, the default barrier for $i=1$ can be determined. On the basis of this result, the default barrier for $2 \leq i \leq n$ can be calculated as follows:
$$p_{i}^{j}=\int_{\bar{g}{i-1}^{j}}^{\infty} f{i-1}^{j}(x) \Phi\left(\frac{\tilde{g}{i}^{j}-x}{\sqrt{\Delta{1}}}\right) \mathrm{d} x$$
The conditional density function $f_{i}^{j}(x), x>\tilde{g}{i}^{j}$ can be calculated by solving the following equation: $$f{i}^{j}(x)=\int_{\bar{g}{i-1}^{j}}^{\infty} f{i-1}^{j}(u) \frac{1}{\sqrt{2 \pi \Delta_{i}}} \mathrm{e}^{-(x-u)^{2} / 2 \Delta_{i}} \mathrm{~d} u$$
The inductive calculated default barriers $\tilde{g}_{i}^{j}$ are not constant over time, but are time dependent. This is due to the usage of observable (risk-neutral) default probabilities for the calibration process.

## 金融代写|金融数学作业代写Financial Mathematics代考|VALUATION BASED ON OBSERVABLE MARKET DATA

Hull 和 White (2001) 的方法将直接和间接可观察的市场数据定义为外生的。由于这种方法，该模型通常被认为是相当实用和适用的。CDS 的估值遵循三个阶段的程序：

1. 计算相关合同代理人（即保护卖方和参考实体）的风险中性违约概率。为此，可以利用参考实体发行的上市债券或与参考实体具有相同违约风险的公司的债券。
2. 计算保护卖方和参考实体之间的默认相关性。Hull and White (2001) 建议使用市场数据。
3. 计算与违约掉期相关的预期未来支付流（即违约时的保费支付和损失补偿）。

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

Hull 和 White (2001) 基于首次通过时间方法对违约进行建模。鉴于这种方法，如果（特定于债务人的）默认障碍变量会发生默认G~吨j（在哪里j代表被分析债务人的指数）达到默认障碍G~吨j.G~吨j被称为公司的信用指数j有时吨. Hull 和 White (2001) 对这个变量提出了两种不同的解释。首先，他们将此变量定义为公司总资产价值的函数，其次将其定义为离散的信用评级。更具体的估值并不重要，因为在此模型中不会计算违约概率。相反，它是从可观察的市场数据中获取的外生输入参数。风险中性违约概率用于校准违约障碍G~吨j.

\mathrm{P}^{}\left{\tau_{j}=t_{i} \mid \tau_{j}>t_{i-1}\right}=1-\int_{\bar{g}{ i}^{j}}^{\infty} f{i}^{j}(x) \mathrm{d} x\mathrm{P}^{}\left{\tau_{j}=t_{i} \mid \tau_{j}>t_{i-1}\right}=1-\int_{\bar{g}{ i}^{j}}^{\infty} f{i}^{j}(x) \mathrm{d} x

τj是公司的默认时间j
F一世j是信用指数的（条件）密度函数G~吨j和τj>吨一世−1

F1j(X)=12圆周率Δ1和−X2/2Δ1 p1j=披(G~1jΔ1)

p一世j=∫G¯一世−1j∞F一世−1j(X)披(G~一世j−XΔ1)dX

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