Understanding CDS Indice Behavior During Periods Of Market Stress & Predict Their Prices Using Machine Learning Techniques : What Happened With Credit Default Swaps In 2008?

Understanding CDS Indice Behavior During Periods Of Market Stress & Predict Their Prices Using Machine Learning Techniques : What Happened With Credit Default Swaps In 2008?

Trading and Investing

Abstract 

Single-name and index CDS have been an important part of any active portfolio management strategies as they provide the most liquid means for institutional investors to gain or hedge credit exposure. In 2022, volumes in CDX and iTraxx reached an all-time high of nearly USD 36 trillion. This motivated us to study the characteristics of these instruments by first investigating their valuation models and implied default probabilities in the market. We analyze these implied probabilities through different periods of stress and also study the impact on the term structure defined by different tenors. Using this information we propose a relative-value strategy as an example of a possible application of our model. This strategy provides additional insight compared to the direct analysis of spread rates. 

In the second part of the project, we explore different forecasting methods, ranging from classical time series models to Machine-Learning techniques, on the North American Investment Grade and High Yield indices. Overall, the Time Series models did not perform well on the determinants. Deep Learning models such as LSTM and LSTM-Merton outperformed basic Machine Learning models. 

1 Introduction 

Credit Default Swaps (CDS) are over-the-counter (OTC) derivatives contracts that provide protection against the risk of default by managing credit risk. CDS can be purchased by the lender and acts as a form of insurance designed to protect the lender. [7] Default protection buyer makes fixed quarterly payments called CDS spreads to default protection sellers. CDSs can be used for market speculation, hedging or as a form of arbitrage. Credit Default Swap Indices (CDX) are superior to single name CDS in terms of liquidity, standardization, operational efficiency, cost efficiency and transparency.

Credit indices allow credit market participants to take a view on the overall credit quality and directions of the underlying basket. [16]Differences between the credit index and a single-name CDS lie in the time to maturity for the credit index and the single-named CDS, categorization of the credit events and in the fact that a contract on a credit index does not terminate after a credit event of a member but rather it functions like a new contract on the remaining entities with a reduced notional amount. Through our thesis project, we wish to understand the modeling and valuation, forecasting and trading of the two most liquid Credit Default Swap Indices – CDX.NA.HY and CDX.NA.IG. [16]Forecasting of CDS Spreads is supportive to the dynamics of pricing credit derivatives and to portfolio management, benefiting in terms of hedging purposes and active management. 

[15]Credit indices have expanded dramatically over recent decades, with volumes rising, trading costs decreasing, and visibility growing across financial markets. CDS Indices also played a huge role during the Great Recession of 2008, European Sovereign Debt Crisis of 2010, Taper Tantrum of 2013, Brexit vote of 2016 and [4]COVID related market turmoil of 2020, making both valuation and forecasting of CDS index prices important and interesting. CDS and CDS Indices are quite important in terms of trading volumes during the times of market stress, for instance, COVID period of 2020 witnessed an increase in volume of CDS traded more than that of interest rate swaps (IRS) and currency swaps (FX).

Furthermore, North American Investment-Grade indices were the most heavily traded indices during the COVID period. We discuss each of the events of the crisis in detail in the data section of our thesis, highlighting how the performance of Investment Grade and High Yield indices was impacted and how the recovery looked like. 

The first part of this project is to explore the implied default probabilities in the market. Particularly, we will be working with Credit Default Indexes (CDX) on corporate bonds and their behavior during periods of stress. First, we will determine how the market participants trade and price the CDX products.

From these market prices, we can use the most common models, where the default probability is an input, to back out the implied probabilities in the market prices. We can then compare the implied probabilities with the real default data of corporations to have as a reference. In the second part of our project, we will present and discuss Machine Learning methods to forecast and trade the North American Investment Grade and High Yield indices. 

Our approach 

1. Select a corporate CDX 

2. Collect data for underlying securities and CDX prices 

3. Choose a CDS valuation model and back out implied default probabilities 

4. Compare implied default probabilities to real default data of corporations 

5. Test CDX price relationship to other variables 

6. Machine Learning and CDX 

2 Literature Review 

2.1 Single name CDS 

Credit default swap (CDS) is an agreement between a protection buyer and a protection seller to exchange the credit risk of an issuer (reference entity). Buying protection is analogous to holding an option: the protection buyer pays a periodic fee and profits if a credit event happens. Selling protection is similar to shorting an option: the protection seller collects the fee and profits if the reference entity has stable or improving credit while the swap is outstanding. 

There are 4 parameters that define a credit default swap: 

1. Issuer : CDS contracts specify a reference obligation (a specific loan or bond) which defines the issuing entity through the bond details. The reference entity tends to be a senior unsecured bond, but bonds at other levels can also be used. 

2. Notional amount: The amount of credit risk being transferred. Negotiated and accepted between the protection buyer and seller. 

3. Spread: The annual payments, quoted in basis points paid annually. It is also called fixed rate, coupon, or price. 

4. Maturity: The expiration of the contract, usually on the 20th of March, June, September, or December. The five year contract is usually the most liquid. 

2.2 CDS Indices 

A CDS index is a basket of single-name credit default swaps. The names of the index are determined using their geographical location, liquidity profile, respective business sectors, as well as the rules governing the index. IHS Markit creates and administers standardized CDS indices, which include CDX and iTraxx indices. An overview of the main indices and their characteristics is shown in figure 1 [14] 

Figure 1: Index CDS Characteristics 

2.2.1 Market Participants and Global CDS Market 

The market is composed of different types of participants: 

1. IHS Markit – takes full ownership of the indices from licensing, calculation, managing, to publishing rules, index annexes and other relevant documents 

2. Licensed Participants: contribute to CDS index ecosystem by trading, generating liquidity, or providing central clearing 

(a) Dealers: supplies access and liquidity 

2.2 CDS Indices  

(b) Clearing Houses: eliminates counterparty risk and reduces operational costs 

(c) Swap Execution Facilities: provides CDS index pricing and trade execution platform (d) Third parties: integrates CDS indices into their platform, thus facilitates trading among different parties in the market 

3. Institutional Investors: hedge or speculate market via CDS indices 

4. ISDA: works with IHS Markit and banks to construct global policy for the CDS indices 

In terms of firm types, the most recent studies [2] identify different sectors interacting in the global markets, as shown in figure 2. 

Figure 2: CDS Global Market 

We can also see that index products are the most popular in terms of market share. In these products, the dominant participants are insurance companies, pension funds, banks, and hedge funds [4].The evolution of their position is shown in figure 3. 

Figure 3: Index CDS positions by firm type 

As of the end of the second quarter of 2019, the majority of single-name CDS and index CDS gross notional outstanding were between central counterparties (CCPs) and dealers. 

Figure 4: Single-name and Index CDS Notional Outstanding by Counterparty Type

2.3 Trading CDS and CDS indices in the market  

2.2.2 CDS Index mechanics 

Each CDS index can be considered as a separate credit default swap contract with a fixed set of credits and annual coupon. One way to invest in CDX index is to pay or receive a regular payment (quarterly) of this fixed coupon on a pre-stated notional. In addition, CDX and iTraxx products can also be traded in the market, with the traded level influenced by supply and demand. 

2.2.2.1 On-the-run vs Off-the-run 

Different from a perpetual index such as S&P 500, CDS indices have fixed compositions and maturities. Approximately every six months, a new series of indices is generated with a new set of underlying constituents and maturity date, to update changes in the credit market and allow investors the flexibility in managing the constant duration. ”On-the-run” refers to an index that is newly launched, whereas the existing indices continue to trade (as ”off-the-run”) until maturity. The on-the-run indices are often more liquid than those off-the-run. While it is not obligatory, investors tend to rotate out of off-the-run series to new positions in the on-the-run indices. 

2.2.2.2 CDS Indices Default Trigger 

The composition of the index remains unchanged throughout its lifetime, except when a credit event is triggered for one of the constituents. When a single name experiences a credit event, or has to be removed for some other reason, then a new version of the series is created for all the indices (on the run and off the run but not yet matured) that contained this name. The previous versions cannot be traded. The only change to the index is that the notional factor is reduced by the weight of the deleted name. In terms of payoff, a one-time payment is done to the contract buyer proportional to the weight of the constituent times the notional amount. 

2.3 Trading CDS and CDS indices in the market 

Similar to the cash market where bonds can trade either on yields or price, the CDS indices match that of the underlying cash instruments: some indices trade on spread, and others on price. [14] 

Market participants can use CDS and CDS indices in numerous scenarios: hedge long cash credit positions, build structured instruments, or speculate on the market. The ability to isolate credit risk makes CDS a good instrument for bond managers to hedge their long bond positions as well as tactically adjust credit exposure without involving the transfer of other risk types. In order to hedge, investors need to enter a short risk position with on-the-run CDX, as opposed to off-the-run, which pushes the on-the-run indices to have a more positive basis to its theoretical value.

The difference in preferences of long-risk and short-risk investors also supports this. Long-risk investors prefer to roll down the curve, and therefore hold more off-the-run indices. In contrast, the short-risk investors want to avoid overpaying for a shorter maturity index, thus they hold more on-the-run indices. 

Investment managers also use CDS indices as part of active strategies. There are two popular trade types: (1) basket arbitrage and (2) balance sheet arbitrage. Basket-arbitrage or carry arbitrage is a strategy often employed by buy-side funds to gain exposure. Balance sheet arbitrage requires keeping on-the-run while hedging with off-the-run indices, which is extremely balance sheet intensive so it is mostly done by sell-side firms. This is the strategy behind the London Whale incident in 2012. 

In terms of trading volume between Single-name and index CDS, while Single-name have higher transaction counts, the dollar magnitude of market activities of index CDS is almost 10 times larger. 

2.3 Trading CDS and CDS indices in the market  

Figure 5: Single-name CDS Market Activity and Transaction Count 

Figure 6: Index CDS Market Activity and Transaction Count 

Among the CDS indices, iTraxx Europe and CDX IG were the most actively traded, followed by iTraxx Europe Crossover and CDX HY. 

2.3 Trading CDS and CDS indices in the market  

Figure 7: Index CDS Market Activity by Credit Indices (US$ trillions) 

2.3.1 Trading observations during COVID 19 crisis 

After the 2008 global financial crisis (GFC), market participants had questioned the role of CDS in hedging credit risk in future periods. The surge in CDS trading volume in COVID-19 sell-off reassured that the market could ramp up when needed. 

The COVID period coincided with the March 2020 CDS index roll when a new index series was launched. While it was expected to have a temporary increase in reported positions as participants roll to the new on-the-run series, the magnitude observed in March 2020 was much larger than typical values. 

Figure 8: Record Global Trading in CDS 

In contrast to the spikes in standard indices, positions in non-standard indices and single-name CDS remained relatively unchanged. This reflected market participants’ preference for standard CDS index products as the quick, relatively inexpensive access to broad market protection against large scale credit deterioration, as opposed to single names or non-standard indices. Investors also have to weigh the trade-off between liquidity and specificity when choosing index or single-name CDS during a crisis. For standard index CDS, while bid-ask spreads rise from an average of 1-2 basis points (bps) pre-pandemic to over 10 bps, it quickly rebound after. [13] 

2.4 CDS valuation  

2.4 CDS valuation 

Bloomberg’s CDS valuation tool is capable of handling single-name CDS as well as CDX. It offers three alternatives [15] 

1. ISDA Standard Model [10] (I Model): regarded as the market standard for valuation. Its key assumption is using a flat CDS curve. 

2. ISDA Fair Value Model (V model): extension of the ISDA Standard Model that allows using a full term structure of par spreads, as well as a way to customize spread curve parameters, like coupon frequency, schedule, and day count. 

3. Bloomberg Fair Value Model [10, 15] (B Model): Similar to the ISDA Fair Value Model, but with an alternative numerical implementation. 

2.4.1 Valuation Theory 

This section is based on the information provided in the document Credit Derivatives Handbook (JP Morgan, 2006). 

In a simplified way, a CDS contract can be thought of as a scenario analysis where the underlying credit either survives or defaults. The protection seller (long risk) profits if the credit survives, and discounts the annual payments by this scenario (the fee leg). On the other hand, the protection buyer is expecting the credit to default, and discounts the expected payment (notional minus recovery rate) by the probability of this scenario (the contingent leg). This means that the CDS buyer pays a ’fee leg’ to the CDS seller as a premium until maturity or default, whichever comes first and on the other side, the buyer is compensated using the protection leg in case of default. 

As in the common swaps, at inception both legs’ value is equal, so the contract is worth zero at inception. Therefore we have: 

PV(Fee Leg) = PV(Contingent Leg) 

Which can be rewritten as: 

(P s_i−1 − P s_i) · DF_i 

Where: 

S_nXⁿ i=1 

∆_i· P s_i· DF_{i + Accr = (1 − R)}Xⁿ i=1 

• S_n: spread for protection to period n 

• ∆_i: length of time period i in years 

• P s_i: probability of survival to time i 

• DF_i: risk-free discount factor to time i 

• R: recovery rate on default 

• Accr: accrual on default, determined by Accr = S_nPⁿ i=1 

Note that: 

∆_i 

₂· (P s_i−1 − P s_i) · DF_i 

(P s_i−1 − P s_i) = 1 − P s_i − (1 − P s_i−1) 

Therefore this term represents the probability of not surviving a period i adjusted by the probability of having defaulted before that period, seen from the initial time, from now on referred to as probability of default (P d). 

2.5 Implied default probabilities  

2.5 Implied default probabilities 

Hull and White suggest that the risk-neutral default probability of a bond can be backed out from the difference between the bond yield and a default-free bond yield (i.e. Treasury bond) For a single bond of a company, the difference between the yield on zero-coupon Treasury bond and Corporate bond is the present value of the cost of default. Given a default probability of p, the present value of the expected default is Notional × p × e^{−T reasury yield}. Therefore, the default probability can be calculated as 

_{p =}ZCB T reasury bond yield − ZCB corp bond yield 

Notional × e^{−T reasury rate} 

We will attempt to apply an analogous method using the valuation framework presented in the last section. 

2.5.1 Hazard Rates 

The most common way to calculate survival probabilities in practice is to use Hazard Rates (λ) which is the conditional probability of default in a period, or more simply, the probability of the company defaulting over the period given that it has not defaulted up to the start of the period. The relationship is expressed formally as: 

P s_{i =}Yⁿ i=1 

(1 − λ_i) 

And, as we showed before, the probability of default can be related to hazard rates as: 

P d_i = P s_i−1 − P s_i = λ_i 

ⁱY⁻¹ j=1 

(1 − λ_j ) 

In the case of the ISDA Model, the simplest model with flat CDS curves, the implication is that the hazard rate is constant. Then we have: 

P s_i = (1 − λ)ⁱ 

Note that this does not imply a constant probability of survival over time, but a decreasing one. If we then replace this in our original valuation formula, we have: 

S_nXⁿ i=1 

∆_i· (1 − λ)ⁱ· DF_{i + Accr = (1 − R)}Xⁿ i=1 

((1 − λ)ⁱ⁻¹ − (1 − λ)ⁱ) · DF_i 

We can see that the nonlinearity of the probability of survival over different periods of cash flows makes it impossible to find a closed-form expression for the hazard rate. Therefore, we will be using numerical methods to solve for the implied hazard rate. 

2.6 Real Default Probabilities 

We can estimate real default probabilities based on the credit rating of the underliers of a CDX index. The first issue to consider here is that the model works with risk-adjusted probabilities, meanwhile the ones provided from the data are actual physical probabilities, which are not directly comparable. This issue is beyond the scope of our project, so we still provide a comparison of both to have an idea of the magnitude of default probabilities. 

We consider a naive model to have an initial reference of what should be the default probability of the constituents in the index. This naive model assumes that the defaults during a period are not correlated, and that each period is independent from each other. Note, these are very strong assumptions, but will help provide a sense of the default rates. 

So, for each of the index constituents, there are only two cases, the issuer either defaults or does not default during a given period. Therefore, the number of defaults in a period can be simplified as a binomial distribution: 

2.6 Real Default Probabilities  

p^k(1 − p)^n−k 

Where: 

P r(X = k) = ϕ(k) = 

 n k 

1. n is the number of constituents 

2. k is the number of defaults in a period 

3. p is the probability of default in a period 

Further, from our model we know that the value of the contingent leg can be expressed as: 

P V_{contingent =}Xⁿ 

i=1 

If we consider only a coupon, we have: 

E[Notional · payoff_i· DF_i] 

coupon_i = E[Notional · payoff_i· DF_i] = Notional · Φ_i−1,i · DF_i 

Where Φ_i−1,i is the probability of default in a period. Our model treats the index as a single entity. If we decompose this into a constituent level we have: 

coupon_{i = E[}X^K k 

payoff_i,k · DF_{i] =} X^K k 

_{Notional ·}^k_K· ϕ(k) · DF_i 

If we use the last expressions for a coupon, we can see that: 

Φ_{i−1,i =}X^K k 

k 

_K· ϕ(k) 

And from our model we know that Φ_i−1,i = λ, therefore: 

_{λ =}X^K k 

_{K· ϕ(k) =} ¹_KX^K_{kkϕ(k) =} ¹_K· K · p = p k 

So the value of the estimated default probability will provide us with a reference for the hazard rate we are trying to back out of the market spreads. 

3 Data 

3.1 Implied Default Probabilities Data 

3.1.1 Model Validation Data 

Real-time and historical information about single-name CDS and CDX spreads can become extracted from the Bloomberg terminal. 

Additionally, the Credit Swap Valuation tool provides access to more details related to a particular contract. In figure 9, we show the valuation of the current on-the-run Markit IG CDX (series 39) for a standard one-year contract. 

Figure 9: CDSW Screen Capture 

Also relevant for our calculations is the screen that provides detailed estimated cash flows for the contract, as shown in figure 15. 

Figure 10: CDSW Cash Flows Screen Capture 

This CDSW Tool information was extracted manually and used to verify our calculations taking into account the methodology differences explained in section 4.1. 

3.1.2 Historical Data 

The historical series were obtained from Refinitiv. We queried for the following series: 

1. Closing Mid Spread for the North American Investment Grade Index (CDX.NA.IG) for all the available tenors (1Y, 2Y, 3Y, 5Y, 7Y, 10Y) 

2. Closing Mid Price for the North American High Yield Index (CDX.NA.HY) for all the available tenors (1Y, 2Y, 3Y, 5Y, 7Y, 10Y) 

3. The USD Zero Rate Curve (0USDZ=R) historical values for all tenors 

The USD Zero Rate Curve (USDZ) was used as a proxy for the discount rate, as the official ISDA Curve used for discounting was not available for longer than the last 6 months. We evaluated the difference in terms of hazard rates obtained using the USDZ curve and the current ISDA SOFR curve and the difference was found only in the fifth decimal place of the hazard rate, meaning less than 1 basis point of difference in our analysis. 

3.2 Machine Learning Models Data 

We use a combination of data from Bloomberg and Refinitive depending on the models. In the first part of our project, we use data ranges from 2007 to 2022 to have a complete view of CDS and CDS indices’ behavior during both GFC and COVID crisis. For the second part of the project, our goal is to use Machine Learning methods to predict CDS spread using data from August, 2005 to February, 2023. 

Table 1: Summary of the datasets used 

Here, 

1. Spot rate is 5-year Treasury constant maturity rate 

2. Term-structure slope is difference between 10-year Treasury Constant maturity rate and the 2-year Treasury Constant maturity rate 

3. Volatility is represented by Chicago Board Options Exchange volatility index (VIX index) 4. Equity’s market value is represented by Standard &Poor’s 500 stock market index (S&P 500) 

3.2.1 Data Sources 

We tried to pull data from multiple sources like Bloomberg, CapitalIQ, yahoo finance and Thomson Reuters Datastream (Refinitiv). Each data source had a different learning curve, eventually we are using CDX.NA.IG data from Bloomberg, CDX.NA.HY data from Refitiv & spot rate, term structure slope, S&P500 and VIX index data from yahoo finance. 

3.2.2 Data Performance 

Figure 11: Performance of the North America CDX IG, quoted in spreads 

Figure 12: Performance of the North America CDX HY, quoted in prices 

3.2.3 Important Shocks in the Data 

Both the North American indices have similar behavior over the entire time frame. Our dataset begins from the second quarter of 2009 and we observe a persistent downward trend in both the North American indices, pointing towards the recovery caused due to the damages from the 2007 subprime crisis. 

1. Global Financial Crisis: During the global financial crisis (GFC), the global financial markets witnessed a period of extreme stress, making the recovery much slower than any of the past recessions. [8]Major causes of the crisis included increased borrowing through banks and investors during favorable market conditions and policy/regulation (regulations on subprime and MBS products were too lax) errors coupled with major risk taking. This, undoubtedly, led to a crisis situation where banks incurred huge losses and millions of people lost their way of livelihood. 

In terms of credit easing, the trail for the financial crisis can be traced back to 2000 to 2003 [23] when Federal Reserves lowered the fed fund rate significantly to soften the blow from the dot com bubble and 9/11 attacks. Lower rates led to increased borrowing and funds being invested in Mortgage backed securities. 

2. London Whale: JP Morgan ignored multiple red flags regarding the London Whale trade which led to the crisis. In December, 2011, warnings related to the synthetic credit position were ignored and when losses were incurred the position wasn’t unwinded but the people responsible doubled down on the position. This is a classic example of disposition effect seen in the market and the active traders where there is a tendency to hold on to the losing stocks and sell off of the winning stocks. By April, 2012, the position was risky enough that potential loss on the same was described as $500 million. [24]At the end, there was a loss of around $6 billion. 

3. Taper Tantrum: To mitigate the effects of 2008 financial crisis which impacted the whole world’s economy, Federal Reserve introduced Quantitative Easing which involved bulk purchase of securities 

from the market to inject liquidity, keeping the policing rates at minimum. [25]Eventually during 2013, the Federal Reserve stopped Quantitative Easing and started to increase the interest rates, leading to a panic in the markets and a spike in the stock prices of many companies. As we already know that if interest rates increase, bond yields increase, therefore, with the gradual tapering of interest rates by the Federal Reserve in 2013, yield started to increase giving the equity holders a completely risk free tool for investing leading to the crash of the stock market. 

4. Chinese Crash: The Chinese Stock market crash of 2015-2016 started in the early February of 2016 and witnessed aftershocks in July and August (24th of August, [21] termed as ”Black Monday”). The trail for the stock market selloff can be traced back to the fall in the SSE Composite Index which fell 43%, leading to the devaluation of Yuan. The stock prices declined globally as a consequence, leading to the investors selling shares. Many other events played a part in the crash, including, fall in petroleum prices, the Greek debt default in June 2015, effects of the end of quantitative easing in the United States in October of 2014 and a sharp rise in the bond yields in early 2016. 

5. Fed Hike: There was a spike in the overnight money market rates in the mid of September of 2019, which exhibited a significant volatility alongside a large drop in the reserves due to corporate tax and increase in net Treasury issuance. [11] Multiple factors contributed to the sharp rate movements in the repo markets which were associated with the pressures in the fed funds market. Federal funds rate rose its target range by a quarter from 1.5% to 1.75%, making the sixth increase since 2015. 

6. Covid-19: Covid-19 financial crisis refers to the global recession caused by the Covid-19 pandemic. The recession began with the stock market crash witnessing a 20% drop by late February 2020 and March 2020. [22]Even though the economy started to recover by April 2020, there were signs of high unemployment globally and the impact of the recession was seen for a long time throughout 2020 in multiple industries including hospitality, tourism, oil and energy. 

3.2.4 Data Processing & Transformations 

We compare the state of the art Machine Learning models like LSTM (Long short term memory) with the linear models including statistical models like AR with different lags, VAR (vector autoregression) and ARIMA (Autoregressive integrated moving average). 

For the linear models, we do the following transformations 

1. We check for stationarity using Augmented Dickey Fuller test, there are unit roots present as part of spot rate, term structure slope and index spread/prices, first differencing is done to remove the unit root. The images are added as part of the Appendix. 

2. Normality is ensured before running Ridge and Lasso regression 

3. PACF and ACF plots were made which suggested that we added a single lag to our regressions. Again, these are part of the Appendix for both the indices 

Figure 13: data plots before transformation 

Figure 14: data plots after transformation 

Figure 15: data plots after transformation for HY 

4 Index CDS Implied Default Probabilities 

4.1 Valuation Model 

We use the ISDA Standard Model, this is the market standard and default model used in the Bloomberg Terminal. Therefore, we expect this model to yield the most accurate results in terms of the market sentiment regarding implied default probabilities. 

We used the Bloomberg Credit Swap Valuation Tool to verify our calculations for different dates, but we introduced one modification to the standard methodology. The Bloomberg Credit Swap Tool follows the market mechanic that considers a standard fixed coupon payment for the duration of the contract (100 bps in the case of the NA IG index). But the value of the payments should be given by the current spread. To correct for this, the market considers the ”Upfront Premium”, a cash amount that is paid immediately to the seller (or the buyer in case of negative amounts). 

We can show the methods are equivalent by decomposing the sum of the cash flows: 

_{P V (CF s) =} Xⁿ 

i=1 

Where (S_n + X) = 1%, then we have: 

∆_i· P s_i· DF_i· (S_n + X) 

_{P V (CF s) =} Xⁿ i=1 

∆_i· P s_i· DF_i· S_{n +}Xⁿ i=1 

∆_i· P s_i· DF_i· X 

P V (CF s) = P V (F ee) + Upfront 

Therefore, when we consider only the spread as the coupon payment, we can ignore the upfront premium used to balance the value of the contract. This simplifies the calculations while providing equivalent results. Considering all, the model used to calculate the implied probability of default follows: 

S_nXⁿ i=1 

∆_i· P s_i· DF_i + Accr_{1 = (1 − R)}Xⁿ i=1 

(P s_i−1 − P s_i) · DF_i + Accr₂ 

Here Accr₁ comes from the assumption that the default is occurring mid-period, and Accr₂ is the compensation provided to the buyer because she must pay for the full-period accrual in the first coupon payment. This can be expressed as: 

Accr₁ = S_nXⁿ i=1 

∆_i 

₂· (P s_i−1 − P s_i) · DF_i 

Accr₂ = S_n · ∆_i−1,t 

Recall that P s_i is a function of the hazard rate (λ), which will be the unknown for our model. In our model selection, where hazard rates are constant, they are related by the equation: 

P s_i = (1 − λ)ⁱ 

Now we can simplify the original model notation from the literature review: 

S_nXⁿ i=1 

∆_i· (1 − λ)ⁱ· DF_{i + Accr = (1 − R)}Xⁿ i=1 

((1 − λ)ⁱ⁻¹ − (1 − λ)ⁱ) · DF_i + Accr₂ 

P V_{f ee}(λ) + Accr₁ = P V_contingent(λ) + Accr₂ 

Then, the minimization problem we solve is stated as: 

min_λ(P V_{f ee}(λ) + Accr₁ − P V_contingent(λ) − Accr₂)² 

_{s.t. λ ≥ 0}(1) 20

4.2 Results 

4.2.1 Real Default Probabilities 

To obtain an estimation of the default probabilities of an IG issuer defaulting in the next year, we use the transition tables provided by S&P[17]. We consider the average probability of investment grade ratings moving to default in one year. 

The average probability is 0.51%, which provides us with a lower bound estimate for our hazard rate calculation. By examining figure 16, we can see that this is in line with the results we have from our model, where the difference can be attributed to two main sources: the difference between risk-adjusted and physical probabilities, and the correlation that is present in the number of defaults. 

4.2.2 Implied Hazard Rates 

We calculated the implied hazard rates for the North American Investment Grade Index CDS (CDX.NA.IG). In figure 16, we show the historical series for the different tenors available. In the same figure, we highlighted the periods of shocks to the index that we identified in section 3.2.3. 

Figure 16: Historical Implied Hazard Rates 

We can see that during the financial crisis, the hazard rates of all tenors elevated up to ranges of 4% to 5%. Since then, the hazard rates have been in the range of 0.5% to 2% (depending on the tenor), except during particular periods of shock. 

4.2.2.1 Term Structure of Hazard Rates 

We can analyze the data by looking at transversal samples of historical values. In a similar way as bonds for different maturities define a term structure of interest rates, the different tenors of Index CDS define a term structure of hazard rates. An example of this is shown in figure 17, where we can see that both hazard rates and spread term structures are similar in shape, but differ in slope. We will analyze the implications of this in section 4.2.3. 

Figure 17: Spreads and Hazard Rates Term Structure for 2022-09-28 

4.2 Results  

As we have identified different shocks, it is interesting to study the behavior of the term structure during these stress periods. We use the information in figure 18 to qualitatively characterize the effects of shocks in the hazard rate term structure. 

In the left column of figure 18, we show all of the curves belonging to the particular stress period (identified in the left Y axis). The intensity of the lines reflects more data points superposed in the same location. This provides us with a general idea of the curve shape during the period and also the dispersion during that time. 

We can see that during the financial crisis, the term structure was almost flat, and even sometimes downward sloping. Since that period, we have only seen term structures with positive slopes. Note also the difference in the level of the term structure, which even during the following stress periods has not reached similar levels as of the financial crisis. Finally, we observe that the only time that the term structure has come close to being as flat as in the financial crisis was during the REPO market crisis in 2020, reflecting the severity of the liquidity impact in the market. 

The right column of figure 18 shows in green the term structure of the period immediately before the shock. For example, the bottom-right panel shows in green all the curves between February 2019 and March 2020. The average term structure during the following shock period is shown in black, this provides a reference of the effect the shock had on the hazard term structure. 

It is clear that shocks generate the whole term structure to raise to higher levels than those seen in normal periods. This applies to every case except for the Taper Tantrum, which may be explained by the fact that markets were already in turmoil before this event. 

Figure 18: Hazard rates term structure in stress periods 

4.2.3 Trading Application: Relative Value Strategy 

If we refer back to figure 17 we can notice that the slopes between spreads and implied hazard rates point to different signals. We can analyze this in a similar way as the interest rate curve analyzes forward rates between different periods. 

For spreads we have the standard forward rate formula: 

spread_{fwd = [} (1 + spread_t)^t 

(1 + spread_t−1)^t−1]¹ 

^∆t,t−1 − 1 

11 We can generate an equivalent signal based on hazard rates, as we showed in section 2.5.1, the probability of default in a period is defined by: 

P d_i−1,i = (1 − λ_i−1)ⁱ⁻¹ − (1 − λ_i)ⁱ 

We can call this our ”hazard forward rate”, as it determines the implied forward rate in a given period. A note before proceeding is that this is only an approximation and not the actual probability of default implied in a period. To do this correctly, we would need to modify our model specification to allow for varying hazard rates and bootstrap the hazard rate structure on each date. 

We can continue with this approximation and address the next issue, the periods between contracts are not the same so the probabilities are not comparable. To correct for this we add a transformation similar to the forward rates by raising to the power of the inverse of the period length. Then our signal is defined by: 

hazard_fwd = [(1 − λ_i−1)ⁱ⁻¹ − (1 − λ_i)ⁱ]¹ 

^∆t,t−1 

Note that the units of this expression have no real interpretation, but it is the relative value we are interested in. We observe the results of this for the same date as figure 17 in figure 19. 

Figure 19: Forward Spreads and Forward Hazard Rates 

From this, we can note that the spreads would suggest a relative value trade between the 6 and 8.5 periods, but the hazard rates show that the periods that have the same default rate are the 4 and 6. We show how the historical forward hazard rates look from a historical perspective in figure ??. 

4.2 Results  

Figure 20: Historical Forward Hazard Rates 

We focus on the difference between the 6-year and 4-year forwards in the last 10-year period in figure 21. The trend seems to mean reverting, so a relative value trade could be set up using this signal. To do this, if the value of the signal is above the mean, the trader would have to take a long position on the 6-year forward period and a short position on the 4-year (the opposite in case the signal is below the mean). To take a long forward, we require to buy the contract with longer maturity and sell the contract with shorter maturity. This trade overall would involve 4 positions. 

Figure 21: Historical Forward Hazard Rates 6y-4y Difference 

5 Machine Learning and CDX 

In this section of our paper, we wish to replicate the research done on forecasting and trading two most liquid CDS Indices, CDX.NA.IG and CDX.NA.HY using Merton-LSTM and LSTM model. Merton-LSTM is a deep learning model formed by integrating LSTM with Merton determinants. In the paper, the authors discuss that Merton-LSTM performs better than LSTM, gated recurrent unit (GRU), multilayer perceptron network (MLP), support vector machine (SVM) and a typical stochastic series model in forecasting High Yield and Investment Grade CDS indices by generating highest annualized Sharpe ratios and lowest maximum losses.

By forecasting credit indices, market participants are able to get a comprehensive picture of the appropriate hedging models and profitable trading strategies to manage credit risk along with being a crucial indicator of the financial conditions of the whole economy. We wish to test the following hypothesis for forecasting CDS indices through this part of our paper: 

1. Linear models including auto-regressive/vector auto-regressive models perform poorly 2. Merton-LSTM performs better than LSTM model 

3. Adding commodities like gold, natural gas and electricity prices as factor improves forecasting (given the strong spillover effects of commodity indices over credit indices) [6] 

5.1 Merton Model 

5.1.1 Background 

In 1974, Merton developed a framework for pricing credit risk based on the idea that a company is financed by equity (shared percentage of the firm coupled with unlimited upside) and debt (fixed return, low risk). Merton structural models use the equity price of the firm in order to determine the probability of default.

We understand that equity of a company comes with an unlimited upside to make profits along with high risk and debt comes with fixed returns along with security of payments. For a company to fall apart, debt gets repaid first and then the shareholders are paid back. During bankruptcy, the company is liquidated and debt is unable to be repaid using equity. Using the Merton’s model, one can calculate the probability of default of a company, company’s debt during the lifetime of the trade and the credit spread. 

5.1.2 Mathematics 

We assume that we are dealing with no-dividend stocks for now and all the debt is financed by zero-coupon bonds. Formulation for the Merton determinants are as follows 

Value(t=0) = Equity(t=0) + Debt(t=0) 

Debt(T) = min(Debt(0)^∗(1 + i)^T, Value(T)) 

Equity(T) = max(Value(T) − Debt(T), 0) 

[19]We come across two cases in such a situation, when a company does well and when it doesn’t. When a company is doing well, equity is valued at Value(T) – Debt(T) and debt will be easily repaid taking into consideration the interest rate as Debt(0)^∗(1 + i)^T. However, when the company is doing poorly, equity is priced at 0 as the company is liquidated and debt is the remaining Value(T). 

5.1 Merton Model  

Figure 22: Payoffs of Merton’s model, Equity and Debt as call/put options 

here, 

1. Equity(0) = c (Premium for a call option) 

2. Debt(T) = k (Strike Price) 

3. i^∗Debt(0) = p (Premium for a put option) 

4. Value(T) = S_T (Share Price at Time T) 

5. Value(0) = S₀ (Share Price at Time 0) 

Using the payoff diagrams, we can use Black Scholes Model to get the probability of default at time T. Value(t=0) = Equity(t=0) + Debt(t=0) 

Debt(T) = min(Debt(0)^∗(1 + i)^T, Value(T)) 

Equity(T) = max(Value(T) − Debt(T), 0) 

Equity(0) = Value(0)N(d₁) − Debt(T)e^rT N(d₁) 

i^∗Debt(0) = Debt(T)e^−rT N(−d₂) − Value(0)N(−d₁) 

d_{1 =}ln( ^S0 

_k) + (r + 0.5σ²T 

_σ^√_T) 

d₂ = d_{1 − σ}^√T 

To calculate the probability of default, we know that put option will be in the money such that P(Value(T) < Debt(T)) = N(−d₂) and i^∗Debt(0) = Debt(T)e^−rT − Debt(0). This implies that Debt(c) = Debt(T)e^−bT and the credit spread is b − r where b is the implied credit spread. 

5.1.3 Merton’s Determinants 

[3]Most fundamental factors that determine credit spread changes include 

1. Spot Rate: It has been pointed out that higher spot rate reduces the probability of default by increasing the risk-neutral drift of the entity’s value process. Negative correlation has been found in research between credit spreads and spot rates. 

2. Term-structure slope: Default probability might increase with slope steepening. Litterman discovered that term structure slope can be one of the primary factors driving the influence of CDS spreads. 

5.2 LSTM Model  

3. Volatility: One of the major theoretical determinants of the CDS spreads is firm’s volatility, intuitively, it makes sense such that if there are fluctuations at the firm level, the probability of hitting the barrier-to default increases. Considering equity volatility to be a proxy for the firm-wide volatility or fluctuations, one can argue that an upward trend in the equity volatility can lead or is proportional to an upward trend in CDS spreads. 

4. Firm Value (Probability or Magnitude of a downward jump): Considering the impact of firm value on CDS Spreads, one can argue that if the firm value goes down, the probability of default will increase given that hitting the default barrier will become more likely. Therefore, using equity value as proxy for firm value, [1]structural models of CDX suggest that downward trends in the firm/equity level are accompanied by upward trends in the CDS spread levels. 

5.2 LSTM Model 

Long Short Term Memory (LSTM) Model, an artificial neural network used in the fields of artificial intelligence and deep learning. Similar to recurrent neural networks (RNN), LSTM models have feedback connections, making them better at time series modeling in comparison to Neural Networks. LSTM [20] networks are well-suited for classifying, processing and making predictions based on time series data. Moreover, LSTM networks are better than recurrent neural networks (RNNs) as they overcome the vanishing gradient problems and learn long-short term dependencies more easily. 

Figure 23: LSTM architecture and the structure of an LSTM memory cell 

5.3 Merton-LSTM 

Using the Merton’s model. One infers that a company’s default happens stochastically and independent of the market conditions/perturbations. Whereas LTSM model, though, good at predicting CDS Spreads, majorly data driven and forecasts CDS spreads based on past CDS Spreads. Therefore, seen that Merton’s model is company specific and doesn’t take market conditions into consideration while calculating the probability of default and LSTM only analyzes the data, making predictions on the basis of historical events without considering any financial and economic condition of the company. In order to make better predictions of the CDS spreads, it only makes sense to combine both these models in a way that financial determinants can be integrated with the neural network. 

5.4 Experimental Design  

Figure 24: Forecast of CDS spread at time t + 1 

where 

x^t: vectors consisting of Merton determinants (Spot rate, Term slope, Volatility and Equity Market Value) CDX^t: CDS spreads at time t 

CDX^t+1: Forecasted CDS spread at t + 1 

5.4 Experimental Design 

5.4.1 Methodology 

We start our forecasting part by utilizing classical time series models, models like AR(1), ARIMA, VAR, OLS, Lasso, and Ridge. In this section we will outline our work for each model. 

For the OLS model, we started our work by running the augmented Dickey Fuller test, to check if our dependent and independent variables contain unit roots. We found out that our dependent variable CDX, and four of our independent variables, namely: Spot Rate, Term Structure, Natural Gas, and Momentum. Therefore, we had to differentiate the data ,and add a lag of the dependent variable to the list of the regressors that we have, figure 23 and 24 show the time series plot of the variables.

After taking care of the unit root issue, we calculated the correlations between the variables; the goal is to make sure that we don’t have perfect multicollinearity between regressors, figure 25 shows the correlation between variables. After transforming the data to insure stationary, we ran the OLS model. Regression results are in figure 26, it’s worth mentioning that some of the variables are not statistically significant. 

Figure 25: Time Series of the variables 

5.4 Experimental Design  

Figure 26: Time Series of the variables 

Figure 27: Heat Map that highlights the correlations between the variables 

Figure 28: Regression Results 

At day t, we try to predict the CDX IG spread at t + 1 using ML and DL models and a different set of variables. From March 22, 2005 to February 21, 2023, we collected daily data on CDX IG prices and Merton determinants, which was divided into 80% for training data and 20% testing data. Specifically, our training data covers the period from March 22, 2005 to August 7, 2019, while our testing data covers the period from 

5.5 Results  

August 8, 2019 to February 21, 2023. It’s worth noting that the testing dataset includes the Covid crisis, which is expected to be a challenging prediction task for both ML and DL models. Our benchmark for ML models is the following : 

• Support Vector Machine Regression : the algorithm tries to find a hyperplane that maximizes the margin between the predicted values and the actual target values. The hyperplane is chosen such that it lies as close as possible to the points in the training set while also keeping the errors (the difference between the predicted and actual values) as small as possible. 

• Random Forest Regression : it is a popular ensemble learning method used for regression tasks. The algorithm works by creating a forest of decision trees on randomly selected subsets of the data and then averaging the results of these trees to make predictions. 

About the LSTM model, we have chosen two hidden layers with 300 nodes, an Adam optimizer to train the models and a callback function to prevent the overfitting. To facilitate the training process for both models, we have standardized the training data to have a mean of 0 and a standard deviation of 1. To achieve this, we have utilized the sample mean and standard deviation of the training data. Subsequently, we have applied the same transformation of the mean and standard deviation of the training set on the test set. We will compute the errors using the root mean squared errors (RMSE) for the test set. 

5.5 Results 

We start the forecast results with the OLS prediction. As expected, the OLS predictions did not perform well; given the shocks in time series, the OLS won’t be able to do a good job in terms of forecasting. Figure 27 shows the in-sample time series of the CDX IG, out of sample time series, and the forecast for CDX IG. As shown in the forecast result, the OLD performs poorly, especially during shocks. 

Figure 29: OLS Forecast 

The next model we ran, AR(1), we decided to choose AR(1), instead of higher AR models, based on the PACF, shown in the appendix. The PACF shows the direct effect that a lag has on the dependent variable. The graph shows the first lag above the significance line, other variables did not cross the significance line. Figure 28 shows the prediction results for the AR(1) model. The AR(1) predictions are almost a straight line, which indicates the poor performance of the model. 

5.5 Results 

Figure 30: AR(1) Forecast 

The third Time Series forecast approach we used was the ARIMA model. Starting off, we ran the Auto ARIMA function, which gives us the best ARIMA order. The Auto ARIMA gives a score based on the lowest BIC, among the models. The best ARIMA order we got is (5,0,2). Figure 29, shows the plot of the forecast against the testing data; the forecast is poor, just as expected. 

Figure 31: ARIMA Forecast 

The fourth Time Series model we ran, is the vector Autoregression model. The variables selected in this model are based on their significance in the OLS model we ran. Also, we ran Lasso model, but with changing the alpha, the idea behind that is to see which variables are decaying faster to zero, as the alpha level increases, the results are in figure 30. Aside from the lag, Momentum, VIX, and SP500 show significance in the OLS model, and decay relatively slower in Lasso, compared with other variables. The forecast results of the VAR model are in figure 31. 

5.5 Results  

Figure 32: ARIMA Forecast 

Figure 33: VAR Forecast 

Our fifth model we used is Lasso, but before running the more we normalize the data to insure model conversion. The model performance was not good, either, figure 32 shows the results of the forecast. 

5.5 Results  

Figure 34: Lasso Forecast 

Our last model to forecast CDX IG before pivoting to the Machine Learning forecast approach, is Ridge model, the performance of the model was not good either, just as we expected the model to perform. The forecast results are in figure 35. 

Figure 35: Ridge Forecast 

Overall, the Time Series models did not perform well in terms of forecasting, as we expected. However, we find them useful in terms of figuring out which feature would be important when we finally feed it to the ML models that we are going to use next. Figure 36, shows the root mean square error (RMSE) and mean absolute error (MAE), for the six time series models we ran. We notice that VAR, Lasso, and Ridge, outperformed the rest of the models, in terms of RMSE. 

Figure 36: Ridge Forecast 

Our second forecast for the Time Series approach is trying to forecast CDX HY. We used a similar process as forecasting CDX IG in terms of dealing with unit roots, differencing the time series, checking for perfect multicollinearity, before running the OLS. We also ran, OLS, AR(1), ARIMA, VAR, 

5.5 Results  

Lasso, and Ridge. Figure 37-38, show our time series plot for the variables. Figure 39, shows the correlation heat map that we use to rule out perfect multicollinearity. Figure 40-45 shows the forecast plot for each model. 

Figure 37: Plot of the Data 

Figure 38: Plot of the Data 

5.5 Results  

Figure 39: Variables Corr 

Figure 40: OLS Forecast 

5.5 Results  

Figure 41: AR(1) Forecast 

Figure 42: ARIMA Forecast 

Figure 43: VAR Forecast 

5.5 Results  

Figure 44: Lasso Forecast 

Figure 45: Ridge Forecast 

Similar to the CDX IG case, the forecast was poor. We knew using Time Series for prediction would be a challenge, however, the idea was to use these time series models as a benchmark for the more advanced forecast that we have done through the machine learning approach. 

Now for the DL models, we have worked directly in predicting the spread so the RMSE is about the spread and not any transformations used in linear models. Here are the DL model results: 

Table 2: Summary of the models used for predicting CDX IG spread at date t + 1 from August 2019 to February 2023 

We have the following observations : 

5.5 Results  

• DL models outperform our ML models. Indeed, one potential issue with SVM and RF models when applied to time series data is that the training samples may not conform to the same underlying probability distribution. In other words, some of the data points among the n samples may deviate from the expected probability distribution, leading the models to learn noise rather than meaningful patterns. LSTM suffers less from this issue as it can handle time series data by retaining information about past inputs through a memory mechanism. This enables the LSTM to capture long-term dependencies in the data and make accurate predictions. 

• LSTM models that only use the spread at date to outperform those that use the spread along with Merton determinants and other variables such as gold and gas prices. This finding may seem counterintuitive, but our results indicate that adding more variables does not improve the forecasting performance of the model. In fact, it appears that the additional variables introduce noise into the model, making it more difficult for the LSTM to capture the purest information about the spread. 

To test the last point, we use an interesting feature from RF where it computes the feature importance. It is typically calculated using the Mean Decrease Impurity (MDI) method. During the training process, the random forest builds multiple decision trees using bootstrap samples of the training data and a random subset of features at each node. To measure feature importance, the MDI method calculates the decrease in impurity or purity of the target variable that results from splitting on each feature across all decision trees in the forest. The feature importance score for each feature is calculated as the average of the total decrease in impurity or purity over all trees in the forest. Features with high scores are deemed more important because they contribute more to the overall reduction in impurity or increase in purity across the forest. 

Figure 46: Feature importance in random forest 

The graph is normalized so that the sum for all the feature importance is 1. It shows that when predicting for t + 1, the CDX IG spread at date t is the most crucial variable among our prediction variables and has an importance of 99.5%. The addition of Merton determinants does not provide any significant information. 

Another forecasting objective is to predict the values for day t + h, where h represents a horizon date selected from the range [1, 3, 5, 7, 10, 14, 21, 28]. We compared the DL model and noted that the RMSE increases on the test set as it becomes more challenging to predict the values for t + h at date t. Our observations still reveal that LSTM performs better than LSTM Merton. 

1 3 5 7 10 14 21 28 

LSTM Merton 3.462642 4.809693 5.673161 6.484545 7.432075 8.536648 10.405896 12.004688 LSTM 3.345445 4.586897 5.587412 6.319453 7.367975 8.355636 10.142439 11.555155 

Table 3: RMSE of LSTM and LSTM-Merton models used in the time period from August 2019 to February 2023 

6 Future Work 

6.1 Factor Models 

Much research has been done till date to understand the possibility of market factors influencing the credit default swaps. Factors, including, commodity spot prices, foreign exchanges spot prices, equity index prices, ESG factor returns and jump to default risks have been some of the most important determinants influencing the CDX prices and spreads. Factor modeling coupled with time series analysis seems to be a novel and interesting idea for trading, hedging and research perspectives. 

[12]Engman and Alander found strong correlations between commodity spot prices, foreign exchange spot prices, equity index prices and credit default swaps. Takei, examined whether Environment, social, and corporate governance have an effect on CDS spreads (Spglobal,2022)[18]. Takei used ENV, SOC, and GOV factor returns; those factors represent market sentiment of ESG scores. Each factor is essentially a portfolio that generates returns when the market compensates firms with higher ESG scores.

Takei found a significant negative correlation between Environmental factor (ENV) and CDS spreads; however, Social (SOC) and corporate governance (GOV) factors weren’t significant. [9] Benjamin Yibin Zhang, Hao Zhou & Haibin Zhu, wanted to see if CDS spreads could be explained by Equity Volatility and Jump Risks of Individual Firms (Bank for International Settlements,2005). They authors find that their model contains volatility risk as a factor, alongside with jump risk, by using these two factors, their model could explain as much as 69% of total variation of CDS spreads. However, when Macroeconomic factors are added into the mix, total variation explained by the model increases to 77%. 

6.1.1 Electricity Price volatility 

Electricity prices can be a good commodity factor for forecasting CDX prices and spreads given the volatility. We will explore this as future work since limited data is available on a monthly basis, therefore, complex machine learning models are hard to train, however, Ordinary Least Squares can be tested. 

6.2 Trading Strategies  

Figure 47: Consumer Price Index: OECD Groups: Fuel, Electricity, and Gasoline for the United States

6.2 Trading Strategies 

Trading Strategies help in analyzing the economic performance of the structural models in Investment grade and High Yield markets to create arbitrage. There are two classic types of strategies-static and dynamic. 

6.2.1 Static Strategies 

Investments in the credit indices are done regardless of the effective economic and market conditions and trading signals from the structural models’ forecasts are not used or taken into account. 

6.2.2 Dynamic Strategies 

Investments in the credit indices are done taking into account the trading signals obtained from the structural models’ forecasts. 

1. Binary Spread Strategy: Binary Spread strategy is used to predict the credit spread changes in the correct direction regardless of the accuracy of the predictions. 

2. Scaled Spread Strategy: Scaled Spread strategy is used to predict not only the correct direction of the credit spread but is also concerned with the magnitude of the forecast. 

6.3 Additional Merton’s Determinants 

1. [5]Leverage (ratio of amount of debt outstanding and equity value) ^Debt 

_{Equity Value} : Credit Spreads are 

expected to increase as leverage ratio approaches unity. Therefore, this is a good Merton’s Determinant which can be tested in the future given enough data on the debt markets is available. 

2. Business Climate: Expected recovery rate of the firm is dependent on the business climate of the firm and hence changes in credit spreads are dependent on it, irrespective of the probability of default. Therefore, it could be a good determinant from a firm’s credit valuation perspective. 

6.3 Additional Merton’s Determinants  

3. Jump-to-default Risk: Jump to default happens when a company defaults suddenly before the market has had the time to account for its increased default risk. Magnitude of jump risks could impact the credit spread changes and hence they are an important Merton’s determinant. 

Written by

Alexandre Caduc 

Divya Garg 

Juan Mora 

Quynh Anh Pham 

Mohammed Alshowaikhat 

Understanding CDS Indice Behavior During Periods Of Market Stress & Predict Their Prices Using Machine Learning Techniques : What Happened With Credit Default Swaps In 2008?

Acknowledgements 

We would like to express our deepest gratitude to our academic advisor Professor Nancy Wallace for all her support and mentorship in this project. Professor Wallace guided us through idea generation, steered us in the right direction, helped us pass through hurdles during data finding and research process. Her enthusiasm inspired us to explore this novel topic and stay on track of the research. 

We also would like to thank Emmanuel Vallod, for his constructive feedback and industry insights on the credit derivatives market. 

Finally, we are grateful for the support of Aaron Nebres, Lab Manager at the Master of Financial Engineering at Berkeley, who helped us get access to all the data sources needed for the project. 

Appendix 

• Firstly, Figure 48: Summary of descriptive statistics from 8th April, 2009 to 20th January, 2020
• Figure 49: Correlations 
• Figure 50: Correlations 
• Fourthly, Figure 51: Augmented Dickey Fuller Test 
• Figure 52: Augmented Dickey Fuller Test 
• Figure 53: PACF plots 
• Additionally, Figure 54: PACF plots 
• Figure 55: ACF plots 
• Figure 56: ACF plots 

Understanding CDS Indice Behavior During Periods Of Market Stress & Predict Their Prices Using Machine Learning Techniques : What Happened With Credit Default Swaps In 2008?

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[22] Wikipedia. Covid-19. url: https://en.wikipedia.org/wiki/COVID-19_recession. 

[23] Wikipedia. Global Financial Crisis 2007-2008. url: https://en.wikipedia.org/wiki/2007%5C%E2% 5C%80%5C%932008_financial_crisis%5C#Easy_credit_conditions. [24] Youtube. London Whale Crisis. url: https://www.youtube.com/watch?v=6-ga2PbREK4. [25] Youtube. Taper Tantrum. url: https://www.youtube.com/watch?v=TAR23e_hyTo.

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