---
title: "Credit Default Swaps"
book: "Markets II: Rates, FX and Credit"
subject: quant
language: en
chapter: 23
exercises: 8
source: https://one-course.com/books/quant/2/en/chapter/23-credit-default-swaps
---

# Chapter 23 — Credit Default Swaps

Lehman Brothers filed for bankruptcy on 15 September 2008. [Credit default swaps](#def-m2-credit-default-swaps-cds) on it with a gross notional of about USD 72 billion were registered, and only about a third of that protection could have been bought to hedge Lehman bonds; the rest was held by investors who had no bonds to hand over. On 10 October fourteen dealers ran a two-stage auction on behalf of 358 parties; it priced Lehman’s senior bonds at 8.625 per 100, and every [protection seller](#def-m2-credit-default-swaps-cds) that had signed up paid 91.375 per 100 of notional, in cash. Offsetting positions shrank the money that changed hands to about USD 5.2 billion. This chapter explains the contract that made those payments: its standard form since 2009, how it is quoted and priced with a [hazard rate](#def-m2-credit-default-swaps-hazard), who decides that a default has happened, how the auction sets the price, and why the cheapest bond decides the payout.

## 23.1 The standard contract

**Definition 23.1 (Credit default swap, protection buyer and seller, reference entity).**

A *credit default swap* (CDS) is a bilateral contract in which the *protection buyer* pays a regular premium on a notional amount until maturity or until a [credit event](#def-m2-credit-default-swaps-event) of a named borrower, the *reference entity*, whichever comes first; after a [credit event](#def-m2-credit-default-swaps-event) the *protection seller* pays the buyer the notional times one minus the recovery value of the entity’s debt, fixed by settlement.

A CDS is insurance on a borrower’s debt, with two differences that matter: the buyer need not own the debt, and the contract can be traded and offset like any swap. The buyer is short the credit and the seller long, so a seller of protection has the same exposure to default as a bondholder without having to fund the bond. On USD 10 million of protection at 100 basis points a year, the buyer pays USD 25 000 a quarter; if the debt is worth 40 per 100 after default, the seller pays USD 6 million ([Figure 23.1](#fig-m2-credit-default-swaps-flows)).

![The cash flows of a credit default swap. The buyer pays a running premium until a credit event or maturity; after a credit event, determined by a committee, the seller pays the notional times one minus the price of the entity’s debt set by an auction. Schematic.](https://one-course.com/images/onecourse/chapters/quant-2/m2-credit-default-swaps/fig-3299c77dfc75.svg)

***Figure 23.1.** The cash flows of a [credit default swap](#def-m2-credit-default-swaps-cds). The buyer pays a running premium until a [credit event](#def-m2-credit-default-swaps-event) or maturity; after a [credit event](#def-m2-credit-default-swaps-event), determined by a committee, the seller pays the notional times one minus the price of the entity’s debt set by an auction. Schematic.*

**Definition 23.2 (Standard coupon, upfront payment).**

A *standard coupon* is a fixed running premium, the same for all contracts of a class, which the market trades instead of a coupon negotiated for each trade. The *upfront payment* is the amount, in per cent of notional, that one party pays the other when the trade starts so that the contract with the standard coupon is fair at the market’s spread; the [protection buyer](#def-m2-credit-default-swaps-cds) pays it when the spread is above the coupon and receives it when below.

Until 2009 each trade carried its own coupon, the par spread of the day, so two offsetting trades done a week apart left two streams of different premiums that could not be netted or cleared. From 8 April 2009, contracts on North American companies moved to standard terms: a fixed coupon of 100 basis points for investment-grade names and 500 for high yield, with the difference paid upfront; maturities on the quarterly roll dates, 20 March, June, September and December; premiums accruing from the last roll date before the trade, the stub settled with the upfront; and restructuring no longer a [credit event](#def-m2-credit-default-swaps-event) for these contracts. The same change created the committees and hard-wired the auctions of [Section 23.3](#sec-23-3). Contracts on the same name and maturity became identical, and so could be netted and, later, cleared by a central counterparty.

**As of September 2026 — The size of the CDS market.**

At the end of 2025, by the BIS statistics as summarised by ISDA: notional outstanding USD 11.0 trillion, up 21.7% in a year, of which USD 4.6 trillion single-name and USD 6.4 trillion on several names (indices and tranches, next chapter); gross market value USD 273.7 billion; 70.2% of the notional, USD 7.7 trillion, cleared by central counterparties.

## 23.2 Upfront and running

To convert a spread into an [upfront payment](#def-m2-credit-default-swaps-std), and to price a contract at all, one needs a model of when the entity defaults.

**Definition 23.3 (Hazard rate, survival probability).**

The *hazard rate* $\lambda(t)$ of a [reference entity](#def-m2-credit-default-swaps-cds) is its instantaneous rate of default at time $t$ given survival to $t$: the probability of default in $[t, t + dt]$, given no default before $t$, is $\lambda(t)\,dt$. The *survival probability* to $T$ is $Q(T) =
\exp\bigl(-\int_0^T \lambda(u)\,du\bigr)$, which is $e^{-\lambda T}$ for a flat hazard rate.

A CDS has two legs. The premium leg pays $c$ a year, quarterly, while the entity survives, plus the premium accrued up to a default; its value per unit of coupon is the risky *annuity* $A$ (chapter 9’s annuity, with each payment weighted by the probability of surviving to it). The protection leg pays $1 - R$ at default, with $R$ the [recovery rate](https://one-course.com/books/quant/2/en/chapter/21-corporate-bonds#def-m2-corporate-bonds-covenant) of chapter 21. With a flat rate $r$ and a flat hazard $\lambda$, paying defaults at the end of the quarter in which they occur and counting half a quarter of accrued premium,

$$
A = \sum_{k=1}^{4T} \tfrac14\, e^{-r t_k}\Bigl(Q(t_k) + \tfrac12\bigl(Q(t_{k-1}) - Q(t_k)\bigr)\Bigr),
\qquad
P = (1-R)\sum_{k=1}^{4T} e^{-r t_k}\bigl(Q(t_{k-1}) - Q(t_k)\bigr),
$$

with $t_k = k/4$. The par spread is $P/A$, the running premium that makes the contract worth zero.

**Proposition 23.4 (The credit triangle).**

With a flat [hazard rate](#def-m2-credit-default-swaps-hazard) $\lambda$, a constant recovery $R$ and premiums paid continuously, the par spread is

$$
s = \lambda\,(1 - R),
$$

whatever the interest rate and the maturity.

**Proof.** With premiums paid continuously, the premium leg is worth $s\int_0^T e^{-(r+\lambda)t}\,dt$ and the protection leg $(1-R)\int_0^T \lambda\,e^{-(r+\lambda)t}\,dt$, since default occurs in $[t, t+dt]$ with probability $\lambda e^{-\lambda t}dt$. The integrals differ only by the constant factor $\lambda$, so the legs are equal when $s =
\lambda(1-R)$. ∎

With quarterly payments the triangle is no longer exact, but close: at a spread of 100 basis points and a recovery of 40%, the [hazard rate](#def-m2-credit-default-swaps-hazard) that prices the five-year contract is 1.6667%, against $0.01/0.6 = 1.6667\%$ from the triangle, and the five-year [survival probability](#def-m2-credit-default-swaps-hazard) is 92.0%.

**Proposition 23.5 (Upfront from a quoted spread).**

If a contract with [standard coupon](#def-m2-credit-default-swaps-std) $c$ is quoted at a spread $s$, and $\lambda_s$ is the flat [hazard rate](#def-m2-credit-default-swaps-hazard) at which $s$ is the par spread, the [upfront payment](#def-m2-credit-default-swaps-std) per unit notional, paid by the [protection buyer](#def-m2-credit-default-swaps-cds), is

$$
U = (s - c)\,A(\lambda_s).
$$

**Proof.** At $\lambda_s$ the protection leg equals $s\,A(\lambda_s)$, by the definition of the par spread. The contract with coupon $c$ is worth $P - cA = (s - c)A$ to the buyer, which the buyer pays upfront to make the trade fair. ∎

**Example 23.6 (Two quotes, two conventions).**

With a flat rate of 4% and a recovery of 40%, the five-year risky annuity is 4.332 at a spread of 100 basis points and 4.165 at 200. An investment-grade name quoted at 200 basis points, on the 100 basis point coupon, costs the buyer $1\% \times 4.165 =
4.16\%$ upfront, USD 416 000 on USD 10 million. A high-yield name quoted at 300 basis points on the 500 coupon pays the buyer 8.01% upfront; at 800 the buyer pays 9.98%, and at 1 200, 20.28%. Investment-grade names are quoted in spread, high-yield names in points upfront.

[Figure 23.2](#fig-m2-credit-default-swaps-upfront) shows the upfront against the spread for both coupons: it crosses zero at the coupon, and it bends because the annuity shrinks as the [hazard rate](#def-m2-credit-default-swaps-hazard) rises. The slope near a spread is the annuity, so USD 10 million of five-year protection near 100 basis points gains about USD 4 300 for each basis point the spread widens. [Figure 23.3](#fig-m2-credit-default-swaps-survival) shows what the spreads mean for survival.

![Upfront payment of a five-year contract, in per cent of notional, against the quoted spread, for the two standard coupons (flat rate 4%, recovery 40%). The upfront is zero where the spread equals the coupon, and negative, paid to the buyer, below it; the dots are . Illustrative; data: the chapter’s tutorial.](https://one-course.com/images/onecourse/chapters/quant-2/m2-credit-default-swaps/fig-9a86ed35a5df.svg)

***Figure 23.2.** [Upfront payment](#def-m2-credit-default-swaps-std) of a five-year contract, in per cent of notional, against the quoted spread, for the two [standard coupons](#def-m2-credit-default-swaps-std) (flat rate 4%, recovery 40%). The upfront is zero where the spread equals the coupon, and negative, paid to the buyer, below it; the dots are [Example 23.6](#ex-m2-credit-default-swaps-upfront). Illustrative; data: the chapter’s tutorial.*

![Survival probabilities implied by five-year spreads of 100, 300 and 800 basis points with a flat hazard rate and a recovery of 40%. The hazard rates are close to spread divided by 1 - R: about 1.7%, 5% and 13% a year. These are risk-neutral probabilities, which include a premium for bearing default risk. Illustrative; data: the chapter’s tutorial.](https://one-course.com/images/onecourse/chapters/quant-2/m2-credit-default-swaps/fig-9fd15e3aa696.svg)

***Figure 23.3.** Survival probabilities implied by five-year spreads of 100, 300 and 800 basis points with a flat [hazard rate](#def-m2-credit-default-swaps-hazard) and a recovery of 40%. The [hazard rates](#def-m2-credit-default-swaps-hazard) are close to spread divided by $1 - R$: about 1.7%, 5% and 13% a year. These are risk-neutral probabilities, which include a premium for bearing default risk. Illustrative; data: the chapter’s tutorial.*

The probabilities are those that price the contract, not forecasts. Chapter 21 split a bond’s spread into expected loss and a premium for risk and illiquidity; the same split holds here, so a [hazard rate](#def-m2-credit-default-swaps-hazard) read from a spread is higher than the default rate history would suggest.

## 23.3 Credit events and determinations committees

**Definition 23.7 (Credit event, determinations committee).**

A *credit event* is one of the events listed in a CDS contract whose occurrence triggers the protection payment, such as the [reference entity](#def-m2-credit-default-swaps-cds)’s bankruptcy or failure to pay. A *determinations committee* (DC) is the body of dealers and investors that decides, for all standard contracts at once, whether and when a credit event has occurred and how it is settled.

ISDA’s disclosure material lists the [credit events](#def-m2-credit-default-swaps-event) a transaction may specify: bankruptcy, failure to pay, restructuring, obligation acceleration, obligation default and repudiation or moratorium; the contract says which apply. Bankruptcy and failure to pay are the common ones for companies, restructuring matters for banks and sovereigns, and whether a given event qualifies is often a question of interpretation. Before 2009 each pair of counterparties decided for itself, and two offsetting trades could be triggered differently.

The 2009 reform created a committee for each of five regions (the Americas, Asia excluding Japan, Japan, Australia and New Zealand, and Europe, the Middle East and Africa), with, at its creation, ten dealers and five other members voting and ISDA as secretary. Anyone party to a covered trade may ask a question; the committee decides whether a [credit event](#def-m2-credit-default-swaps-event) has occurred and when, whether to hold an auction, which obligations may be delivered, and whether the entity has a successor. Most decisions need 80% of the members, failing which an external panel decides, and they bind all covered trades. A 60-day look-back replaced each trade’s own start date, so that offsetting trades done on different days are triggered by the same events.

The rules evolve with the cases. In 2013 the Dutch state nationalised the bank SNS and expropriated its subordinated bonds, an event the definitions of the time did not clearly cover. The 2014 definitions, in force from 22 September 2014, added a *governmental intervention* [credit event](#def-m2-credit-default-swaps-event) for financial entities outside the United States and let [protection buyers](#def-m2-credit-default-swaps-cds) deliver whatever assets a bail-in or restructuring turns the bonds into.

## 23.4 Settlement auctions

After a [credit event](#def-m2-credit-default-swaps-event), the payment is $1 - R$ per unit of notional, and someone must set $R$. Physical settlement, the original method, has the buyer deliver bonds of the entity and receive par; but when most protection is held by investors without bonds, as with Lehman, those buyers must buy bonds to deliver, which pushes their price up and the payout down. Cash settlement needs one agreed price for bonds that may barely trade. The auction provides it.

**Definition 23.8 (Auction final price).**

The *auction final price* of a [credit event](#def-m2-credit-default-swaps-event) is the price per 100 of par of the entity’s deliverable debt, set by the settlement auction, at which every CDS covered by the auction is cash settled and every physical settlement request made in the auction is executed.

The auction has two stages. In the first, each participating dealer submits a bid and an offer, no more than 2 apart, for a fixed quotation size of bonds, and physical settlement requests, for itself and its clients, to buy or sell bonds at the final price, each no larger than the party’s CDS position and in the same direction. The administrators remove crossing pairs, highest bid against lowest offer, until no bid reaches an offer, and average the better half of the remaining bids and offers, rounded to the nearest eighth: the *inside market midpoint*. The net of the requests is the *open interest*. A dealer whose quote crossed on the wrong side of the midpoint, given the direction of the open interest, pays a penalty.

In the second stage, participants submit limit orders on the other side of the open interest; the orders are filled best price first until the open interest is met, and the last order filled sets the final price. The price may not be more than 1 above the midpoint when the open interest is to sell, or 1 below when it is to buy.

![The first stage of the Lehman Brothers auction, 10 October 2008: each of the fourteen dealers’ bid and offer for USD 5 million of bonds. One pair crossed (a bid and an offer of 10); the average of the better half of the rest gave an inside market midpoint of 9.75. The open interest was USD 4.92 billion to sell, and the second stage set the final price at 8.625. Data: Helwege, Maurer, Sarkar and Wang, FRBNY Staff Report 372, Box 1.](https://one-course.com/images/onecourse/chapters/quant-2/m2-credit-default-swaps/fig-76d528d400ec.svg)

***Figure 23.4.** The first stage of the Lehman Brothers auction, 10 October 2008: each of the fourteen dealers’ bid and offer for USD 5 million of bonds. One pair crossed (a bid and an offer of 10); the average of the better half of the rest gave an inside market midpoint of 9.75. The open interest was USD 4.92 billion to sell, and the second stage set the final price at 8.625. Data: Helwege, Maurer, Sarkar and Wang, FRBNY Staff Report 372, Box 1.*

**Example 23.9 (Lehman Brothers).**

In the first stage, the highest bid, 10, met an offer of 10 and both were removed; the better seven of the remaining thirteen bids and offers averaged 9.804, rounded to 9.75. Requests to sell bonds totalled USD 5.69 billion and to buy USD 0.77 billion: an open interest of USD 4.92 billion to sell, capping the final price at 10.75. The dealer whose bid of 10 crossed, above the midpoint while the open interest was to sell, paid $5\,\text{million} \times (10 - 9.75)\% = \text{USD}~12\,500$. In the second stage, 453 bids to buy were submitted, from 10.75 down to 0.125; bids at 8.625 or higher added up to the open interest, and the final price was 8.625 ([Figure 23.4](#fig-m2-credit-default-swaps-lehman)).

The auction does two things at once. It sets a single price, so an investor hedged with a single-name contract against an index contract is not left with different recoveries on each. And it turns physical settlement into a trade of the net: a [protection buyer](#def-m2-credit-default-swaps-cds) that owns bonds and wants to be rid of them asks to sell them at the final price and ends with par, while everyone else settles in cash. For Lehman, the USD 72 billion of gross protection became USD 5.2 billion of net payments, because most dealers had bought and sold protection on it.

The price is not guaranteed to match the bond market. Lehman’s bonds traded at 34 on the day of the bankruptcy, 13 the day before the auction and about 10 on the day; the final price, pushed down by the large open interest to sell, was a little lower. In the Fannie Mae auction of 2008, strong demand in the first stage pushed the final price of subordinated debt above that of the senior debt.

## 23.5 The cheapest-to-deliver option

A CDS protects not one bond but a class of the entity’s debt: any obligation of the required seniority, currency and form is deliverable. After a default, bonds of the same seniority are claims on the same estate and should trade at similar prices, but not always: a long bond with a low coupon and a short one with a high coupon can differ before the bankruptcy settles, and in a restructuring the obligations may be treated differently. A [protection buyer](#def-m2-credit-default-swaps-cds) that settles physically delivers the cheapest of them, and in the auction the sellers of bonds deliver the cheapest too, so the final price is set by the cheapest deliverable: the [cheapest-to-deliver](https://one-course.com/books/quant/2/en/chapter/6-bond-futures#def-m2-bond-futures-ctd) option of chapter 6, now in credit. The option is worth most when deliverables differ in price, which is when it matters most: restructurings that leave some bonds long-dated and low-coupon, or bail-ins that convert bonds into other assets.

## 23.6 Tutorial: pricing a CDS and replaying an auction

**Goal.** Price a CDS with a flat [hazard rate](#def-m2-credit-default-swaps-hazard), convert quoted spreads into [upfront payments](#def-m2-credit-default-swaps-std) for both [standard coupons](#def-m2-credit-default-swaps-std), check the credit triangle, and replay the Lehman auction. **End state:** Figures [23.2](#fig-m2-credit-default-swaps-upfront), [23.3](#fig-m2-credit-default-swaps-survival), [23.4](#fig-m2-credit-default-swaps-lehman) and [23.5](#fig-m2-credit-default-swaps-orders), Examples [23.6](#ex-m2-credit-default-swaps-upfront) and [23.9](#ex-m2-credit-default-swaps-lehman) and the numbers of the weekend problem.

1. **Pricing**: the two legs, par spread, upfront, and the [hazard rate](#def-m2-credit-default-swaps-hazard) implied by a spread. `@dataclass (frozen=True ) class Cds : maturity: float coupon: float # 0.01 or 0.05 for standard contracts recovery: float = 0.40 freq: int = 4 def legs (cds: Cds, lam: float , r: float ) -> tuple [float , float ]: """(risky annuity: value of 1 a year of coupon, protection leg value).""" dt, annuity, protection = 1.0 / cds.freq, 0.0 , 0.0 for k in range (1 , round (cds.maturity * cds.freq) + 1 ): q0, q1 = math.exp(-lam * (k - 1 ) * dt), math.exp(-lam * k * dt) df = math.exp(-r * k * dt) annuity += dt * df * (q1 + 0.5 * (q0 - q1)) protection += (1.0 - cds.recovery) * df * (q0 - q1) return annuity, protection def par_spread (cds: Cds, lam: float , r: float ) -> float : a, p = legs(cds, lam, r) return p / a def upfront (cds: Cds, lam: float , r: float ) -> float : """Paid by the protection buyer at the start (negative: received), per unit notional.""" a, p = legs(cds, lam, r) return p - cds.coupon * a def hazard_from_spread (cds: Cds, spread: float , r: float ) -> float : lo, hi = 1e-9 , 5.0 for _ in range (200 ): mid = 0.5 * (lo + hi) lo, hi = (mid, hi) if par_spread(cds, mid, r) < spread else (lo, mid) return 0.5 * (lo + hi) def upfront_from_spread (cds: Cds, spread: float , r: float ) -> float : return upfront(cds, hazard_from_spread(cds, spread, r), r)` **Listing 23.1.** A CDS with a flat hazard rate: legs, par spread, upfront and the implied hazard. code/firm/cds/firm_cds.py
2. **The auction**: inside market midpoint, open interest, final price and cash settlement. `def inside_market_midpoint (bids: list [float ], offers: list [float ]) -> float : """Remove crossing bid/offer pairs, average the better half of the remaining bids and offers, round to the nearest 1/8.""" b, o = sorted (bids, reverse=True ), sorted (offers) while b and o and b[0 ] >= o[0 ]: b.pop(0 ) o.pop(0 ) h = math.ceil(len (b) / 2 ) return round ((sum (b[:h]) + sum (o[:h])) / (2 * h) * 8 ) / 8 def open_interest (requests: list [tuple [str , float ]]) -> float : """Net physical settlement requests: positive to sell bonds, negative to buy.""" return sum (size if side == " sell " else -size for side, size in requests) def final_price (imm: float , oi: float , orders: list [tuple [float , float ]]) -> float : """Second stage: limit orders (price, size) fill the open interest, best price first; the last order filled sets the price, which may not move more than 1 point past the midpoint.""" if oi == 0 : return imm if oi > 0 : # sellers: bids to buy, highest first, cap imm + 1 filled, price = 0.0 , imm for p, size in sorted (orders, key=lambda x: -x[0 ]): filled, price = filled + size, p if filled >= oi: break return min (price, imm + 1.0 ) filled, price = 0.0 , imm # buyers: offers to sell, lowest first, floor imm - 1 for p, size in sorted (orders): filled, price = filled + size, p if filled >= -oi: break return max (price, imm - 1.0 ) def cash_settlement (notional: float , price: float ) -> float : """Paid by the protection seller to the buyer: (100 - price)% of notional.""" return notional * (100.0 - price) / 100.0` **Listing 23.2.** The two-stage settlement auction. code/firm/cds/firm_cds.py
3. **Run** `cds_demo.tutorial()` , `cds_demo.problem()` and `fig_cds.py` ; build and test the C++20 and Rust versions in `code/firm/cds/cpp` and `code/firm/cds/rust` .

**What to change next.** Replace the flat hazard by a curve bootstrapped from one-, three-, five- and seven-year spreads, and see how much the five-year upfront changes; then let the [recovery rate](https://one-course.com/books/quant/2/en/chapter/21-corporate-bonds#def-m2-corporate-bonds-covenant) vary and check that the upfront barely moves when the [hazard rate](#def-m2-credit-default-swaps-hazard) is recalibrated to the same spread.

## 23.7 Build: the CDS pricer

**Purpose.** The miniature firm quotes and marks CDS on its bond issuers: upfront and running conversions, [hazard rates](#def-m2-credit-default-swaps-hazard) for its credit models, and the auction’s arithmetic when a name defaults. Like the bond calculator of chapter 3, it exists in Python, C++20 and Rust, with the same tests.

**Interface.** `Cds(maturity, coupon, recovery, freq)`; `legs(cds, lam, r)`; `par_spread`; `upfront`; `hazard_from_spread(cds, spread, r)`; `upfront_from_spread`; `inside_market_midpoint(bids, offers)`; `open_interest(requests)`; `final_price(imm, oi, orders)`; `cash_settlement(notional, price)`.

**Rules.** Flat hazard and flat continuously compounded rate; quarterly premiums with half a period of accrued premium on default; defaults paid at the end of the quarter; auction prices per 100 of par, midpoint rounded to an eighth, second-stage price capped one point beyond the midpoint; the dealers’ first-stage quotes are not carried into the second stage.

**Acceptance tests.** `code/firm/cds/tests/`, `cpp/firm_cds_test.cpp`, `rust/src/lib.rs`: the par spread round-trips through the implied hazard; the upfront is zero at the coupon and changes sign across it; the Lehman first stage gives 9.75 and an open interest of 4 920; the cap binds at 10.75.

**Stretch.** Piecewise-constant hazard curves bootstrapped from a term structure of spreads; exact dates and accrual on the roll-date calendar; a CDS on an index (next chapter); the risk of a CDS book by name and tenor (One Quant Book 6).

Sources and further reading

- J. Helwege, S. Maurer, A. Sarkar and Y. Wang, “Credit default swap auctions”, Federal Reserve Bank of New York Staff Report 372, May 2009.
- Paul, Weiss, “The Big Bang Protocol and a new structural framework for credit default swaps”, client memorandum, 24 March 2009.
- F. Carruzzo, “New ISDA 2014 Credit Derivatives Definitions”, Harvard Law School Forum on Corporate Governance, August 2014.
- ISDA, Disclosure Annex for Credit Derivative Transactions.
- ISDA, “Key trends in the size and composition of OTC derivatives markets in the second half of 2025”, July 2026.

## 23.8 Exercises

**Exercise 23.1 ★.**

An investor buys USD 10 million of protection at 100 basis points a year, paid quarterly. What does it pay each quarter, and what does it receive if the entity defaults and the [auction final price](#def-m2-credit-default-swaps-final) is 40?

**Solution of Exercise 23.1.**

USD 25 000 a quarter ($10 \text{ million} \times 1\% / 4$); after the default, $10
\text{ million} \times (100 - 40)\% = \text{USD}~6$ million, and the premiums stop.

**Exercise 23.2 ★.**

A name’s five-year spread is 300 basis points and the recovery is taken as 40%. Use the credit triangle to estimate its [hazard rate](#def-m2-credit-default-swaps-hazard) and its five-year [survival probability](#def-m2-credit-default-swaps-hazard).

**Solution of Exercise 23.2.**

$\lambda \approx 0.03 / 0.6 = 5\%$ a year; $Q(5) = e^{-0.25} = 77.9\%$. It is a risk-neutral probability, not a forecast.

**Exercise 23.3 ★.**

List four questions a [determinations committee](#def-m2-credit-default-swaps-event) decides after a possible [credit event](#def-m2-credit-default-swaps-event).

**Solution of Exercise 23.3.**

Whether a [credit event](#def-m2-credit-default-swaps-event) has occurred, and when; whether to hold an auction; which obligations are deliverable; whether the entity has a successor, and which. It also answers questions of interpretation of the contract.

**Exercise 23.4 ★★.**

A five-year contract on an investment-grade name is quoted at 50 basis points. With the chapter’s assumptions, what is the upfront on USD 10 million, and who pays it?

**Solution of Exercise 23.4.**

The upfront is $-2.21\%$: the [protection buyer](#def-m2-credit-default-swaps-cds) receives about USD 221 000, because it will pay a coupon of 100 basis points for protection worth 50.

**Exercise 23.5 ★★.**

Why did [standard coupons](#def-m2-credit-default-swaps-std) make it possible to net and clear CDS, when contracts with each trade’s own par spread did not?

**Solution of Exercise 23.5.**

Netting needs identical contracts: two trades on the same name and maturity but with different coupons leave a residual stream of premiums that cannot be cancelled. With a [standard coupon](#def-m2-credit-default-swaps-std) the difference is paid once, upfront, and the running legs are identical, so opposite trades cancel and a central counterparty can hold one net position per name and maturity.

**Exercise 23.6 ★★.**

After a default two bonds of the same seniority are deliverable, one trading at 30 and one at 35. What does a [protection buyer](#def-m2-credit-default-swaps-cds) with USD 10 million of protection receive by physical settlement, and which price should the auction reflect?

**Solution of Exercise 23.6.**

It delivers the bond at 30 and receives par: USD 10 million for bonds worth USD 3 million, a payout of USD 7 million. The auction’s final price should reflect the cheapest deliverable, about 30, since sellers of bonds in the auction deliver it.

**Exercise 23.7 ★★★.**

*Coding.* With `firm_cds`, find the [hazard rate](#def-m2-credit-default-swaps-hazard) implied by five-year spreads of 100, 300 and 800 basis points, and compare each with the credit triangle. Explain the gap.

**Solution of Exercise 23.7.**

1.6667%, 5.0001% and 13.335%, against 1.6667%, 5.0000% and 13.333% from the triangle. The small gap, growing with the spread, comes from paying premiums and defaults quarterly rather than continuously; counting half a period of accrued premium makes the discrete legs nearly match the continuous ones.

**Exercise 23.8 ★★★.**

*Find the flaw.* “The CDS spread is the market’s probability of default per year: a name at 300 basis points has a 3% chance of defaulting each year.” Correct it.

**Solution of Exercise 23.8.**

The spread is about the [hazard rate](#def-m2-credit-default-swaps-hazard) times the loss given default, $\lambda(1-R)$: at 300 basis points and 40% recovery the [hazard rate](#def-m2-credit-default-swaps-hazard) is about 5% a year, not 3%. And it is a risk-neutral rate, which includes a premium for bearing default risk and illiquidity; actual default rates are usually lower.

## 23.9 Problem: The Auction

**Problem 23.1.**

Weekend problem — the two stages of a settlement auction

Replay the Lehman Brothers auction of 10 October 2008 with the first-stage submissions of [Figure 23.4](#fig-m2-credit-default-swaps-lehman): fourteen dealers’ bids and offers for USD 5 million of bonds, and physical settlement requests (USD millions) to buy 130, 612 and 30 and to sell 755, 870, 141, 480, 464, 1 470, 390, 574, 191, 170 and 187. Only the range and median of the second-stage orders were published; use the illustrative book of limit bids to buy in [Figure 23.5](#fig-m2-credit-default-swaps-orders), consistent with the published result. Four illustrative participants settle: A, a pension fund, holds USD 20 million of the bonds and USD 20 million of protection; B, a hedge fund, bought USD 50 million of protection and holds no bonds; C, a dealer, sold USD 100 million of protection and bought USD 60 million; D, an insurer, sold USD 30 million of protection.

![The second stage of the auction on an illustrative book of limit bids to buy bonds: the bids are filled from the highest price down until they meet the open interest of USD 4.92 billion to sell, which happens at 8.625, the published final price. Bids below 7.5 are off the chart. Illustrative book; open interest and result: FRBNY Staff Report 372.](https://one-course.com/images/onecourse/chapters/quant-2/m2-credit-default-swaps/fig-c9305a2a1d1a.svg)

***Figure 23.5.** The second stage of the auction on an illustrative book of limit bids to buy bonds: the bids are filled from the highest price down until they meet the open interest of USD 4.92 billion to sell, which happens at 8.625, the published final price. Bids below 7.5 are off the chart. Illustrative book; open interest and result: FRBNY Staff Report 372.*

**Part I — The first stage.**

1. Which bid and offer cross, and are removed?
2. Average the better half of the remaining bids and offers, and round it to the inside market midpoint.
3. Compute the open interest and its direction.
4. Which dealer pays a penalty, and how much?
5. Why was the open interest to sell bonds?

**Part II — The second stage.**

6. Which orders are submitted, and what is the highest price allowed?
7. Fill the book from the highest bid and find the final price.
8. What would the final price be if all bids were at 12?
9. If only USD 1 billion had been bid at 8.625, what would the final price be?
10. By how much did the final price differ from the midpoint, and why?

**Part III — Settlement.**

11. A cash settles: what does it receive, and what is its bond worth at the final price?
12. A instead asks to sell its USD 20 million of bonds in the auction: what does it receive in all?
13. What does B receive?
14. What does C pay?
15. D asks to buy USD 30 million of bonds in the auction: what does it pay, and what does it own afterwards?

**Part IV — Judgement.**

16. Why did USD 72 billion of gross protection turn into only USD 5.2 billion of payments?
17. The bonds traded at about 10 on the day of the auction. Why could the final price be lower?
18. What would have happened with physical settlement alone, with more protection than bonds?
19. State the *named result* : the final price and each participant’s settlement.
20. In one sentence: what does the auction add to a CDS?

**Solution of Problem 23.1.**

**1.** The highest bid, 10, and the lowest offer, 10: they cross and are removed. **2.** Seven of the thirteen remaining: bids 9.5, 9.5, 9.25, 9.25, 9, 9, 8.875 and offers 10, 10, 10, 10.25, 10.75, 10.875, 11 average 9.804, rounded to 9.75. **3.** Sells USD 5 692 million, buys USD 772 million: USD 4 920 million to sell. **4.** The dealer that bid 10, HSBC, above the midpoint while the open interest was to sell: $5\text{ million} \times (10 - 9.75)\% = \text{USD}~12\,500$. **5.** [Protection buyers](#def-m2-credit-default-swaps-cds) that held bonds asked to sell them at the final price, to end with par, and they outweighed [protection sellers](#def-m2-credit-default-swaps-cds) asking to buy. **6.** Limit bids to buy bonds, at no more than $9.75 + 1 = 10.75$. **7.** Cumulative sizes 50, 200, 500, 1 400 and 2 800 million down to 8.75, then 5 400 million at 8.625, which meets 4 920: the final price is 8.625. **8.** 10.75, the cap. **9.** 3 800 million at 8.625 is not enough; with the 1 800 million at 8.5 the book reaches 5 600: the final price is 8.5. **10.** 1.125 below the midpoint: the open interest to sell was large, and the bids needed to absorb it were lower. **11.** $20 \text{ million} \times 91.375\% = \text{USD}~18.275$ million; its bonds are worth USD 1.725 million at 8.625. **12.** USD 18.275 million from the CDS and USD 1.725 million for the bonds: USD 20 million, par, and no bonds left. **13.** USD 45.6875 million. **14.** On its net USD 40 million sold, USD 36.55 million. **15.** USD 27.4125 million on the CDS and USD 2.5875 million for the bonds: USD 30 million in all, for USD 30 million face of bonds, as if it had been delivered them. **16.** Most of the protection was held by dealers that had both bought and sold it; in the auction each paid or received only its net position. **17.** The open interest to sell was USD 4.92 billion; absorbing that supply took bids below the bond market’s last prices. **18.** Holders of protection without bonds would have had to buy them, pushing the price up and their payout down, and contracts on the same name would have settled at different prices. **19.** Named result: *the auction* sets the final price at 8.625; A receives USD 18.275 million and keeps bonds worth 1.725 (or USD 20 million with no bonds), B receives USD 45.6875 million, C pays USD 36.55 million, and D pays USD 30 million in all for USD 30 million face of bonds. **20.** It gives every contract on a name one recovery price, set by trading the net demand for its bonds, so settlement no longer depends on finding bonds.

## 23.10 Interview questions

**Interview question 23.1 ★ trader, researcher.**

What is a [credit default swap](#def-m2-credit-default-swaps-cds)? Who pays what, and when?

**Solution of Interview question 23.1.**

A contract in which the [protection buyer](#def-m2-credit-default-swaps-cds) pays a running premium on a notional to the seller, quarterly, until maturity or a [credit event](#def-m2-credit-default-swaps-event) of the [reference entity](#def-m2-credit-default-swaps-cds); after a [credit event](#def-m2-credit-default-swaps-event) the seller pays the notional times one minus the recovery price set by the auction. The buyer need not own the debt.

*What the interviewer is looking for: the two legs, the trigger, settlement.*

**Interview question 23.2 ★ trader, bank.**

Why are CDS traded with a fixed coupon and an [upfront payment](#def-m2-credit-default-swaps-std)?

**Solution of Interview question 23.2.**

Since 2009 contracts carry a fixed coupon (100 or 500 basis points for North American names) and the difference from the market spread is paid upfront. Identical running legs make trades on the same name and maturity fungible: they can be netted, compressed and cleared.

*What the interviewer is looking for: fungibility, netting, clearing.*

**Interview question 23.3 ★★ researcher.**

Derive the relation between a CDS spread, the [hazard rate](#def-m2-credit-default-swaps-hazard) and the [recovery rate](https://one-course.com/books/quant/2/en/chapter/21-corporate-bonds#def-m2-corporate-bonds-covenant).

**Solution of Interview question 23.3.**

With a flat hazard $\lambda$, recovery $R$ and continuous premiums, the premium leg is $s\int e^{-(r+\lambda)t}dt$ and the protection leg $(1-R)\lambda\int
e^{-(r+\lambda)t}dt$; equating gives $s = \lambda(1-R)$. Discrete payments change it slightly; the hazard is risk-neutral.

*What the interviewer is looking for: the two legs and the cancellation.*

**Interview question 23.4 ★★ trader, researcher.**

How does a CDS settlement auction work, and why is it needed?

**Solution of Interview question 23.4.**

First stage: dealers submit bids and offers for a fixed size, and physical settlement requests; crossing pairs are removed and the better half averaged into the inside market midpoint; the net requests are the open interest. Second stage: limit orders on the other side fill the open interest, best first, and the last order filled sets the final price, within one point of the midpoint. It is needed because protection can exceed deliverable bonds and bonds trade thinly after default; it gives one price for all contracts.

*What the interviewer is looking for: both stages, the cap, the reason.*

**Interview question 23.5 ★★ trader, researcher.**

What is the CDS–bond basis, and how would you trade a negative basis?

**Solution of Interview question 23.5.**

The basis is the CDS spread minus the bond’s spread over the funding rate (for a floating-rate bond, its spread; for a fixed bond, an asset-swap or Z-spread). If negative, buy the bond, fund it in repo and buy protection: you earn the difference and are hedged against default. The risks are funding and repo haircuts, the gap between the bond’s price and the auction price, delivery options, and the mark-to-market if the basis widens.

*What the interviewer is looking for: the package, funding, and the residual risks.*

**Interview question 23.6 ★★★ developer.**

Design a system that values a book of 50 000 CDS positions every night and settles it after a [credit event](#def-m2-credit-default-swaps-event).

**Solution of Interview question 23.6.**

Store trades with their standard terms; nightly, bootstrap hazard curves per name and recovery from market quotes, value each trade and aggregate risk by name and tenor, in parallel across names; on a [credit event](#def-m2-credit-default-swaps-event), watch the committee’s decisions and the auction result, then compute each trade’s settlement at the final price, net by counterparty and through the clearing house, and book the payments; test against past auctions.

*What the interviewer is looking for: curves per name, parallel valuation, the event workflow.*