### 金融代写|金融数学作业代写Financial Mathematics代考|RISK NEUTRAL PRICING

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## 金融代写|金融数学作业代写Financial Mathematics代考|Default Status Tree

Nowadays, financial products are valued as the discounted value the expected cash flows under a risk neutral probability measure. We can discount the payments at the risk free rate using risk neutral valuation in the lines of Jarrow and Turnbull (1995). This implies that the default probabilities in the risk neutral world will be relevant.

We discretize the remaining time to maturity of the CDS, i.e., the time interval $[0 \mathrm{~T}]$, into $n$ intervals with equal distance $\Delta t$ between the grid points. For $n=3$, Figure $1.6$ pictures the evolution of the default status of the reference bond.

At $t_{0}$ the company is alive (nondefault, $N D$ ) with probability 1 . At $t_{1}$, the company can default $(D)$ or survive $(N D)$. If it defaults, this state remains the same for the future. If the firm did not default, the tree will further branch out with two possible states ( $D$ and $N D$ at $t_{2}$ ). Default at any $t_{i}$ acts as an “absorbing state” for the rest of the tree.

Let us refer back to our example in Table 1.1, where the cash flows for the buyer of a 3 year CDS with annual payments were summarized. Time was discretized in $n=6$ steps of $0.5$ years. Bankruptcy can occur on every node (column). In case of $N D$ at time $T$, the last payment occurs at $t_{4}$ or at $T$ depending on the contract specification.

In case of $D$, the payment of the premium payments stops but a last accrued premium may contractually be due at the time of default. This accrued fee is netted with the payment the protection seller has to pay in case of $D$. For the buyer, the $t_{3}$-value of the settlement of the CDS is $F(1-R)$. Depending on the contract specifications the par value can be augmented with the accrued interest from $t_{2}$ to $t_{3}$. In most general terms, the payment becomes $(F+A I)(1-R)-A F$.

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

The default status tree (Figure $1.6$ ) neglected the probabilities. The definition of the various default and survival probabilities makes the literature not very transparent. Deciphering notation is most of the time the hardest part of understanding the model.

First let us define the probability of default at each node in the tree, $p_{(i)}$, as the probability to default during period $i$. The firm still was alive at $t_{i-1}$, it defaults at $t_{i}$. Whenever the default probabilities are kept constant over time, such as in Figure 1.7, we will drop the subscript. In Figure $1.7 p=5 \%$.

These default probabilities define a number of conditional probabilities characterizing the various states of the default process.
For the first period the marginal probabilities apply:
$$\begin{gathered} P\left(D_{1}\right)=p_{(1)}=5 \% \ P\left(N D_{1}\right)=\left(1-p_{(1)}\right)=95 \% \end{gathered}$$
For the second period,
$$\begin{gathered} P\left(D_{2} \mid N D_{1}\right)=\left(1-p_{(1)}\right) p_{(2)}=4.75 \% \ P\left(N D_{2} \mid N D_{1}\right)=\left(1-p_{(1)}\right)\left(1-p_{(2)}\right)=90.25 \% \end{gathered}$$
For the third period, we get
$$\begin{gathered} P\left(D_{3} \mid N D_{2}\right)=\left(1-p_{(1)}\right)\left(1-p_{(2)}\right) p_{(3)}=4.51 \% \ P\left(N D_{3} \mid N D_{2}\right)=\left(1-p_{(1)}\right)\left(1-p_{(2)}\right)\left(1-p_{(3)}\right)=85.74 \% \end{gathered}$$
Two series of probabilities emerge:

• On the one hand, we find the probabilities of survival until time $t_{i}$, which we will denote $\pi_{i}: 95 \%, 90.25 \%$, and $85.74 \%$.
• On the other hand, we obtain the probability of default at time $t_{i}$, conditional upon survival until the previous period: $5 \%, 4.75 \%$, and $4.51 \%$. We will denote these probabilities by $p_{i}$.

In general, we can express these conditional probabilities for $t_{i}$ as
$$\left{\begin{array}{l} p_{1}=p_{(1)} \ p_{i}=P\left(D_{i} \mid N D_{i-1}\right)=p_{(i)} \prod_{j=2}^{i}\left(1-p_{(j-1)}\right) \quad \text { for } i=2,3, \ldots, n \ \pi_{i}=P\left(N D_{i} \mid N D_{i-1}\right)=\prod_{j=1}^{i}\left(1-p_{(j)}\right) \quad \text { for } i=1,2, \ldots, n \end{array}\right.$$
If the default probability is constant over the tree, these formulas for $i=1,2, \ldots, n$ collapse to
\begin{aligned} &p_{i}=P\left(D_{i} \mid N D_{i-1}\right)=p(1-p)^{i-1} \ &\pi_{i}=P\left(N D_{i} \mid N D_{i-1}\right)=(1-p)^{i} \end{aligned}
Note that the following relationships hold:

• Probability of survival until time $t_{i}$ is equal to 1 minus the sum of the conditional probabilities of default in the previous periods: $\pi_{i}=1-\sum_{j=1}^{i} p_{j}$.
• Probability of survival until time $t_{i}$ is equal to the probability of survival until the previous time minus the conditional probability to default in period $i: \pi_{i}=\pi_{i-1}-p_{i}$.
• Probability to survival until time $t_{i}$ is equal to the probability of survival until the previous period multiplied by the factor 1 minus the default probability (given that the default probability is constant): $\pi_{i}=\pi_{i-1}(1-p)$ or even more general $\pi_{i}=\pi_{i-j}$ $(1-p)^{j}$ with $j<i$ and $j \in N$.

## 金融代写|金融数学作业代写Financial Mathematics代考|Risk Neutral Pricing of the CDS

The periodic premium payments are generally labeled in the literature as the floating leg. The contingent terminal payment is known as the fixed leg of the CDS.
1.5.5.1 Under the Discrete Default Process
1.5.5.1.1 Floating Leg At the payment dates $t_{g} t_{2 g}, \ldots, t_{n}$ the protection buyer pays the premium $P=s F(g \Delta t$ ), until (and not including) default.

The expected payment for time $t_{i}$, seen today, is $P\left(N D_{i} \mid N D_{i-1}\right) P+P\left(D_{i} \mid N D_{i-1}\right) 0=\pi_{i} P$. Consequently, the present value of the expected payments is $\sum_{i=1}^{n / g} d f_{i g} \pi_{i g} P$ where $d f_{i}$ denotes the discount factor for the time $t_{\text {ig }}$ cash flow.

1. Note that the discount function can be expressed in various ways:
• Cheng (2001) uses the general zero bond price formulation: $B\left(t_{0}, t_{n}\right)$ i.e., the time $t_{0}$ value of a default free zero bond maturing at $t_{n}$.
• Brooks and Yan (1998) use simple one period forward rates: $d f_{i}=\frac{d f_{i-1}}{1+r_{w-1} \times W_{i}}$. NAD is the number of accrued days and NTD the total number of days, defining the applicable day count convention.
• Continuous interest rates would yield: $\mathrm{e}^{-r_{i} t_{i}}$.
• Scott (1998), Aonuma and Nakagawa (1998), Houweling and Vorst (2005), Longstaff, Mithal and Neis (2005) use the stochastic interest formulation: $\mathrm{e}^{-\int_{0}^{t} r(u) \mathrm{d} u}$.Also with the probabilities we can juggle a bit. Note that, as Figure $1.8$ illustrates, the survival probability, $\pi_{b}, i<n$ equals the sum of the survival probability at the end of the contract, $\pi_{n}$, plus the sum of the probabilities of defaulting between $t_{i}$ and $t_{n}$.
• $$• \begin{gathered} • \pi_{i}=\pi_{n}+\sum_{j=i}^{n} p_{j} \ • \sum_{i=1}^{n / g} d f_{i g} \pi_{i g} P \text { then becomes } \sum_{i=1}^{n / g} d f_{i g}\left(\pi_{n / g}+\sum_{j=i+1}^{n / g} p_{j}\right) P • \end{gathered} •$$
• which is a formulation Brooks and Yan (1998) use.

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

• 一方面，我们发现直到时间的生存概率吨一世, 我们将表示圆周率一世:95%,90.25%， 和85.74%.
• 另一方面，我们获得了违约概率吨一世, 以生存到前一时期为条件：5%,4.75%， 和4.51%. 我们将这些概率表示为p一世.

$$\left{p1=p(1) p一世=磷(D一世∣ñD一世−1)=p(一世)∏j=2一世(1−p(j−1)) 为了 一世=2,3,…,n 圆周率一世=磷(ñD一世∣ñD一世−1)=∏j=1一世(1−p(j)) 为了 一世=1,2,…,n\对。 一世F吨H和d和F一种在l吨pr这b一种b一世l一世吨是一世sC这ns吨一种n吨这在和r吨H和吨r和和,吨H和s和F这r米在l一种sF这r一世=1,2,…,nC这ll一种ps和吨这 p一世=磷(D一世∣ñD一世−1)=p(1−p)一世−1 圆周率一世=磷(ñD一世∣ñD一世−1)=(1−p)一世$$

• 直到时间的生存概率吨一世等于 1 减去前期违约条件概率的总和：圆周率一世=1−∑j=1一世pj.
• 直到时间的生存概率吨一世等于直到前一次的生存概率减去该期间违约的条件概率一世:圆周率一世=圆周率一世−1−p一世.
• 生存到时间的概率吨一世等于直到前一时期的生存概率乘以因子 1 减去违约概率（假设违约概率是常数）：圆周率一世=圆周率一世−1(1−p)甚至更一般圆周率一世=圆周率一世−j (1−p)j和j<一世和j∈ñ.

## 金融代写|金融数学作业代写Financial Mathematics代考|Risk Neutral Pricing of the CDS

1.5.5.1 在离散违约过程下
1.5.5.1.1 浮动腿 在付款日吨G吨2G,…,吨n保护买方支付保费磷=sF(GΔ吨)，直到（不包括）默认值。

1. 请注意，折扣函数可以用多种方式表示：
• Cheng (2001) 使用一般的零债券价格公式：乙(吨0,吨n)即时间吨0到期的违约免费零债券的价值吨n.
• Brooks and Yan (1998) 使用简单的一期远期汇率：dF一世=dF一世−11+r在−1×在一世. NAD 是累计天数，NTD 是总天数，定义了适用的天数计算惯例。
• 连续利率将产生：和−r一世吨一世.
• Scott (1998)、Aonuma 和 Nakagawa (1998)、Houweling 和 Vorst (2005)、Longstaff、Mithal 和 Neis (2005) 使用随机利率公式：和−∫0吨r(在)d在.另外，我们可以稍微调整一下概率。请注意，如图1.8说明，生存概率，圆周率b,一世<n等于合约结束时的生存概率之和，圆周率n，加上之间违约概率的总和吨一世和吨n.
• $$• \开始{聚集} • \pi_{i}=\pi_{n}+\sum_{j=i}^{n} p_{j} \ • \sum_{i=1}^{n / g} d f_{ig} \pi_{ig} P \text { 然后变成 } \sum_{i=1}^{n / g} d f_{ig}\left (\pi_{n / g}+\sum_{j=i+1}^{n / g} p_{j}\right) P • \结束{聚集} •$$
• 这是 Brooks 和 Yan (1998) 使用的公式。

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