risk topic 1
TRANSCRIPT
-
8/17/2019 Risk Topic 1
1/87
Risk Management
Topic One – Credit yield curves and credit derivatives
1.1 Implied probability of default and credit yield curves
1.2 Credit default swaps
1.3 Credit spread and bond price based pricing
1.4 Pricing of credit derivatives
1
-
8/17/2019 Risk Topic 1
2/87
1.1 Implied probability of default and credit yield curves
The price of a corporate bond must reflect not only the spot rates
for default-free bonds but also a risk premium to reflect default risk
and any options embedded in the issue.
Credit spreads : compensate investor for the risk of default on the
underlying securities
2
-
8/17/2019 Risk Topic 1
3/87
• The spread increases as the rating declines. In general, it also
increases with maturity (for BBB-rating or above).
• The spread tends to increase faster with maturity for low credit
ratings than for high credit ratings.
3
-
8/17/2019 Risk Topic 1
4/87
Term structures of forward probabilities of default
Year Cumulative de-fault probabil-ity (%)
Forward default prob-ability in year (%)
1 0.2497 0.2497
2 0.9950 0.7453
3 2.0781 1.0831
4 3.3428 1.2647
5 4.6390 1.2962
0.2497% + (1 − 0.2497%) × 0.7453% = 0.9950%
0.9950% + (1 − 0.9950%) × 1.0831% = 2.0781%
P [τ def ≤ 2] = cumulative default probability up to Year 2
= 0.9950%;
P [τ def ≤ 2|τ def > 1] = forward default probability of default in Year 2
= 0.7453%.
4
-
8/17/2019 Risk Topic 1
5/87
Probability of default assuming no recovery
Define
y(T ) : Yield on a T -year corporate zero-coupon bond
y∗(T ) : Yield on a T -year risk-free zero-coupon bond
Q(T ) : Probability that corporation will default between time zero
and time T
τ : Random time of default
• The value of a T -year risk-free zero-coupon bond with a principal
of 100 is 100e−y∗(T )T while the value of a similar corporate bond
is 100e−y(T )T .
5
-
8/17/2019 Risk Topic 1
6/87
Assuming zero recovery upon default, there is a probability Q(T )
that the corporate bond will be worth zero at maturity and a prob-ability 1 − Q(T ) that it will be worth 100. The value of the risky
bond is
{Q(T ) × 0 + [ 1 − Q(T )] × 100}e−y∗(T )T = 100[1 − Q(T )]e−y
∗(T )T .
Since the yield on the risky bond is y(T ), so
100e−y(T )T = 100[1 − Q(T )]e−y∗(T )T .
The T -year survival probability is given by
S (T ) = 1 − Q(T ) = e−[y(T )−y∗(T )]T .
Note that the probability Q(T ) is the risk neutral probability sinceit is inferred from prices of traded securities.
6
-
8/17/2019 Risk Topic 1
7/87
As a summary, assuming zero recovery upon default, the survival
probability as implied from the bond prices is seen to be
S (T ) = 100e−y(T )T
100e−y∗(T )T
= price of defaultable bond
price of default free bond
= e−credit spread×T ,
where credit spread = y(T ) − y∗(T ). Here, the T -year credit spread
is the difference in the yield of the risky zero-coupon and its riskfree
counterpart, both with maturity T .
Alternative proof
Assuming zero recovery and independence of the interest rate pro-
cess and default event, and letting τ be the random default time,
we then have
price of risky bond = E [100e−∫ T
0 ru du1{τ >T }] (zero recovery)
= E [100e−∫ T
0 ru du]E [1{τ >T }] (independence)= price of riskfree bond × S (T ).
7
-
8/17/2019 Risk Topic 1
8/87
Example
Suppose that the spreads over the risk-free rate for 5-year and a 10-
year BBB-rated zero-coupon bonds are 130 and 170 basis points,respectively, and there is no recovery in the event of default. The
default probabilities can be inferred from the term structure of credit
spreads as follows:
P [τ ≤ 5] = Q(5) = 1 − e−0.013×5 = 0.0629
P [τ ≤ 10] = Q(10) = 1 − e−0.017×10 = 0.1563.
The probability of default between five years and ten years is Q(5; 10)
where
Q(10) = Q(5) + [(1 − Q(5)]Q(5; 10)
or
P [τ ≤ 10|τ > 5] = Q(5; 10) = 0.01563 − 0.0629
1 − 0.0629 .
8
-
8/17/2019 Risk Topic 1
9/87
Credit spreads and default intensities (hazard rates)
The default intensity (hazard rate) at time t is defined so that λ(t)∆t
is the probability of default between time t and t + ∆t conditional
on no earlier default. If S (t) is the cumulative probability of the
company surviving to time t (no default by time t), then
probability of default occurring within (t, t + ∆t]
= S (t) − S (t + ∆t) = S (t)λ(t)∆t.
Taking the limit ∆t → 0, we obtain
dS (t)
S (t) = −λ(t) dt with S (0) = 1,
so that
S (t) = e− ∫ t0 λ(u) du = e−λ(t)t = 1 − Q(t),where Q(t) is the probability of default by time t and λ(t) is the
average default intensity between time 0 and time t.
9
-
8/17/2019 Risk Topic 1
10/87
• The average default intensity λ̄(t) can be visualized as the credit
spread over (0, t) since
S (t) = e−[y(t)−y∗(t)]t = e−
∫ t0 λ(u) du = e−λ̄(t)t, t ∈ [0, T ].
• The unconditional default probability density q(t) is defined sothat q(t)∆t gives the probability of default that occurs within
(t, t + ∆t). Let F (t) be the distribution function of the random
default time τ , where
F (t) = P [τ ≤ t],
we then have q(t) = F ′(t).
• Recall that q(t)∆t = S (t)λ(t)∆t so that
q(t) = e−∫ t
0 λ(u) duλ(t) = S (t)λ(t), t ≥ 0,
where S (t) = 1 − F (t). Also, the probability of surviving until
time t, conditional on survival up to s, where s ≤ t, is given by
P [τ > t|τ > s] = S (t)
S (s) =
e−∫ t
0 λ(u) du
e−∫ s
0 λ(u) du = e−
∫ ts λ(u) du.
10
-
8/17/2019 Risk Topic 1
11/87
Recovery rates
Amounts recovered on corporate bonds as a percent of par valuefrom Moody’s Investor’s Service are shown in the table below.
Class Mean (%) Standard derivation (%)
Senior secured 52.31 25.15
Senior unsecured 48.84 25.01
Senior subordinated 39.46 24.59
Subordinated 33.17 20.78
Junior subordinated 19.69 13
.85
The amount recovered is estimated as the market value of the bond
one month after default.
• Seniority of the bond among outstanding bonds issued by the
same issuer is an important determinant of the recovery rate of that bond. Bonds that are newly issued by an issuer must have
seniority below that of existing bonds issued earlier by the same
issuer.
11
-
8/17/2019 Risk Topic 1
12/87
Finite recovery rate
• In the event of a default, the bondholder receives a proportion
R of the bond’s no-default value. If there is no default, then thebondholder receives 100.
• The bond’s no-default value is 100e−y∗(T )T and the probability
of a default is Q(T ). The value of the bond is
[1 − Q(T )]100e−y∗(T )T + Q(T )100Re−y∗(T )T
so that
100e−y(T )T = [1 − Q(T )]100e−y∗(T )T + Q(T )100Re−y
∗(T )T .
The implied probability of default in terms of yields and recovery
rate is given by
Q(T ) = 1 − e−[y(T )−y
∗(T )]T
1 − R .
12
-
8/17/2019 Risk Topic 1
13/87
Numerical example on the impact of different assumptions of recov-
ery rates on default probability estimation
Suppose the 1-year default free bond price is $100 and the 1-yeardefaultable XY Z corporate bond price is $80.
(i) Assuming R = 0, the probability of default of XY Z as implied
by the two bond prices is
Q0(1) = 1 − 80100
= 20%.
(ii) Assuming R = 0.6, we obtain
QR(1) =
1 − 801001 − 0.6 =
20%
0.4 = 50%.
The ratio of Q0(1) : QR(1) = 1 : 11−R.
13
-
8/17/2019 Risk Topic 1
14/87
Calculation of default intensity with non-zero recovery rate
Consider a 5-year risky corporate bond that pays a coupon of 6%
per annum (paid semiannually)
• Yield on the corporate bond is 7% per annum (with continuouscompounding)
• Yield on a similar risk-free bond is 5% per annum (with contin-
uous compounding)
The yields imply that
(i) price of the riskfree bond= 3e−0.05×0.5 + 3e−0.05×1 + . . . + 3e−0.05×4.5 + 103e−0.05×5
= 104.09.
(ii) price of the risky bond
= 3e−0.07×0.5
+ 3e−0.07×1
+ . . . + 3e−0.07×4.5
+ 103e−0.07×5
= 95.34.
The present value of expected loss from default over the 5-year life
of the bond = 104.09 − 95.34 = 8.75.
14
-
8/17/2019 Risk Topic 1
15/87
Let Q denote the constant unconditional probability of default per
year. Assuming that defaults can happen at times 0.5, 1.5, 2.5, 3.5
and 4.5 year (immediately before coupon payment dates), we cancalculate the expected loss from default in terms of Q.
Calculation of loss from default on a bond in terms of the default probability per year, Q. Notional principal = $100.
Time Default Recovery Risk-free Loss given Discount PV of expected
(years) probability amount($) value($) default($) factor loss($)
0.5 Q 40 106.73 66.73 0.9753 65.08Q
1.5 Q 40 105.97 65.97 0.9277 61.20Q
2.5 Q 40 105.17 65.17 0.8825 57.52Q
3.5 Q 40 104.34 64.34 0.8395 54.01Q
4.5 Q 40 103.46 63.46 0.7985 50.67Q
Total 288.48Q
15
-
8/17/2019 Risk Topic 1
16/87
Consider the 3.5 year row in the table.
• The expected value of the riskfree bond at Year 3.5 (time toexpiry is 1.5 years) is
3 + 3e−0.05×0.5 + 3e−0.05×1.0 + 103e−0.05×1.5 = 104.34.
• The amount recovered if there is a default is 40, so the loss
given default is 104.34 − 40 = 64.34.
• The present value of this loss = 64.34× e−0.05×3.5 × Q = 64.34×
0.8395 × Q = 54.01Q.
The total expected loss is 288.48Q. Setting this equal to 8.75, we
obtain Q = 3.03%.
16
-
8/17/2019 Risk Topic 1
17/87
Generalization - term structure of default probabilities
Suppose we have bonds maturing in 3, 5, 7, and 10 years, we coulduse the first bond to estimate a default probability per year for the
first 3 years, the second bond to estimate default probability per
year for years 4 and 5, the third bond for years 6 and 7, and the
last bond for years 8, 9 and 10.
For example, suppose λ[0,3] is the default intensity in the first 3
years, which has been obtained from an earlier calculation based
on 3-year risky and riskfree bonds. We compute λ[3,5] using 5-year
bonds by following the sample calculations as shown in the above,
except that the default intensity at times 0.5, 1.5 and 2.5 are set
to be the known quantity λ[0,3]. The default intensity at times 3.5
and 4.5 are set to be λ[3,5], a quantity to be determined.
17
-
8/17/2019 Risk Topic 1
18/87
Construction of a credit risk adjusted yield curve is hindered by
1. The general absence in money markets of liquid traded instru-
ments on credit spread. In recent years, for some liquidly traded
corporate bonds, we may have good liquidity on trading of credit
default swaps whose underlying is the credit spread.
2. The absence of a complete term structure of credit spreads as
implied from traded corporate bonds. At best we only have
infrequent data points.
18
-
8/17/2019 Risk Topic 1
19/87
The default probabilities estimated from historical data are much
less than those derived from bond prices
For example, from historical data published by Moody’s, an A-rated
company has average cumulative default rate Q(7) of 0.0091 =
0.91%. The average 7-year default intensity λ̄(7) is determined by
S (7) = 1 − 0.0091 = 0.9909 = e−λ̄(7)×7
so that
λ(7) = − 17
ln 0.9909 = 0.0013 = 0.13%.
On the other hand, based on bond yields published by Merrill Lynch,
the average Merrill Lynch yield for A-rated bonds was 6.274%.
The average riskfree rate was estimated to be 5.505%. As an ap-
proximation, the average 7-year default intensity is0.06274 − 0.05505
1 − 0.4 = 0.0128 = 1.28%.
Here, the recovery rate is assumed to be 0.4.
19
-
8/17/2019 Risk Topic 1
20/87
Seven-year average default intensities (% per annum).
Rating Historical default Default intensity Ratio Difference
intensity from bonds
Aaa 0.04 0.67 16.8 0.63
Aa 0.06 0.78 13.0 0.72
A 0.13 1.28 9.8 1.15
Baa 0.47 2.38 5.1 1.91
Ba 2.40 5.07 2.1 2.67
B 7.49 9.02 1.2 1.53
Caa 16.90 21.30 1.3 4.40
• Corporate bonds are relatively illiquid and bond traders demand
an extra return to compensate for this.
• Bonds do not default independently of each other. This gives
rise to risk that cannot be diversified away, so bond traders
should require an expected excess return for bearing the risk.
20
-
8/17/2019 Risk Topic 1
21/87
Implied default probabilities (equity-based versus credit-based)
• Recovery rate has a significant impact on the defaultable bond
prices. The forward probability of default as implied from the
defaultable and default free bond prices requires estimation of
the expected recovery rate (an almost impossible job).
• The industrial code mKMV estimates default probability using
stock price dynamics – equity-based implied default probability.
For example, the JAL stock price dropped to 1 in early 2010.
Obviously, the equity-based default probability over one year horizon
is close to 100% (stock holders receive almost nothing upon JAL’s
default). However, the credit-based default probability as implied by
the JAL bond prices is less than 30% since the bond par payments
are somewhat partially guaranteed even in the event of default.
21
-
8/17/2019 Risk Topic 1
22/87
1.2 Credit default swaps
The protection seller receives fixed periodic payments from the pro-
tection buyer in return for making a single contingent payment cov-
ering losses on a reference asset following a default.
protection
seller
protection
buyer
140 bp per annum
Credit event payment
(100% recovery rate)
only if credit event occurs
holding a
risky bond
22
-
8/17/2019 Risk Topic 1
23/87
Protection seller
• earns premium income with no funding cost
• gains customized, synthetic access to the risky bond
Protection buyer
• hedges the default risk on the reference asset
1. Very often, the bond tenor is longer than the swap tenor. In
this way, the protection seller does not have exposure to the full
period of the bond.
2. Basket default swap – gain additional premium by selling default
protection on several assets.
23
-
8/17/2019 Risk Topic 1
24/87
A bank lends 10mm to a corporate client at L + 65bps. The bank
also buys 10mm default protection on the corporate loan for 50bps.
Objective achieved by the Bank through the default swap:
• maintain relationship with the corporate borrower
• reduce credit risk on the new loan
Corporate
Borrower Bank Financial
House
Risk Transfer
Interest and
Principal
Default Swap
Premium
If Credit Event:
par amount
If Credit Event:obligation (loan)
Default swap settlement following Credit Event of Corporate Borrower
24
-
8/17/2019 Risk Topic 1
25/87
Settlement of compensation payment
1. Physical settlement:
The defaultable bond is put to the Protection Seller in return
for the par value of the bond.
2. Cash compensation:
An independent third party determines the loss upon default
at the end of the settlement period (say, 3 months after the
occurrence of the credit event).
Compensation amount = (1 − recovery rate) × bond par.
25
-
8/17/2019 Risk Topic 1
26/87
Selling protection
To receive credit exposure for a fee (simple credit default swaps) or
in exchange for credit exposure to better diversify the credit portfolio
(exchange credit default swaps).
Buying protection
To reduce either individual credit exposures or credit concentrations
in portfolios. Synthetically to take a short position in an asset
which are not desired to sell outright, perhaps for relationship or
tax reasons.
26
-
8/17/2019 Risk Topic 1
27/87
Funding cost arbitrage
Should the Protection Buyer look for a Protection Seller who has a
higher/lower credit rating than himself?
50bps
annual premium
A-rated institution
as Protection Seller
AAA-rated institution
as Protection Buyer
Lender to the
AAA-ratedInstitution
LIBOR-15bps
as fundingcost
BBB risky
reference asset
Lender to the
A-rated Institution
coupon
= LIBOR + 90bps
funding cost of
LIBOR + 50bps
27
-
8/17/2019 Risk Topic 1
28/87
The combined risk faced by the Protection Buyer:
• default of the BBB-rated bond
• default of the Protection Seller on the contingent payment
Consider the S&P’s Ratings for jointly supported obligations (the
two credit assets are uncorrelated)
A+ A A− BBB+ BBBA+ AA+ AA+ AA+ AA AAA AA+ AA AA AA− AA−
The AAA-rated Protection Buyer creates a synthetic AA−asset with
a coupon rate of LIBOR + 90bps − 50bps = LIBOR + 40bps.
This is better than LIBOR + 30bps, which is the coupon rate of a
AA−asset (net gains of 10bps).
28
-
8/17/2019 Risk Topic 1
29/87
For the A-rated Protection Seller, it gains synthetic access to a
BBB-rated asset with earning of net spread of
• Funding cost of the A-rated Protection Seller = LIBOR + 50bps
• Coupon from the underlying BBB bond = LIBOR + 90bps
• Credit swap premium earned = 50bps
29
-
8/17/2019 Risk Topic 1
30/87
In order that the credit arbitrage works, the funding cost of the
default protection seller must be higher than that of the default
protection buyer.
Example
Suppose the A-rated institution is the Protection Buyer, and assume
that it has to pay 60bps for the credit default swap premium (higher
premium since the AAA-rated institution has lower counterpartyrisk).
spread earned from holding the risky bond
= coupon from bond − funding cost
= (LIBOR + 90bps) − (LIBOR + 50bps) = 40bps
which is lower than the credit swap premium of 60bps paid for
hedging the credit exposure. No deal is done!
30
-
8/17/2019 Risk Topic 1
31/87
Counterparty risk in CDS
Before the Fall 1997 crisis, several Korean banks were willing to
offer credit default protection on other Korean firms.
US commercial
bank
Hyundai
(not rated)
Korea exchange
bank
LIBOR + 70bp
40 bp
Higher geographical risks lead to higher default correlations.
⋆ Higher geographic risks lead to higher default correlations.
Advice: Go for a European bank to buy the protection.
31
-
8/17/2019 Risk Topic 1
32/87
How does the inter-dependent default risk structure between the
Protection Seller and the Reference Obligor affect the credit swap
premium rate?
1. Replacement cost (Seller defaults earlier)
• If the Protection Seller defaults prior to the Reference En-
tity, then the Protection Buyer renews the CDS with a new
counterparty.
• Supposing that the default risks of the Protection Seller and
Reference Entity are positively correlated, then there will be
an increase in the swap rate of the new CDS.
2. Settlement risk (Reference Entity defaults earlier)
• The Protection Seller defaults during the settlement period
after the default of the Reference Entity.
32
-
8/17/2019 Risk Topic 1
33/87
Hedge strategy using fixed-coupon bonds
Portfolio 1
• One defaultable coupon bond C ; coupon c, maturity tN .• One CDS on this bond, with CDS spread s
The portfolio is unwound after a default.
Portfolio 2
• One default-free coupon bond C : with the same payment dates
as the defaultable coupon bond and coupon size c − s.
The default free bond is sold after default of the defaultable coun-
terpart.
33
-
8/17/2019 Risk Topic 1
34/87
Comparison of cash flows of the two portfolios
1. In survival, the cash flows of both portfolio are identical.
Portfolio 1 Portfolio 2 t = 0 −C (0) −C (0)t = ti c − s c − st = tN 1 + c − s 1 + c − s
2. At default, portfolio 1’s value = par = 1 (full compensation by
the CDS); that of portfolio 2 is C (τ ), τ is the time of default.
The price difference at default = 1 − C (τ ). This difference is
very small when the default-free bond is a par bond.
Remark
The issuer can choose c to make the bond be a par bond such that
the initial value of the bond is at par.
34
-
8/17/2019 Risk Topic 1
35/87
This is an approximate replication.
Recall that the value of the CDS at time 0 is zero. Let B(0, tN )
denote the price of a zero-coupon default-free bond. Neglecting
the difference in the values of the two portfolios at default, the
no-arbitrage principle dictates
C (0) = C (0) = B(0, tN ) + cA(0) − sA(0).
Here, (c − s)A(0) is the sum of present value of the coupon pay-ments at the fixed coupon rate c − s. The equilibrium CDS rate s
can be solved:
s = B(0, tN ) + cA(0) − C (0)
A(0) .
B(0, tN ) + cA(0) is the time-0 price of a default free coupon bond
paying coupon at the rate of c.
35
-
8/17/2019 Risk Topic 1
36/87
Cash-and-carry arbitrage with par floater
A par floater C ′
is a defaultable bond with a floating-rate coupon
of ci = Li−1 + s par, where the par spread s par is adjusted such that
at issuance the par floater is valued at par.
Portfolio 1
• One defaultable par floater C ′
with spread s par over LIBOR.
• One CDS on this bond: CDS spread is s.
The portfolio is unwound after default.
36
-
8/17/2019 Risk Topic 1
37/87
Portfolio 2
• One default-free floating-coupon bond C ′: with the same pay-ment dates as the defaultable par floater and coupon at LIBOR,
ci = Li−1.
The bond is sold after default.
Time Portfolio 1 Portfolio 2 t = 0 −1 −1t = ti Li−1 + s par − s Li−1t = tN 1 + LN −1 + s par − s 1 + LN −1τ (default) 1 C ′(τ ) = 1 + Li(τ − ti)
The hedge error in the payoff at default is caused by accrued interest
Li(τ − ti), accumulated from the last coupon payment date ti to the
default time τ . If we neglect the small hedge error at default, then
s par = s.
37
-
8/17/2019 Risk Topic 1
38/87
Remarks
• The non-defaultable bond becomes a par bond (with initial value
equals the par value) when it pays the floating rate equals LI-
BOR. The extra coupon s par paid by the defaultable par floater
represents the credit spread demanded by the investor due to
the potential credit risk. The above result shows that the creditspread s par is just equal to the CDS spread s.
• The above analysis neglects the counterparty risk of the Pro-
tection Seller of the CDS. Due to potential counterparty risk,
the actual CDS spread will be lower.
38
-
8/17/2019 Risk Topic 1
39/87
Valuation of Credit Default Swap
• Suppose that the probability of a reference entity defaultingduring a year conditional on no earlier default is 2%. That is,
the default intensity is assumed to be the constant 2%.
• Table 1 shows the survival probabilities and forward default prob-
abilities (i.e., default probabilities as seen at time zero) for each
of the 5 years. The probability of a default during the first year
is 0.02 and the probability that the reference entity will survive
until the end of the first year is 0.98.
• The forward probability of a default during the second year is
0.02 × 0.98 = 0.0196 and the probability of survival until the endof the second year is 0.98 × 0.98 = 0.9604.
39
-
8/17/2019 Risk Topic 1
40/87
Table 1 Forward default probabilities and survival probabilities
Time (years) Default probability Survival probability
1 0.0200 0.9800
2 0.0196 0.9604 = 0.982
3 0.0192 0.9412 = 0.983
4 0.0188 0.9224 = 0.984
5 0.0184 0.9039 = 0.985
P [3 < τ ≤ 4]
= forward default probability of default during the fourth year (as
seen at current time)
= P [τ > 3] × P [3 < τ ≤ 4|τ > 3]
= survival probability until end of Year 3 × conditional probabilityof default in Year 4
= 0.983 × 0.02 = 0.9412 × 0.02 = 0.0188.
40
-
8/17/2019 Risk Topic 1
41/87
Assumptions on default and recovery rate
We will assume the defaults always happen halfway through a year
and that payments on the credit default swap are made once a year,
at the end of each year. We also assume that the risk-free (LIBOR)
interest rate is 5% per annum with continuous compounding and
the recovery rate is 40%.
Expected present value of CDS premium payments
Table 2 shows the calculation of the expected present value of the
payments made on the CDS assuming that payments are made at
the rate of s per year and the notional principal is $1.
For example, there is a 0.9412 probability that the third paymentof s is made. The expected payment is therefore 0.9412s and its
present value is 0.9412se−0.05×3 = 0.8101s. The total present value
of the expected payments is 4.0704s.
41
-
8/17/2019 Risk Topic 1
42/87
Table 2 Calculation of the present value of expected payments.
Payment = s per annum.
Time
(years)
Probability
of survival
Expected
payment
Discount
factor
PV of expected
payment
1 0.9800 0.9800s 0.9512 0.9322s
2 0.9604 0.9604s 0.9048 0.8690s
3 0.9412 0.9412s 0.8607 0.8101s
4 0.9224 0.9224s 0.8187 0.7552s
5 0.9039 0.9039s 0.7788 0.7040s
Total 4.0704s
42
-
8/17/2019 Risk Topic 1
43/87
Table 3 Calculation of the present value of expected payoff. No-
tional principal = $1.
Time (years)
Probability of default
Recovery rate
Expected payoff ($)
Discount factor
PV of expected payoff ($)
0.5 0.0200 0.4 0.0120 0.9753 0.0117
1.5 0.0196 0.4 0.0118 0.9277 0.0109
2.5 0.0192 0.4 0.0115 0.8825 0.0102
3.5 0.0188 0.4 0.0113 0.8395 0.0095
4.5 0.0184 0.4 0.0111 0.7985 0.0088
Total 0.0511
For example, there is a 0.0192 probability of a payoff halfway through
the third year. Given that the recovery rate is 40%, the expected
payoff at this time is 0.0192 × 0.6 × 1 = 0.0115. The present valueof the expected payoff is 0.0115e−0.05×2.5 = 0.0102.
The total present value of the expected payoffs is $0.0511.
43
-
8/17/2019 Risk Topic 1
44/87
• When default occurs in mid-year, the Protection Buyer has to
pay the premium accrued half year (between the last premium
payment date and default time).
Table 4 Calculation of the present value of accrual payment.
Time
(years)
Probability
of default
Expected
accrual
payment
Discount
factor
PV of ex-
pected accrual
payment
0.5 0.0200 0.0100s 0.9753 0.0097s
1.5 0.0196 0.0098s 0.9277 0.0091s
2.5 0.0192 0.0096s 0.8825 0.0085s
3.5 0.0188 0.0094s 0.8395 0.0079s
4.5 0.0184 0.0092s 0.7985 0.0074s
Total 0.0426s
44
-
8/17/2019 Risk Topic 1
45/87
As a final step we evaluate in Table 4 the accrual payment made in
the event of a default.
• There is a 0.0192 probability that there will be a final accrual
payment halfway through the third year.
• The accrual payment is 0.5s.
• The expected accrual payment at this time is therefore 0.0192 ×
0.5s = 0.0096s.
• Its present value is 0.0096se−0.05×2.5 = 0.0085s.
• The total present value of the expected accrual payments is
0.0426s.
From Tables 2 and 4, the present value of the expected payment is
4.0704s + 0.0426s = 4.1130s.
45
-
8/17/2019 Risk Topic 1
46/87
Equating expected CDS premium payments and expected compen-
sation payment
From Table 3, the present value of the expected payoff is 0.0511.
Equating the two, we obtain the CDS spread for a new CDS as
4.1130s = 0.0511
or s = 0.0124. The mid-market spread should be 0.0124 times the
principal or 124 basis points per year.
In practice, we are likely to find that calculations are more extensive
than those in Tables 2 to 4 because
(a) payments are often made more frequently than once a year
(b) we might want to assume that defaults can happen more fre-
quently than once a year.
46
-
8/17/2019 Risk Topic 1
47/87
Impact of expected recovery rate R on credit swap premium s
Recall that the expected compensation payment paid by the Pro-tection Seller is (1 − R)× notional. Therefore, the Protection Seller
charges a higher s if her estimation of the recovery rate R is lower.
Let sR denote the credit swap premium when the recovery rate is
R. We deduce that
s10
s50=
(100 − 10)%
(100 − 50)% = 90%
50% = 1.8.
Remark
A binary credit default swap pays the full notional upon default
of the reference asset. The credit swap premium of a binary swap
depends only on the estimated default probability but not on the
recovery rate.
47
-
8/17/2019 Risk Topic 1
48/87
Marking-to-market a CDS
• At the time it is negotiated, a CDS, like most swaps, is worth
zero. Later, it may have a positive or negative value.
• Suppose, for example the credit default swap in our example
had been negotiated some time ago for a spread of 150 basis
points, the present value of the payments by the buyer would be
4.1130 × 0.0150 = 0.0617 and the present value of the payoff would be 0.0511.
• The value of swap to the seller would therefore be 0.0617 −
0.0511, or 0.0166 times the principal.
• Similarly the mark-to-market value of the swap to the buyer of
protection would be −0.0106 times the principal.
48
B sk t d f lt s s
-
8/17/2019 Risk Topic 1
49/87
Basket default swaps
The credit event to insure against using the kth-to-default credit
default swap is the event of the kth default. A premium or spread
s is paid as an insurance fee until maturity or the event of the
kth default, whichever comes first. If the kth default occurs before
swap’s maturity, the Protection Buyer puts the defaulting bond to
the Protection Seller in exchange for the face value of the bond.
Sum of the kth-to-default swap spreads, k = 1, 2, . . . , n, for n obligors
in total in the basket is greater than the sum of the individual spreadsof the same set of n obligors:
nk=1
sk >n
i=1
si.
Why? Apparently, both sides insure exactly the same set of risks:
the n defaults in the basket. At the time of the first default, the
left side stops paying the huge spread s1 while on the plain-vanilla
side one just stops paying the spread si of the first default that falls
on obligor i.
49
B d h i f h fi d f l (F D)
-
8/17/2019 Risk Topic 1
50/87
Bounds on the swap premiums for the first-to-default (FtD) swaps
under low default correlation
Assuming all 3 obligors have the same dollar exposure, we have
fee on CDS on ≤ fee on FtD ≤ portfolio of worst credit swap CDSs on all
credits
s̄C ≤ s̄FtD ≤ s̄A + s̄B + s̄C
With low default probabilities and low default correlation, we have
s̄FtD ≈ s̄A + s̄B + s̄C .
To see this, by assuming zero default correlation, the probability of
at least one default is
p = 1 − (1 − pA)(1 − pB)(1 − pC )
= pA + pB + pC − ( pA pB + pA pC + pB pC ) + pA pB pC
so that
p pA + pB + pC for small pA, pB and pC .
50
-
8/17/2019 Risk Topic 1
51/87
1.3 Credit spread and bond price based pricing
Market’s assessment of the default risk of the obligor (assuming
some form of market efficiency – information is aggregated in the
market prices). The sources are
• market prices of bonds and other defaultable securities issuedby the obligor
• prices of CDS’s referencing this obligor’s credit risk
How to construct a clean term structure of credit spreads from
observed market prices?
51
-
8/17/2019 Risk Topic 1
52/87
Based on no-arbitrage pricing principle, a model that is based upon
and calibrated to the prices of traded assets is immune to simple
arbitrage strategies using these traded assets.
Market instruments used in bond price-based pricing
• At time t, the defaultable and default-free zero-coupon bond
prices of all maturities T ≥ t are known. These defaultable
zero-coupon bonds have no recovery at default.
• Information about the probability of default over all time hori-
zons as assessed by market participants are fully reflected when
market prices of default-free and defaultable bonds of all matu-
rities are available.
52
-
8/17/2019 Risk Topic 1
53/87
Risk neutral probabilities
The financial market is modeled by a filtered probability space (Ω,
(F t)t≥0, F , Q), where Q is the risk neutral probability measure.
• All probabilities and expectations are taken under Q. Probabili-
ties are considered as state prices.
1. For constant interest rates, the discounted Q-probability of
an event A at time T is the price of a security that pays off
$1 at time T if A occurs.
2. Under stochastic interest rates, the price of the contingent
claim associated with A is E [β(T )1A], where β(T ) is the dis-count factor. This is based on the risk neutral valuation prin-
ciple and the money market account M (T ) = 1β(T )
= e∫ T
t ru du
is used as the numeraire.
53
-
8/17/2019 Risk Topic 1
54/87
Indicator functions
For A ∈ F , 1A(ω) ={ 1 if ω ∈ A0 otherwise .
τ = random time of default; I (t) = survival indicator function
I (t) = 1{τ >t} ={
1 if τ > t0 if τ ≤ t
.
B(t, T ) = price at time t of zero-coupon bond paying off $1 at T
B(t, T ) = price of defaultable zero-coupon bond if τ > t;
I (t)B(t, T ) ={ B(t, T ) if τ > t
0 if τ ≤ t .
54
-
8/17/2019 Risk Topic 1
55/87
Monotonicity properties on the bond prices
1. 0 ≤ B(t, T ) < B(t, T ), ∀t < T
2. Starting at B(t, t) = B(t, t) = 1,
B(t, T 1) ≥ B(t, T 2) > 0 and B(t, T 1) ≥ B(t, T 2) ≥ 0
∀t < T 1 < T 2, τ > t.
Independence assumption
{B(t, T )|t ≤ T } and τ are independent under (Ω, F , Q) (not the true
measure).
55
-
8/17/2019 Risk Topic 1
56/87
Implied probability of survival in [t, T ]– based on market pricesof bonds
B(t, T ) = E
e−
∫ T t ru du
and B(t, T ) = E
e−
∫ T t ru duI (T )
.
Invoking the independence between defaults and the default-free
interest rates
B(t, T ) = E
e−
∫ T t ru du
E [I (T )] = B(t, T )P (t, T )
implied survival probability over [t, T ] = P (t, T ) = B(t, T )
B(t, T ).
56
-
8/17/2019 Risk Topic 1
57/87
• The implied default probability over [t, T ], P def (t, T ) = 1−P (t, T ).
• Assuming P (t, T ) has a right-sided derivative in T , the implied
density of the default time
Q[τ ∈ (T, T + dT ]|F t] = − ∂
∂T P (t, T ) dT.
• If prices of zero-coupon bonds for all maturities are available,
then we can obtain the implied survival probabilities for all ma-
turities (complementary distribution function of the time of de-
fault).
57
-
8/17/2019 Risk Topic 1
58/87
Properties on implied survival probabilities, P (t, T )
1. P (t, t) = 1 and it is non-negative and decreasing in T . Also,
P (t, ∞) = 0.2. Normally P (t, T ) is continuous in its second argument, except
that an important event secheduled at some time T 1 has direct
influence on the survival of the obligor.
3. Viewed as a function of its first argument t, all survival proba-
bilities for fixed maturity dates will tend to increase.
If we want to focus on the default risk over a given time interval in
the future, we should consider conditional survival probabilities.
conditional survival probability over [T 1, T 2
] as seen from t
= P (t, T 1, T 2) = P (t, T 2)
P (t, T 1), where t ≤ T 1 < T 2.
58
-
8/17/2019 Risk Topic 1
59/87
Implied hazard rate (default probabilities per unit time interval length)
Discrete implied hazard rate of default over (T, T + ∆T ] as seen
from time t
H (t , T , T + ∆T )∆T = P (t, T )
P (t, T + ∆T ) − 1 =
P def (t , T , T + ∆T )
P (t , T , T + ∆T ) ,
so that
P (t, T ) = P (t, T + ∆T )[1 + H (t , T , T + ∆T )∆T ].
In the limit of ∆T → 0, the continuous hazard rate at time T as
seen at time t is given by
h(t, T ) = −
∂
∂T ln P (t, T ).
59
-
8/17/2019 Risk Topic 1
60/87
Proof First, we recall
1
P (t , T , T + ∆T ) =
P (t, T )
P (t , T , T + ∆T ).
We have
h(t, T ) = lim∆T →0
H (t , T , T + ∆T )
= lim∆T →0
1 − P (t , T , T + ∆T )
∆T P (t , T , T + ∆T )
= lim∆T →0
1∆T
P (t, T )P (t, T + ∆T )
− 1
= lim∆T →0
− 1
P (t, T + ∆T )
P (t, T + ∆T ) − P (t, T )
∆T
= − 1
P (t, T )
∂
∂T
P (t, T )
= − ∂
∂T ln P (t, T ).
60
-
8/17/2019 Risk Topic 1
61/87
Forward spreads and implied hazard rate of default
For t ≤ T 1 < T 2, the simply compounded forward rate over the
period (T 1, T 2] as seen from t is given by
F (t, T 1, T 2) = B(t, T 1)/B(t, T 2) − 1
T 2 − T 1.
This is the price of the forward contract with expiration date T 1 on
a unit-par zero-coupon bond maturing on T 2. To prove, we consider
the compounding of interest rates over successive time intervals.
1
B(t, T 2) compounding over [t, T 2]
= 1
B(t, T 1) compounding over [t, T 1]
[1 + F (t, T 1, T 2)(T 2 − T 1)] simply compounding over [T 1, T 2]
Defaultable simply compounded forward rate over [T 1, T 2]
F (t, T 1, T 2) = B(t, T 1)/B(t, T 2) − 1
T 2 − T 1.
61
-
8/17/2019 Risk Topic 1
62/87
Instantaneous continuously compounded forward rates
f (t, T ) = lim∆T →0
F (t , T , T + ∆T ) = − ∂
∂T ln B(t, T )
f (t, T ) = lim∆T →0
F (t , T , T + ∆T ) = − ∂ ∂T
ln B(t, T ).
Implied hazard rate of default
Recall
P (t, T 1, T 2) = B(t, T 2)
B(t, T 2)
B(t, T 1)
B(t, T 1)
= 1 + F (t, T 1, T 2)(T 2 − T 1)
1 + F (t, T 1, T 2)(T 2 − T 1) = 1 − P def (t, T 1, T 2),
and upon expanding, we obtain
P def (t, T 1, T 2) [ 1 + F (t, T 1, T 2)(T 2 − T 1)] B(t,T 1)/B(t,T 2)
= [F (t, T 1, T 2)−F (t, T 1, T 2)](T 2−T 1).
62
P (t T T )
-
8/17/2019 Risk Topic 1
63/87
Define H (t, T 1, T 2) = P def (t, T 1, T 2)
(T 2 − T 1)P (t, T 1, T 2) as the discrete implied
rate of default. We then have
H (t, T 1, T 2) = B(t, T 2)
B(t, T 1)
[F (t, T 1, T 2) − F (t, T 1, T 2)]
P (t, T 1, T 2)
= B(t, T 2)
B(t, T 1)[F (t, T 1, T 2) − F (t, T 1, T 2)].
Taking the limit T 2 → T 1, then the implied hazard rate of default at
time T > t
as seen from time t
is the spread between the forward
rates:
h(t, T ) = f (t, T ) − f (t, T ).
Alternatively, we obtain the above relation using
f (t, T ) − f (t, T ) = −
∂
∂T ln
B(t, T )
B(t, T )
= − ∂
∂T ln P (t, T ) = h(t, T ).
63
-
8/17/2019 Risk Topic 1
64/87
The local default probability at time t over the next small time step
∆t1
∆tQ[τ ≤ t + ∆t|F t ∧ {τ > t}] ≈ r(t) − r(t) = λ(t)
where r(t) = f (t, t) is the riskfree short rate and r(t) = f (t, t) is the
defaultable short rate.
Recovery value
View an asset with positive recovery as an asset with an additional
positive payoff at default . The recovery value is the expected value
of the recovery shortly after the occurrence of a default.
64
Payment upon default
-
8/17/2019 Risk Topic 1
65/87
y p
Define e(t , T , T +∆T ) to be the value at time t < T of a deterministic
payoff of $1 paid at T + ∆T if and only if a default happens in
[T, T + ∆T ].
e(t , T , T + ∆T ) = E Q [β(t, T + ∆T )[I (T ) − I (T + ∆T )]|F t] .
Note that
I (T ) − I (T + ∆T ) =
{ 1 if default occurs in [T, T + ∆T ]0 otherwise
,
E Q[β(t, T + ∆T )I (T )] = E Q[β(t, T + ∆T )]E Q[I (T )]
= B(t, T + ∆T )P (t, T ),
E Q[β(t, T + ∆T )I (T + ∆T )] = B(t, T + ∆T ),
and
B(t, T + ∆T ) = B(t, T + ∆T )/P (t, T + ∆T ).
65
-
8/17/2019 Risk Topic 1
66/87
It is seen that
e(t , T , T + ∆T ) = B(t, T + ∆T )P (t, T ) − B(t, T + ∆T )
= B(t, T + ∆T ) P (t, T )
P (t, T + ∆T ) − 1
= ∆T B(t, T + ∆T )H (t , T , T + ∆T )
On taking the limit ∆T → 0, we obtain
rate of default compensation = e(t, T ) = lim∆T →0
e(t , T , T + ∆T )
∆T = B(t, T )h(t, T ) = B(t, T )P (t, T )h(t, T ).
The value of a security that pays π(s) if a default occurs at time s
for all t < s < T is given by
T t π(s)e(t, s) ds =
T t π(s)B(t, s)h(t, s) ds.
This result holds for deterministic recovery rates.
66
Random recovery value
-
8/17/2019 Risk Topic 1
67/87
Random recovery value
• Suppose the payoff at default is not a deterministic function
π(τ ) but a random variable π′ which is drawn at the time of
default τ . π′ is called a marked point process . Define
πe(t, T ) = E Q[π′|F t ∧ {τ = T }].
which is the expected value of π′ conditional on default at T and
information at t.
• Conditional on a default occurring at time T , the price of asecurity that pays π′ at default is B(t, T )πe(t, T ).
• Since the time of default is not known, we have to integrate
these values over all possible default times and weight them
with the respective probability of default occurring.
• The price at time t of a payoff of π′ at τ if τ ∈ [t, T ] is given by T t
πe(t, s) B(t, s)P (t, s) B(t,s)
h(t, s) ds.
67
-
8/17/2019 Risk Topic 1
68/87
Building blocks for credit derivatives pricing
Tenor structure
δk = T k+1 − T k, 0 ≤ k ≤ K − 1
Coupon and repayment dates for bonds, fixing dates for rates, pay-
ment and settlement dates for credit derivatives all fall on T k, 0 ≤
k ≤ K .
68
F da e tal a tities of the odel
-
8/17/2019 Risk Topic 1
69/87
Fundamental quantities of the model
• Term structure of default-free interest rates F (0, T )• Term structure of implied hazard rates H (0, T )• Expected recovery rate π (rate of recovery as percentage of par)
From B(0, T i) = B(0, T i−1)
1 + δi−1F (0, T i−1T i), i = 1, 2, · · · , k, and B(0, T 0) =
B(0, 0) = 1, we obtain
B(0, T k) =k
∏i=11
1 + δi−1F (0, T i−1, T i).
Similarly, from P (0, T i) = P (0, T i−1)
1 + δi−1H (0, T i−1, T i), we deduce that
B(0, T k) = B(0, T k)P (0, T k) = B(0, T k)k∏
i=1
1
1 + δi−1H (0, T i−1, T i).
e(0, T k, T k+1) = δkH (0, T k, T k+1)B(0, T k+1)= value of $1 at T k+1 if a default
has occurred in (T k, T k+1].
69
-
8/17/2019 Risk Topic 1
70/87
Taking the limit δi → 0, for all i = 0, 1, · · · , k
B(0, T k) = exp(
− T k
0f (0, s) ds
B(0, T k) = exp
(− T k
0[h(0, s) + f (0, s)] ds
e(0, T k) = h(0, T k)B(0, T k).
Alternatively, the above relations can be obtained by integrating
f (0, T ) = − ∂
∂T ln B(0, T ) with B(0, 0) = 1
f (0, T ) = h(0, T ) + f (0, T ) = − ∂
∂T
ln B(0, T ) with B(0, 0) = 1.
70
-
8/17/2019 Risk Topic 1
71/87
Defaultable fixed coupon bond
c(0) =K
n=1
cnB(0, T n) (coupon) cn = cδn−1
+ B(0, T K ) (principal)
+ πK
k=1
e(0, T k−1, T k) (recovery)
The recovery payment can be written as
πK
k=1
e(0, T k−1, T k) =K
k=1
πδk−1H (0, T k−1, T k)B(0, T k).
The recovery payments can be considered as an additional couponpayment stream of πδk−1H (0, T k−1, T k).
71
Defaultable floater
-
8/17/2019 Risk Topic 1
72/87
Defaultable floater
Recall that L(T n−1, T n) is the reference LIBOR rate applied over
[T n−1, T n] at T n−1 so that 1 + L(T n−1, T n)δn−1 is the growth factor
over [T n−1, T n]. Application of no-arbitrage argument gives
B(T n−1, T n) = 1
1 + L(T n−1, T n)δn−1.
• The coupon payment at T n equals LIBOR plus a spread
δn−1L(T n−1, T n) + s
par = 1B(T n−1, T n)
− 1
+ s parδn−1.
• Consider the payment of 1
B(T n−1, T n) at T n, its value at T n−1
is
B(T n−1, T n)
B(T n−1, T n) = P (T n−1, T n). Why? We use the defaultable
discount factor B(T n−1, T n) since the coupon payment may be
defaultable over [T n−1, T n].
72
• Seen at t = 0, the value becomes
-
8/17/2019 Risk Topic 1
73/87
B(0, T n−1)P (0, T n−1, T n)
= B(0, T n−1)P (0, T n−1)P (0, T n−1, T n)
= B(0, T n−1)P (0, T n).
Combining with the fixed part of the coupon payment and observing
the relation
[B(0, T n−1) − B(0, T n)]P (0, T n) = B(0, T n−1)
B(0, T n)
− 1B(0, T n)= δn−1F (0, T n−1, T n)B(0, T n),
the model price of the defaultable floating rate bond is
c(0) =
K
n=1 δn−1F (0, T n−1, T n)B(0, T n) + s
parK
n=1 δn−1B(0, T n)
+ B(0, T K ) + πK
k=1
e(0, T k−1, T k).
73
-
8/17/2019 Risk Topic 1
74/87
1.4 Pricing of credit derivatives
Credit default swap revisited
Fixed leg Payment of δn−1s at T n if no default until T n.
The value of the fixed leg is
s
N n=1
δn−1B(0, T n).
Floating leg Payment of 1 − π at T n if default in (T n−1, T n]
occurs. The value of the floating leg is
(1 − π)N
n=1e(0, T n−1, T n)= (1 − π)
N n=1
δn−1H (0, T n−1, T n)B(0, T n).
74
-
8/17/2019 Risk Topic 1
75/87
The market CDS spread is chosen such that the fixed leg and float-
ing leg of the CDS have the same value. Hence
s = (1 − π)
N n=1
δn−1H (0, T n−1, T n)B(0, T n)
∑N n=1 δn−1B(0, T n)
.
Define the weights
wn = δn−1B(0, T n)
N k=1
δk−1B(0, T k)
, n = 1, 2, · · · , N, andN
n=1
wn = 1,
then the fair swap premium rate is given by
s = (1 − π)N
n=1
wnH (0, T n−1, T n).
75
-
8/17/2019 Risk Topic 1
76/87
1. s depends only on the defaultable and default free discount rates,
which are given by the market bond prices. CDS is an example
of a cash product.
2. It is similar to the calculation of fixed rate in the interest rate
swap
s =
N n=1
w′nF (0, T n−1, T n)
where w′n = δn−1B(0, T n)
N k=1
δk−1B(0, T k)
, n = 1, 2, · · · , N.
76
M k d t k t l
-
8/17/2019 Risk Topic 1
77/87
Marked-to-market value
original CDS spread = s′; new CDS spread = s
Let Π = CDSold − CDSnew, and observe that CDSnew = 0, then
marked-to-market value = CDSold = Π = (s − s′)
N n=1
B(0, T n)δn−1.
Why? If an offsetting trade is entered at the current CDS rate s,
only the fee difference (s − s′
) will be received over the life of theCDS. Should a default occurs, the protection payments will cancel
out, and the fee difference payment will be cancelled, too. The
fee difference stream is defaultable and must be discounted with
B(0, T n).
• CDS’s are useful instruments to gain exposure against spread
movements, not just against default arrival risk.
77
-
8/17/2019 Risk Topic 1
78/87
Hedge based pricing – approximate hedge and replication strate-
gies
Provide hedge strategies that cover much of the risks involved in
credit derivatives – independent of any specific pricing model.
Basic instruments
1. Default free bond
C (t) = time-t price of default-free bond with fixed-coupon C
B(t, T ) = time-t price of default-free zero-coupon bond
2. Defaultable bond
C (t) = time-t price of defaultable bond with fixed-coupon c
C ′(t) = time-t price of defaultable bond with floating coupon
LIBOR + s par
78
3. Interest rate swap
-
8/17/2019 Risk Topic 1
79/87
S (t) = swap rate at time t of a standard fixed-for-floating
= B(t, tn) − B(t, tN )
A(t; tn, tN ) , t ≤ tn
where A(t; tn, tN ) =N
i=n+1
δiB(t, ti) = value of the payment stream
paying δi on each date ti.
Proof of the swap rate formula
The floating rate coupon payments can be generated by putting $1
at tn and taking away the floating interests immediately. At tN ,
$1 remains. The sum of the present value of the floating interests
= B(t, tn) − B(t, tN ).
Intuition behind cash-and-carry arbitrage pricing of CDSs
A combined position of a CDS with a defaultable bond C is very
well hedged against default risk.
79
Asset swap packages
-
8/17/2019 Risk Topic 1
80/87
An asset swap package consists of a defaultable coupon bond C with
coupon c and an interest rate swap. The bond’s coupon is swapped
into LIBOR plus the asset swap rate sA
. Asset swap package is soldat par.
Remark Asset swap transactions are driven by the desire to strip
out unwanted structured features from the underlying asset.
Payoff streams to the buyer of the asset swap package
time defaultable bond swap net
t = 0 −C (0) −1 + C (0) −1
t = ti c∗ −c + Li−1 + s
A Li−1 + sA + (c∗ − c)
t = tN (1 + c)∗ −c + LN −1 + sA 1∗ + LN −1 + sA + (c∗ − c)default recovery unaffected recovery
* denotes payment contingent on survival.
80
-
8/17/2019 Risk Topic 1
81/87
s(0) = fixed-for-floating swap rate (market quote)
A(0) = value of an annuity paying at the $1 (calculated based on
observable default free bond prices)
The value of asset swap package is set at par at t = 0, so that
C (0) + A(0)s(0) + A(0)sA(0) − A(0)c swap arrangement
= 1.
The present value of the floating coupons is given by A(0)s(0). The
swap continues even after default so that A(0) appears in all terms
associated with the swap arrangement.
81
-
8/17/2019 Risk Topic 1
82/87
Solving for sA(0)
sA(0) = 1A(0)
[1 − C (0)] + c − s(0).
Rearranging the terms,
C (0) + A(0)sA(0) = [1 − A(0)s(0)] + A(0)c
default-free bond≡ C (0)
where the right-hand side gives the value of a default-free bond with
coupon c. Note that 1 − A(0)s(0) is the present value of receiving
$1 at maturity tN . We obtain
sA(0) = 1
A(0)[C (0) − C (0)].
82
Credit spread options
-
8/17/2019 Risk Topic 1
83/87
The terminal payoff is given by
P sp(r,s,T ) = max(s − K, 0)
where r = riskless interest rates = credit spread
K = strike spread
Discrete-time Heath-Jarrow-Morton (HJM) method
• Follows the HJM term structure approach that models the for-ward rate process and forward spread process for riskless and
risky bonds.
• The model takes the observed term structures of riskfree forward
rates and credit spreads as input information.
• Find the risk neutral drifts of the stochastic processes such that
all discounted security prices are martingales.
83
-
8/17/2019 Risk Topic 1
84/87
Example Price a one-year put spread option on a two-year risky
zero-coupon bond struck at the strike spread K = 0.01.
Let the current observed term structure of riskless interest rates as
obtained from the spot rate curve for Treasury bonds be
r =
( 0.070.08
.
The riskless forward rate between year one and year two is
f 12 = 1.082
1.07 − 1 ≈ 0.09.
The market one-year and two-year spot spreads are
s = ( 0.0100.012 .
84
-
8/17/2019 Risk Topic 1
85/87
The two-year risky rate is 0.0 8 + 0.012 = 0.092. The current price
of a risky two-year zero coupon bond with face value $100 is
B(0) = $100/(1.092)2 = $83.86.
• The discrete stochastic process for the spread under the true
measure is assumed to take the form of a square-root processwhere the volatility depends on
√ s(0)
s(∆t) = s(0) + k[θ − s(0)]∆t ± σ√
s(0)∆t
where k = 0.3, θ = 0.02 and σ = 0.04, ∆t = 1, s(0) = 0.01.
85
-
8/17/2019 Risk Topic 1
86/87
• We need to add an adjustment term γ in the drift term in order
to risk-adjust the stochastic forward spread process
s(t) = s(0) + k[θ − s(0)]∆t + γ ± σ√ s(0)∆t.The adjustment term γ is determined by requiring the discounted
bond prices to be martingales.
• Let B(1) denote the price at t = 1 of the risky bond maturing
at t = 2. The forward defaultable discount factor over year one
and year two is 11 + f 12 + s(1)
, where s(1) is the forward spread
over the period.
s(1) =
{ γ + 0.017γ + 0.009
so that B(1) =
1001+f 12+γ +0.017
100
1+f 12+γ +0.009,with equal probabilities for assuming the high and low values.
86
-
8/17/2019 Risk Topic 1
87/87
We determine γ such that the bond price is a martingale.
B(0) = 83.86 = 1
1 + 0.0 7 + 0.01 ×
1
2 ( 100
1.107 + γ +
100
1.099 + γ .
The first term is the risky defaultable discount factor and the last
term is the expected value of B(1). We obtain γ = 0.0012 so that
s(1) = { 0.01820.0102
.
The current value of put spread option is
1
1.07 ×
1
2[(0.0182 − 0.01) + (0.0102 − 0.01)]L = 0.00393L,
where L is the notional value of the put spread option. Note that
the default free discount factor 1/1.07 is used in the option valuecalculation.