Please help me with the report.
You can do analysis with both “try1” & ”try2”. If possible, please use data from try2 “US 5 CDS”. Maybe there is a conflict between workshop and project requirement, just try to put them together, make the project reasonable.
Reference priority: “Certificate in Quantitative Finance Final Project Brief”> “FP_Workshop_II_CR_IR”
What I have done in the following is from workshop about this project. Content Begins from page 4.
Structure:
# Pricing A Credit Spread Product
The aim of this project is to price a portfolio of basket CDS for 5 reference names. As it’s a portfolio with unknown distribution, we assume copula variables to default times with a marginal term structure of hazard rates.
Samples are implemented by Gaussian and t copulae, and price all k-th to default instruments (1st to 5th).
Spread convergence sample is the low discrepancy sequences (eg; Halton, Sobol)
The results are shown as histograms, scatter plots and sensitivity analysis.
## 2. Two Datasets
### 2.1. A snapshot of credit curves on a particular day.
USD/EUR CDS curve. To get a term structure of hazard rates and exact default times ui → τi.
5YRS
### 2.2. Historical credit spreads time series
The most liquid tenor 5Y for each reference name to get a 5 * 5 default correlation matrix.
## 3.1.
a) Implied default probabilities
b) Term structure of Hazard rates.
## 3.2.
a) Default Correlation matrics(near and rank)
b) Parameter(Calibrate copulae)
c) Pricing by Gaussian and t copulae sparetely
## 3.3. Copula algorithm
a) Generate a vector of correlated uniform random variable
b) For each reference name, use its term structure of hazard rates to calculate exact time of default (or use semi-annual accrual).
c) Calculate the discounted values of premium and default legs for every instrument from 1st to 5th-to-default. Conduct MC separately or use one big simulated dataset.
## 3.4.
a) Average premium and default legs across simulations separately
b) Calculate the fair spread
# 4. Model Validation
## 4.1.
a) The fair spread for kth-to-default Basket CDS should be less than k-1 to default. Why?
## 4.2. Risk and Sensitivity Analysis
a) default correlation among reference names: either stress-test by constant high/low correlation or ± percentage change in correlation from the actual estimated levels.
b) credit quality of each individual name (change in credit spread, credit delta) as well as recovery rate.
## 4.3.
a) Explain historical sampling of default correlation matrix
b) Explain copula fit (uniformity of pseudo-samples) – Correlations Experiment and Distribution Fitting Experiment by histograms
# 5. Copula, CDF and Tails for Market Risk
a) The recent practical tutorial on using copula to generate correlated samples is available at:
https://www.mathworks.com/help/stats/copulas-generate-correlated-samples.html
b) Semi-parametric CDF ftting gives us percentile values with ftting the middle and tails.
Generalised Pareto Distribution applied to model the tails, while the CDF interior is Gaussian kernel-smoothed.
The approach comes from Extreme Value Theory that suggests correction for an Empirical CDF (kernel ftted) because of the tail exceedances.
http://uk.mathworks.com/help/econ/examples/using-extreme-value-theory-and-copulas-to-evaluate-market-risk.html
Definition
According to the lecture Final Project Workshop Part II, we can get a clear understanding of project in the following.
Identify Basket CDS:
Credit Default swap (CDS) can be seen as a protection contract for the buyer to transfer the credit risk of the reference asset with a premium fee regularly. When the credit event happens, the seller needs to pay the face value. Otherwise, the seller will earn a premium.
Basket CDS (BDS) is the structured protection contract of multiple assets. And in the k number of defaults, this BDS will be terminated in the “kth to default”.
Here we get 5 reference stocks, AAPLE, RCL, MACY, COSTCO and JPM.
For CDS buyer:
LGD = (1-R) * 1/5 * N
Where R is recovery rate; N is notional; 5 means 5 reference names
While for CDS seller:
PL=S*N per year
where S is spread.
for “kth to default”, here is what we can get,
1st to 5th to default are recognized as different instruments, which should be calculated separately.
1st: use Gaussian & t copula (Monte Carlo)
5th: will be terminated until it defaults within 5-year period
The price of spread is:
S=DL/PL
where DL is default lag; PL is premium lag
Hazard rate:
Hazard rate depends on exponential distribution of Default time. Joint distribution by using copula method to simulate joint distribution numerically. Will have Simulate uniform variables and need to confirm each uniform in default times,
Getting hazard rates from the ‘today’ credit curve for each reference name in the following.
(photos here are snipped from lecture material)
The hazard rate function h of τ:
F(T)=P()
Where is Stopping time.
F(T) has a density f(T), and we can get,
Need Discount factor. And create curve.
Probabilities of Survival () depends on time .
The cum probabilities of survival:
λ is hazard rate in discrete way; Δt is 1 year in this situation.
(photos here are snipped from lecture material)
Hazard rate parameters
Survival probability
Joint distribution of 5 spread/Hazard rate like graphs shown above, each one will show marginal distribution. Joint them together.
PD (Default probability)
Survival probability: the probability of the asset
Default probability=1-survival probability
According to credit rating, the real-world default probability is calculated from historical default data.
And the risk-neutral default probability is calculated from market data based on CDS. As
Structured credit product, an OTC credit exotic. Product characteristics and pricing methodology give a practical insight in how to price tranched products (pools of securities), with correlation.
If the k number of defaults occur the contract terminates, the protection buyer receives
LGD = (1 − R) × 1 5 × N
Protection seller receives PL = s × 10 million per year. Paid periodically in arrears (vs. upfront fee CDS). Five reference names, 2m notional each, with maturity T = 5 years.
kth to default
To be more specific, we describe each kth to default CDS.
For 1st to default:
Mth to default:
Intensity is a ratio of survival probabilities
This is credit spread in the following:
Implied survival prob
Quotient=(DF*((1-RR) +DT*market spread/10000))
(photos here are snipped from lecture material)
Do hazard rates(E11)=ln(E11/E10) = LN(ISPn/ISP(n-1))
N17=SUM(N11:N15)
(photos here are snipped from lecture material)
Premium leg
Default leg
To compute default probability:
SN=credit spread
L=1-RR
D (0, Tn) =Discount Factor
P(TN-1) =default probability
INHOMOGENOUS POISSON PROCESS(IHP)
For each name, we have a term structure of IHP.
Exponential inter-arrival times
Suppose correlated () , convert to
Exponential CDF: , so
Input u and , are two unknowns.
Marginal default time
1st, Find the yr of default, and
2nd, estimate the year fraction or use accruals.
We will know time simulated u, hazard rate , into ; (inf = max)
Where default occurs if inequality holds and
Validating example
Using absolute values to compare.
EG:
We have 3 ， 0.03，0.07，0.05； u=0.11
|Log(1-0.11)|=0.12,
0.12 not
0.12 not
0.12
so，Default in 3 rd year.
Then, make 2nd default time on page 24.
where hazard rate per period, u simulated is known, m=3, can get ; for 2 years
Copula. Correlation of Default Events
The distribution of default time is for each reference name: an Exponential Distribution parametrized empirically by a set of five hazard rates (piecewise constant, per year).
τName 1 ∼ Exp(1Y , … , λ5Y ), set of λ
τName 2 ∼ Exp(λ1Y , … , λ5Y )
Copula method:
Marginal Distributions for default times τi:
Dependence structure (correlation: linear, rank, calibrated based on MLE by an optimizer)
The joint distribution for k-th to default time across all reference names:
τk ∼ Fk (t1, t2, … , tn) has no closed form.
F(x1, x2, … , xn) ≡ C(u1, u2, … , un)
Joint Student’s t Distribution
x->Uname1; y->Uname2
Student’s t Copula (same distribution, in uniform way)
v->Tv(y); u->Tv(x); v as index is discrete of freedom parameters for t distribution.
CDF “grades of variable”, bound between 0 and 1
Sample:
One round of simulation, generate a correlated set ()
Joint distribution of default time: τk ∼ Fk (t1, t2, … , tn)
Cholesky decomposition: (5*5)
Generate independent Normal RNs and impose correlation by , then convert back
Challenge: estimate correlation rate. correlated; Z: 5 independents normal RNS
Due to historic Credit Spreads bps
Get difference gap between t and t-1. On day to day basis, can find correlation more than 90% commonly,
Distribution Fitting:
Need to Estimate correlation.
Imagine have CDS, put data into uniform. Once get rank correlation, then linearize independently.
X=>pseudo samples
– Historical CDS 5Y
– Historical ΔCDS 5Y
– ΔPD 5Y
– Returns (Delay)
T copula 39-38-40
Use rank correlation matrix input to get LOG-LIKELIHOOD OF T COPULA, repeat t=1 to 20 and ploy
MLE wrd d.f. psrsmeterd for t copula
U historical; Ct; Ct+1;
T copula density
Sampling Gaussian copula: pearson matrix on Δ CDS
Sampling t copula. Difference: rank correlation & chi-squared RN
Why t copula?
43
Structural models and Default Prediction
Structural Models, Basket Credit Derivatives & CDOs
3.1 Structural Models-Empirical Data
3.2 Deriving PDs from Market Prices: Merton Model
SPREAD COMPUTATION
Kth-to-default spread
Par spread
Get s from k=1 to k=5
1st to default
Get table:
2nd to default
This rule is
Model validation
Sensitivity to constant correlation 54
Equal correlation is high, the fair spread
Equal correlation is low, the fair spread
Pricing:
Spread price
For fair pricing the MTM should be no arbitrage and equal to 0.
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