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Valuation of Constant Maturity Credit Default Swapsby
Hidetoshi Nakagawa
Hitotsubashi University, Tokyo
and
Meng-Lan Yueh
National Chengchi University, Taiwan
and
Ming-Hua Hsieh
National Chengchi University, Taiwan
Seminar on Department of Financial and Computational Mathematics
Providence University
8 November, 2011
1
Agenda
• Introduction of CDS/CMCDS
• Pricing of Credit Default Swaps (CDS)
• Pricing of Constant Maturity Credit Default Swaps (CMCDS)
• Determination of Participation Rate
• Examples
• Conclusions
2
Introduction of CDS
• Constant maturity credit default swaps (CMCDS) are a new kind of credit
derivative that provides a natural extension to vanilla credit default swaps
(CDS).
• A vanilla credit default swap (CDS) is a contract that provides insurance
against the default risk of an underlying entity. A CDS contract has a fixed
premium leg and a contingent default leg.
– The fixed premium leg corresponds to the periodic payments made by the
protection buyer to the seller until the maturity of the CDS or until the
occurrence of a credit event.
– The default leg corresponds to the net payment made by the protection
seller to the buyer in case of default.
• The fair spread for CDS is determined by equating the discounted cash flows
of these two legs.
3
ProtectionBuyer ProtectionSeller
ReferenceAsset
Periodic Fees
Payment
Contingent on
Default
Synthetic exposure
-
6 3
4
Introduction of CMCDS
• The premium leg of a CDS has fixed premium payments, however, the
premium leg of a CMCDS is floating and reset periodically.
• The premium payment is linked to a pre-specified benchmark of a fixed
maturity (called”constant maturity”) CDS rate, e.g. the current 5-year CDS of
a particular reference entity.
• CMCDS can be seen as floating premium CDS contracts.
5
Premium Legs of CDS
6
Premium Legs of CMCDS
7
CMCDS v.s CMS (Constant Maturity Interest Rate
Swaps)
• CMCDS is an extension of CMS in the area of credit market.
• In the fixed-income markets, constant maturity swap (CMS) was originally
used by insurance companies and pension funds to hedge their constant
maturity liabilities.
• Rates with constant maturity features generally facilitate the trading of pure
interest rate instruments for a selected maturity, without reducing time to
maturity associated with physical bonds.
• CMS allow investors to take positions on the forward interest rates between
two nominated CMS rates.
8
Strategy Involving CMCDS
• CMCDS transactions are receiving increasing attentions because they allow
investors to take a view on future credit spreads.
• The strategy of combining a CMCDS with an opposite position in a vanilla
CDS provides a means to separate spread risk from default risk.
• The resulting net position is a floating-fixed premium swap, which enables
investors to take exposure to only credit spread risk. In this way, a CMCDS
becomes a pure synthetic contract used for expressing a view on spread
widening or narrowing.
• Curve steepening benefits CMCDS protection sellers.
• Curve flattening benefits CMCDS protection buyers.
9
Nomura Fixed Income Research
(5)
Illustration of Fixed vs. Floating Premium
0 15
432
The "Fixed Leg" (Regular CDS)
The "Contingent Leg"
Year
Credit Event!
0 15
432
The "Floating Leg" (CMCDS)
The "Contingent Leg"
Year
Credit Event!
0 15
432
The "Fixed Leg" (Regular CDS)
The "Contingent Leg"
Year
Credit Event!
0 15
432
The "Floating Leg" (CMCDS)
The "Contingent Leg"
Year
Credit Event!
Source: Nomura
The market value of a CDS fluctuates as the default risk of the reference entity changes. The current level of default risk for a reference entity is reflected in the market spread. Accordingly, the value of an outstanding fixed-spread CDS changes with the market spread of the same maturity. For instance, if the market spread widens, the protection seller of the outstanding fixed-spread CDS contract suffers a mark-to-market loss, because the spread payment he receives is no longer enough to compensate for the amount of default risk implied in the market spread. On the other hand, if the spread tightens, the protection seller would benefit, because he continues to receive the fixed premium which is now greater than the market level. In contrast, movements in general spread levels little affect the value of a CMCDS, since the spread is periodically reset in sync with the market level.
Unlike an ordinary CDS, however, a CMCDS is sensitive to the shape of the spread curve. As discussed above, the gearing factor of a CMCDS is determined based on the expected levels of future spreads of the reference CDS, as reflected in the forward spreads. An upward-sloping spread curve implies that the reference CDS spread is likely to widen in the future. If that happens, the CMCDS spread will also increase. Therefore, the gearing factor, g, for the CMCDS is low when the spread curve is upward sloping and steep. If the spread curve gets steeper after the gearing factor is set, it will benefit the protection seller of a CMCDS (i.e., investor), because the breakeven level of g declines. The opposite is true when the spread curve gets flatter.
We can gain an exposure to just spread risk, but not default risk, by combining a fixed-spread CDS and a CMCDS. The combined position is sometimes called a “credit spread swap.” This in a sense is closer to the original definition of a “swap,” where two parties exchange series of floating payments and fixed payments.
A. Risk Profile
As mentioned above, the value of a CMCDS is much less sensitive to spread movements than an ordinary CDS. For example, the value of a CMCDS is virtually unchanged when the overall spread level widens by 1 basis point, while a 5-year fixed-spread CDS would see its value drop by nearly 5 bps. Graph 1 compares the value changes of a CMCDS and a regular CDS as we shift the spread curve in a parallel manner, assuming a notional value of $10 million and a flat spread curve of 100 bps initially. As we can see in Graph 1, the value of a regular CDS declines as the spread increases. In contrast, the value of a CMCDS is not very sensitive to changes in overall spread levels.
Price Quotes of CMCDS
• Since a vanilla CDS and a CMCDS provide the same protection against the
credit event of a same reference entity, no arbitrage principle thus implies that
the present value of the floating premium leg of a CMCDS must equal to the
present value of the CDS fixed premium leg.
• The equivalence is achieved by expressing the floating premium in a CMCDS
as a percentage of the reference fixed maturity CDS spread.
• The percentage is called the “participation rate”. The participation rate is the
multiplication factor that equates the expected discounted cash flow of
premium leg on a CMCDS with that of an equivalent CDS.
• CMCDS prices are quoted in terms of the participation rate.
10
16 March 2004 CMCDS: The Path to Floating Credit Spread Products Deutsche Bank@
Global Markets Research 5
Exhibit 3: Comparative Spreads for Traditional versus Constant Maturity
CDS structures (indicative levels as of 14 March 2004)
Single Name CMCDS
Reference Credit 5-Yr CDS
Mid Spread (bp)
5Yx5Y CMCDS Bid
Participation Rate (%)
Deutsche Telekom AG 60 65 France Telecom S.A. 60 64 Telecom Italia SpA 73 66 GMAC 185 78 Ford Motor Co. 219 77 Peugeot S.A. 49 71 Renault S.A. 50 74
CMCDO tranches
5Yx5Y CMCDO
Participation Rate (%) Reference Credit
5-Yr CDO
Tranche
Spread (bp)
Expected
S&P Rating Uncapped
Capped At 2x
Current Spread
Triton4 6 (6%x1%) 90 AAA 120 Triton 6 (4.75%x1%) 190 AA 250 Triton 6 (4.4%x1%) 235 A 275 Triton 6 (3.4%x1%) 400 BBB 370 Strauss (6%x1%) 105 AAA 115 130 Strauss (5.5%x1%) 155 AA 165 195 Strauss (4.4%x1%) 265 A 270 320 Strauss (3.4%x1%) 430 BBB 370 430
Source: DB Global Markets
In Exhibit 3, we show spreads and participation rates for selected single-name CMCDS and tranches of CMCDO deals. We also show participation rates for the Strauss capped and uncapped CMCDO tranches. Capped tranches (tranches where the portfolio spread is capped) offer higher participation rates. For each tranche, we also show expected S&P ratings, tranche subordination and thickness. Standard CMCDOs and capped structures are discussed in more detailed in the last section of this document.
4 Triton 6 is a static 100, predominantly US investment grade portfolio while the Strauss portfolio consists largely of European names.
Introdocution
What is CMS?
plain vanilla swap: floating rate = 6M LIBOR (+α)
CMS: floating rate = the swap rate for a swap for a certain period (e.g. 5-year swaprate)
Convexity adjustments are often used for simple and easy evaluation of future 6MLIBOR.
Floating -ratepayer
Fixed -rate payer
Periodic fixed payments
-Periodic floating paymentsfloating rate = the swap ratefor a swap for a certain period(e.g. 5-year SWAP rate)
Figure: The typical structure of CMS.
Nakagawa, Yueh and Hsieh (HU & NCU ) Valuation of CMCDS Nov. 24, 2010 4 / 36
注意事項：本文件係參照發行公司產品內容，委託人須詳細閱讀，自行判斷是否投資，並自行承擔風險，實際損益計算方式以交易日之英文版商品說明書(term sheet)所載條款為準。
UU SS DD 33 00 yy 及及 11 00 yy CC MM SS 利利 差差 走走 勢勢 圖圖
發發 行行 機機 構構 簡簡 介介 商商 品品 條條 件件
投投 資資 熱熱 線線
產產 品品 特特 色色
保保障障配配息息：： 首年固定配息 88..55%%(p.a.)，之後配息利率以 44%%(p.a.)加碼計算，皆優於定存。
倍倍數數給給息息：： 緊盯長短天期利差，利差越大，配息越高。
到到期期保保本本：： 發行機構到期 100%返還本金。
合合法法節節稅稅：： 自然人海外投資免稅，保障高資產客戶節稅需求。
計計 息息 方方 式式
第一年：無須任何條件，年息 8.5%
第二年至第十二年：配息利率 x n/N（計息基礎：30/360）
配息利率：4% + Max【0, 6 x（30y CMS Coupon Fixing – 10y CMS Coupon Fixing）】
30y CMS Coupon Fixing：每季債息起息日之前二個美國政府債券營業日的美元 30 年期 CMS
10y CMS Coupon Fixing：每季債息起息日之前二個美國政府債券營業日的美元 10 年期 CMS
n：於每季計息期間，符合計息條件之日曆天數。 N：360 天
計息條件：（美元 30 年期 CMS – 美元 10 年期 CMS）大於或等於 0。
募集時間：12/27/2004-01/07/2005 或額滿為止 最低申購金額：10,000 美元，以 10,000 美元累加。 發行日：2005.01.17 到期日：2017.01.17 或依發行機構提前買回條件 提前買回條件：發行機構有權自 2005/07/17 起(含)，每季
以票面金額 100%向委託人買回本債券。 配息日：每季配息一次 信託手續費：申購金額之 1.5%，一次計收。 信託管理費：第一年不收取。自第二年起，依客戶實際持
有天數，按年費率 0.2%，於到期時或客戶提前贖回時收取。贖回規定：自發行半年後，委託人可於每月 7 日、17 日申
請。提前贖回以 10 萬美元為最低贖回單位，並以其倍數增加。（提前贖回，可能會有本金折價的風險）
Commonwealth Bank of Australia（澳大利亞聯邦銀行）成立於 1912 年，為澳洲第一大零售、信用卡
發行以及基金管理銀行，全球銀行排名第 78，主要業務包括銀行貸款、 保險及中小企業融資業務。
信用評等：AA- by S&P；Aa3 by Moody’s
其他商品條件及投資風險請詳商品說明書
過去 10 年（1994 至 2004 年）期間，僅有 130 天利差小於 0
2
Notations
• (Ω,F , Q) : a complete probability space, where Q is supposed to be a
risk-neutral measure for the market model.
• τ : the default time of an issuer, which is a random time defined on
(Ω,F , Q).
• (Ft) : the filtration including the market information except for τ . In short, τ
cannot be a stopping time w.r.t. (Ft).
• (Ht) : the filtration generated by only τ . Namely, Ht := στ ∧ s|s ≤ t.
• (Gt) : the smallest filtration generated by (Ft) and (Ht). Namely,
Gt := Ft ∧Ht.
• F (t) := Q (τ ≤ t|Ft).
• S (t) : the (Ft)-survival process of τ ;
S (t) := 1− F (t) = Q (τ > t|Ft) = EQ[
Iτ>t|Ft
]
.
• Assume that S (t) > 0 for any t ≥ 0 and S (t) is continuous in t.
11
• Γt := − log (S (t)) : the (Ft)-hazard process of τ underQ; equivalently,
S (t) = e−Γt .
• rt : the risk-free short rate process that is (Ft)-adapted process.
• D (t, s) (t ≤ s) : the discount factor from s to t:
D (t, s) := exp(
−∫ s
trudu
)
.
• Tj : the time of the jth premium payment to take place, j = 1, 2, · · · .
• T : maturity of the CDS/CMCDS; T = Tn.
• j−1,j : time increment between payment at the (j − 1)th
and jth time
point in units of years.
• δ : a constant time increment we assume; that is, j−1,j = δ for all j.
• N : notional amount.
• R : recovery rate for the underlying obligor (supposed to be constant;
0 ≤ R < 1).
12
• I• : indicator function.
• s (t;Tj, Tk) (t ≤ Tj < Tk) : Gt-measurable variable; the spread that
makes the CDS contract, which has first payment at time Tj+1 and last
payment at time Tk, fair at the valuation time t; i.e. the spread for protection
in (Tj , Tk] at initial time t. In particular, s (t;Tj , Tk) ≡ 0 if τ ≤ t.
• M : constant maturity defined in the floating premium leg of a CMCDS;
M = mδ.
• η(t) the participation rate in the floating premium leg of a CMCDS.
13
CDS Pricing - Premium Leg
• Define PV premt
(
CDS(T0,Tn])
to be the present value at time t of the fixed
premium leg of a CDS for the period (T0, Tn], then
PV premt
(
CDS(T0,Tn])
(1)
= EQ
n∑
j=1
s (t;T0, Tn)×N × δ ×D(t, Tj)× Iτ>Tj
∣
∣
∣Gt
= N × δ × s (t;T0, Tn)×
n∑
j=1
EQ[
D(t, Tj)× Iτ>Tj
∣
∣
∣Gt
]
= N × δ × s (t;T0, Tn)× Iτ>t ×n∑
j=1
EQ[
D(t, Tj)× eΓt−ΓTj |Ft
]
= N × δ × s (t;T0, Tn)×n∑
j=1
EQ[
D(t, Tj)× eΓt−ΓTj |Ft
]
.
14
• The second last equality follows Lemma 1.
• The last equality follows from s (t;T0, Tn)× Iτ>t = s (t;T0, Tn), for
s (t;T0, Tn) = 0 when τ ≤ t by assumption.
15
Lemma 1
• Lemma 1 (Corollary 5.1.1 of Bielecki-Rutkowski) LetX be a Fs-measurable
and Q-integrable random variable. Then for any t ≤ s, we have
EQ[
Iτ>sX |Gt
]
= Iτ>tEQ[
XeΓt−Γs |Ft
]
.
16
CDS Pricing - Default Leg
• Define PV deft
(
CDS(T0,Tn])
to be the present value at time t of the default leg of the CDS,
then
PV deft
(
CDS(T0,Tn])
= EQ[
(1− R)×N ×D(t, τ)× IT0<τ≤Tn
∣
∣
∣Gt
]
= (1− R)×N × EQ[
D(t, τ)× IT0≤τ≤Tn|Gt
]
= (1− R)×N × Iτ>t × EQ
[∫ Tn
T0
D(t, u)eΓt−ΓudΓu
∣
∣
∣Ft
]
.
• The last equality is justified due to Lemma 2.
17
Lemma 2
EQ[
D(t, τ)× IT0≤τ≤Tn|Gt
]
= Iτ>t×EQ
[
∫ Tn
T0
D(t, u)eΓt−ΓudΓu
∣
∣
∣Ft
]
.
• (Proof)
18
EQ[
D(t, τ)× IT0≤τ≤Tn|Gt
]
= EQ[
D(t, T0)EQ[
D(T0, τ)× IT0≤τ≤Tn|GT0
]
∣
∣
∣Gt
]
(2)
= EQ
[
D(t, T0)Iτ>T0EQ
[
∫ Tn
T0
D(T0, u)eΓT0−ΓudΓu
∣
∣
∣FT0
]
∣
∣
∣Gt
]
= Iτ>tEQ
[
D(t, T0)eΓt−ΓT0EQ
[
∫ Tn
T0
D(T0, u)eΓT0−ΓudΓu
∣
∣
∣FT0
]
∣
∣
∣Ft
]
= Iτ>tEQ
[
EQ
[
∫ Tn
T0
D(t, u)eΓt−ΓudΓu
∣
∣
∣FT0
]
∣
∣
∣Ft
]
= Iτ>tEQ
[
∫ Tn
T0
D(t, u)eΓt−ΓudΓu
∣
∣
∣Ft
]
.
19
• The first and the last equality follow from the iterated conditioning property of
conditional expectation.
• The second equality and the third one respectively follow from (5.21) in
Cor.5.1.3 and (5.13) in Cor. 5.1.1 of Bielecki and Rutkowski (2002).
20
Pricing CDS - CDS Spread
• By equating the present values of both legs, we have the fair spread
s (t;T0, Tn) as
s (t;T0, Tn) = Iτ>t ×(1− R)× EQ
[
∫ Tn
T0D(t, u)eΓt−ΓudΓu
∣
∣
∣Ft
]
δ ×∑n
j=1EQ
[
D(t, Tj)× eΓt−ΓTj |Ft
]
=: Iτ>t × s (t;T0, Tn) .
• Remark that s (t;T0, Tn) is Ft-measurable.
21
CMCDS Pricing
• In a CMCDS with first payment in T1 and with final maturity Tn, protection on
a reference credit against default in [T0, Tn] is given from protection seller to
a protection buyer. In exchange for this protection, a “constant maturity” CDS
spread is paid.
• A contract that protects in [T0, Tn] can be decomposed into a stream of
contracts, each single contract protecting in [Tj−1, Tj ] , for j = 1, · · · , n,
with default payment postponed to Tj if default occurs in [Tj−1, Tj ] .
• In each single period [Tj−1, Tj], the rate s (Tj−1;Tj−1, Tj−1 +M) paid
at time Tj makes the exchange fair, so that a contract offering protection on a
reference credit in [T0, Tn] in exchange for payment of spread
s (T0;T0, T0 +M) , · · · , s (Tn−1;Tn−1, Tn−1 +M) at time
T1, · · · , Tn, is fair, i.e. has zero initial value at time t.
22
CMCDS Pricing - Premium Leg
• Define PVpremt
(
CMCDS(T0,Tn])
to be the present value of a premium leg of the CMCDS at
time t as
PVpremt
(
CMCDS(T0,Tn])
= EQ
n∑
j=1
s (Tj−1;Tj−1, Tj−1 +M)×N × δ ×D(t, Tj)× Iτ>Tj
∣
∣
∣Gt
= N × δ ×n∑
j=1
EQ[
s (Tj−1;Tj−1, Tj−1 +M)× Iτ>Tj−1×D(t, Tj)× Iτ>Tj
∣
∣
∣Gt
]
= N × δ ×n∑
j=1
EQ[
s (Tj−1;Tj−1, Tj−1 +M)×D(t, Tj)× Iτ>Tj
∣
∣
∣Gt
]
= N × δ × Iτ>t ×n∑
j=1
EQ[
s (Tj−1;Tj−1, Tj−1 +M)×D(t, Tj)× eΓt−ΓTj |Ft
]
.
The last equality follows from Lemma 1.
23
CMCDS Pricing - Premium Leg
• Since the default leg of a CMCDS is identical to the default leg of a plain
vanilla CDS written on the same reference entity, it follows that the premium
legs of CDS and CMCDS should be identical too.
PV premt
(
CDS(T0,Tn])
= PV premt
(
CMCDS(T0,Tn])
× η (t)
• The question of pricing a CMCDS reduces to finding the value of participation
rate η (t)
η (t) =s (t;T0, Tn)×
∑n
j=1EQ[
D(t, Tj)× eΓt−ΓTj |Ft
]
∑n
j=1EQ
[
s (Tj−1;Tj−1, Tj−1 +M)×D(t, Tj)× eΓt−ΓTj |Ft
] .
• The numerator in the above equation can be directly obtained under
assumptions for stochastic processes of rt and λt.
The denominator seems still complicated.
24
CMCDS Pricing - Participation Rate
• The determination of participation rate is reduced to the calculation of
EQ[
s (Tj−1;Tj−1, Tj−1 +M)×D(t, Tj)× eΓt−ΓTj |Ft
]
(j = 1, · · · , n).
where
s (Tj−1;Tj−1, Tj−1 +M)
=(1−R)× EQ
[
∫ Tj−1+M
Tj−1D(t, u)eΓt−ΓudΓu
∣
∣
∣Ft
]
δ ×∑j+m−1
k=j EQ[
P (t, Tk)eΓt−ΓTk |Ft
]
25
Determination of
EQ[
s (Tj−1;Tj−1, Tj−1 +M)×D(t, Tj)× eΓt−ΓTj |Ft
]
EQ
[
s(
Tj−1;Tj−1, Tj−1 +M)
× D(t, Tj) × eΓt−ΓTj |Ft
]
= EQ
(1 − R) × EQ
[
∫Tj−1+M
Tj−1D(Tj−1, u)e
ΓTj−1−Γu
dΓu
∣
∣
∣FTj−1
]
δ ×∑j+m−1k=j
EQ
[
D(Tj−1, Tk)eΓTj−1
−ΓTk |FTj−1
] × D(t, Tj) × eΓt−ΓTj
∣
∣
∣
∣
∣
Ft
=
(1 − R) × EQ[
∫Tj−1+M
Tj−1D(t, u)eΓt−ΓudΓu
∣
∣
∣Ft
]
δ ×∑j+m−1k=j
EQ[
D(t, Tk)eΓt−ΓTk |Ft
] ×
∑j+m−1k=j
EQ[
D(t, Tk)eΓt−ΓTk |Ft
]
EQ[
∫Tj−1+M
Tj−1D(t, u)eΓt−ΓudΓu
∣
∣
∣Ft
]
× EQ
EQ
[
∫Tj−1+M
Tj−1D(Tj−1, u)e
ΓTj−1−Γu
dΓu
∣
∣
∣FTj−1
]
∑j+m−1k=j
EQ
[
D(Tj−1, Tk)eΓTj−1
−ΓTk |FTj−1
] × D(t, Tj) × eΓt−ΓTj
∣
∣
∣
∣
∣
Ft
= s(
t;Tj−1, Tj−1 +M)
×
∑j+m−1k=j
EQ[
D(t, Tk)eΓt−ΓTk |Ft
]
EQ[
∫Tj−1+M
Tj−1D(t, u)eΓt−ΓudΓu
∣
∣
∣Ft
]
× EQ
EQ
[
∫Tj−1+M
Tj−1D(Tj−1, u)e
ΓTj−1−Γu
dΓu
∣
∣
∣FTj−1
]
∑j+m−1k=j
EQ
[
D(Tj−1, Tk)eΓTj−1
−ΓTk |FTj−1
] ×D(t, Tj) × eΓt−ΓTj
∣
∣
∣
∣
∣
Ft
.
26
Assumptions and Definitions
• Assume that the risk-free short rate process rt follows a stochastic process.
• Assume that the hazard process Γt is absolutely continuous, that is, there
exists a nonnegative (Ft)-progressively measurable process λt, which is
called the (Ft)-hazard rate process of τ underQ, such that for t ≤ s
Γt =
∫ t
0
λudu.
• Define
ϕ(t, s; rt, λt) := EQ
[
D(t, s)eΓt−Γs |Ft
]
= EQ
[
exp
(
−
∫
s
t(ru + λu) du
)
|Ft
]
ψ(t, s; rt, λt) := EQ
[
λsD(t, s)eΓt−Γs |Ft
]
= EQ
[
λs exp
(
−
∫
s
t(ru + λu) du
)
|Ft
]
.
Φ(t, Tj−1, Tj−1 +M; rt, λt) :=
j+m−1∑
k=j
ϕ(t, Tk; rt, λt)
Ψ(t, Tj−1, Tj−1 +M; rt, λt) :=
∫
Tj−1+M
Tj−1
ψ(t, u; rt, λt)du
27
Determination of EQ[
s (Tj−1;Tj−1, Tj−1 +M) ·D(t, Tj) · eΓt−ΓTj |Ft
]
EQ[
s (Tj−1;Tj−1, Tj−1 +M)×D(t, Tj)× eΓt−ΓTj |Ft
]
= s (t;Tj−1, Tj−1 +M)×
∑j+m−1k=j ϕ(t, Tk; rt, λt)
∫ Tj−1+M
Tj−1ψ(t, u; rt, λt)du
× EQ
∫ Tj−1+M
Tj−1ψ(Tj−1, u; rTj−1 , λTj−1)du
∑j+m−1k=j ϕ(Tj−1, Tk; rTj−1 , λTj−1)
×D(t, Tj)×Γt−ΓTj
∣
∣
∣
∣
∣
Ft
.
• Define a new probability measure Qj equivalent to Q by
dQj
dQ:=
D(0, Tj)e−ΓTj
EQ
[
D(0, Tj)e−ΓTj
] .
• From the Bayes’ rule(see Lemma 5.3 of Karatzas and Shreve(1991)), it
28
follows
EQ
[
s(
Tj−1;Tj−1, Tj−1 +M)
× D(t, Tj) × eΓt−ΓTj |Ft
]
= s(
t;Tj−1, Tj−1 +M)
×Φ(t, Tj−1, Tj−1 +M; rt, λt)
Ψ(t, Tj−1, Tj−1 +M; rt, λt)
×1
e−ΓtD(0, t)EQ
e−ΓTj D(0, Tj) ×
Ψ(Tj−1, Tj−1, Tj−1 +M; rTj−1, λTj−1
)
Φ(Tj−1, Tj−1, Tj−1 +M; rTj−1, λTj−1
)
∣
∣
∣
∣
∣
Ft
= s(
t;Tj−1, Tj−1 +M)
×Φ(t, Tj−1, Tj−1 +M; rt, λt)
Ψ(t, Tj−1, Tj−1 +M; rt, λt)
×1
e−ΓtD(0, t)EQ
[
e−ΓTj D(0, Tj)|Ft
]
× EQj
Ψ(Tj−1, Tj−1, Tj−1 +M; rTj−1, λTj−1
)
Φ(Tj−1, Tj−1, Tj−1 +M; rTj−1, λTj−1
)
∣
∣
∣
∣
∣
Ft
= s(
t;Tj−1, Tj−1 +M)
×Φ(t, Tj−1, Tj−1 +M; rt, λt)
Ψ(t, Tj−1, Tj−1 +M; rt, λt)
× ϕ(t, Tj ; rt, λt) × EQj
Ψ(Tj−1, Tj−1, Tj−1 +M; rTj−1, λTj−1
)
Φ(Tj−1, Tj−1, Tj−1 +M; rTj−1, λTj−1
)
∣
∣
∣
∣
∣
Ft
= s(
t;Tj−1, Tj−1 +M)
×Φ(t, Tj−1, Tj−1 +M; rt, λt)
Ψ(t, Tj−1, Tj−1 +M; rt, λt)× ϕ(t, Tj ; rt, λt)
×
∫
R2
Ψ(Tj−1, Tj−1, Tj−1 +M; r, λ)
Φ(Tj−1, Tj−1, Tj−1 +M; r, λ)Qj (rTj−1
∈ dr, λTj−1∈ dλ|Ft).
29
• That is,
EQ[
s (Tj−1;Tj−1, Tj−1 +M)×D(t, Tj)× eΓt−ΓTj |Ft
]
= s (t;Tj−1, Tj−1 +M)×Φ(t, Tj−1, Tj−1 +M ; rt, λt)
Ψ(t, Tj−1, Tj−1 +M ; rt, λt)× ϕ(t, Tj ; rt, λt)
×
∫
R2
Ψ(Tj−1, Tj−1, Tj−1 +M ; r, λ)
Φ(Tj−1, Tj−1, Tj−1 +M ; r, λ)Qj(rTj−1 ∈ dr, λTj−1 ∈ dλ|Ft).
30
Theorem
• Under all the assumptions given above, the value of participation rate η (t) of
CMCDS is given as
η (t) =s (t;T0, Tn)× Φ(t, T0, Tn; rt, λt)
∑n
j=1 s (t;Tj−1, Tj−1 +M)× ϕ(t, Tj ; rt, λt)×Θ(t, Tj−1)
where
Θ(t, Tj−1) :=Φ(t, Tj−1, Tj−1 +M ; rt, λt)
Ψ(t, Tj−1, Tj−1 +M ; rt, λt)
×
∫
R2
Ψ(Tj−1, Tj−1, Tj−1 +M ; r, λ)
Φ(Tj−1, Tj−1, Tj−1 +M ; r, λ)Qj(rTj−1 ∈ dr, λTj−1 ∈ dλ|Ft).
31
Convexity Adjustments
• In some institutional reports on CMCDS, the pricing of CMCDS is discussed
in terms of “convexity adjustment” methodology.
• The conditional expectation of a forward CDS spread is specified as the
product of the conditional forward survival probability and the current CDS
spread with the same maturity as the forward CDS plus some additional term
containing the volatility of the default intensity process.
EQt
[
s (Tj−1;Tj−1, Tj−1 +M)× Iτ>Tj−1
]
= s (t;Tj−1, Tj−1 +M)×Q (τ > Tj−1|Ft) + convexity adjustment
32
• Given Gt, what is the conditional expectation of a forward CDS spread
EQ [s (Tj−1;Tj−1, Tj−1 +M) |Gt]?
EQ [s (Tj−1;Tj−1, Tj−1 +M) |Gt]
= EQ[
Iτ>Tj−1 × s (Tj−1;Tj−1, Tj−1 +M) |Gt
]
= Iτ>t × EQ[
s (Tj−1;Tj−1, Tj−1 +M)× eΓt−ΓTj−1 |Ft
]
.
33
And
EQ[
s (Tj−1;Tj−1, Tj−1 +M)× eΓt−ΓTj−1 |Ft
]
= s (t;Tj−1, Tj−1 +M)×Φ(t, Tj−1, Tj−1 +M ; rt, λt)
Ψ(t, Tj−1, Tj−1 +M ; rt, λt)
×EQ
[
Φ(Tj−1, Tj−1, Tj−1 +M ; rTj−1 , λTj−1)
Ψ(Tj−1, Tj−1, Tj−1 +M ; rTj−1 , λTj−1)× eΓt−ΓTj
∣
∣
∣
∣
∣
Ft
]
34
• Define a new probability measure Qj−1 equivalent to Q by
dQj−1
dQ:=
e−ΓTj−1
EQ
[
e−ΓTj−1
] .
• Then the Bayes’ rule implies
EQ[
s (Tj−1;Tj−1, Tj−1 +M)× eΓt−ΓTj−1 |Ft
]
= s (t;Tj−1, Tj−1 +M)×Φ(t, Tj−1, Tj−1 +M ; rt, λt)
Ψ(t, Tj−1, Tj−1 +M ; rt, λt)
× EQ[
eΓt−ΓTj−1 |Ft
]
× EQj−1
[
Ψ(Tj−1, Tj−1, Tj−1 +M ; rTj−1 , λTj−1)
Φ(Tj−1, Tj−1, Tj−1 +M ; rTj−1 , λTj−1)
∣
∣
∣
∣
∣
Ft
]
.
35
• Finally we have
EQ [s (Tj−1;Tj−1, Tj−1 + M) |Gt]
= Iτ>t · eΓt · s (t;Tj−1, Tj−1 + M) ·Φ(t, Tj−1, Tj−1 + M ; rt, λt)
Ψ(t, Tj−1, Tj−1 + M ; rt, λt)
· EQ[
e−ΓTj−1 |Ft
]
· EQj−1
[
Ψ(Tj−1, Tj−1, Tj−1 + M ; rTj−1, λTj−1
)
Φ(Tj−1, Tj−1, Tj−1 + M ; rTj−1, λTj−1
)
∣
∣
∣
∣
∣
Ft
]
= Iτ>t · s (t;Tj−1, Tj−1 + M) ·EQ
[
e−ΓTj−1 |Ft
]
e−Γt·Φ(t, Tj−1, Tj−1 + M ; rt, λt)
Ψ(t, Tj−1, Tj−1 + M ; rt, λt)
· EQj−1
[
Ψ(Tj−1, Tj−1, Tj−1 + M ; rTj−1, λTj−1
)
Φ(Tj−1, Tj−1, Tj−1 + M ; rTj−1, λTj−1
)
∣
∣
∣
∣
∣
Ft
]
= s (t;Tj−1, Tj−1 + M) · Q (τ > Tj−1|Ft, τ > t)
·Φ(t, Tj−1, Tj−1 + M ; rt, λt)
Ψ(t, Tj−1, Tj−1 + M ; rt, λt)· EQj−1
[
Ψ(Tj−1, Tj−1, Tj−1 + M ; rTj−1, λTj−1
)
Φ(Tj−1, Tj−1, Tj−1 + M ; rTj−1, λTj−1
)
∣
∣
∣
∣
∣
Ft
]
.
36
• The forward spread EQ [s (Tj−1;Tj−1, Tj−1 +M) |Gt] is as
follows:
EQ
[s (Tj−1;Tj−1, Tj−1 + M) |Gt]
= s (t;Tj−1, Tj−1 + M) × Q (τ > Tj−1|Ft, τ > t)
×Φ(t, Tj−1, Tj−1 + M ; rt, λt)
Ψ(t, Tj−1, Tj−1 + M ; rt, λt)× E
Qj−1
[
Ψ(Tj−1, Tj−1, Tj−1 + M ; rTj−1, λTj−1
)
Φ(Tj−1, Tj−1, Tj−1 + M ; rTj−1, λTj−1
)
∣
∣
∣
∣
∣
Ft
]
.
37
Example - Vasicek Mean-Reverting Processes of rt
and λt
• Assume that both the default-free instantaneous spot rate rt and the
instantaneous hazard rate λt follow so-called Vasicek model as follows:
drt = κ(θ − rt)dt+ σdWt,
dλt = κ(θ − λt)dt+ σdWt.
• Suppose all the parameters are positive. Also, let Wt and Wt are
(Ft)-Brownian motions under the measureQ with dWtdWt = ρdt.
38
Example - Vasicek Mean-Reverting Processes of rt
and λt
For t ≤ s and dummy variables a, b, c(> 0), let
Z(t, s; a) :=1
a
(
1− e−a(s−t))
,
Y (t, s; a, b, c) := −
(
b
a−
c2
2a2
)
(s− t)
+
(
b
a−
c2
2a2
)
Z(t, s; a)−c2
4aZ(t, s; a)2.
Then we have
Proof: See Proposition 7.2 of Schonbucher (2003).
39
Example - Vasicek Mean-Reverting Processes of rt
and λt
• Denote by Lt the Radon-Nikodym density process
Lt : = EQ
[
dQj
dQ
∣
∣
∣Ft
]
=1
EQ
[
exp(
−∫ T
0ru + λudu
)] (3)
×EQ
[
exp
(
−
∫ T
0
ru + λudu
)
∣
∣
∣
∣
Ft
]
. (4)
• Since the instantaneous correlation betweenWt and Wt is given by ρ, we
can use anotherQ-standard Brownian motion Wt independent of Wt to
regardWt as
Wt = ρWt +√
1− ρ2Wt. (5)
40
• Then we also have
dLt = Lt
(
−Z(t, T ;κ)σdWt − Z(t, T ; κ)σdWt
)
(6)
= Lt
(
−ρσZ(t, T ;κ) + σZ(t, T ; κ)dWt − σ√
1− ρ2Z(t, T ; κ)dWt
)
.
(7)
41
41-1
Example - Vasicek Mean-Reverting Processes of rt
and λt
• From Girsanov theorem, it follows that the followings are independent
standard Brownian motions on the period [0, Tj ] with respect to “Tj -forward”
measure Qj .
Wjt := Wt +
∫ t
0
ρσZ(u, T ;κ) + σZ(u, T ; κ) du,
Wjt := Wt +
∫ t
0
σ√
1− ρ2Z(u, T ;κ)du.
• In addition, set Wjt := ρW
jt +
√
1− ρ2Wjt , which is also a standard
Brownian motion with respect to Qj and dWjt dW
jt = ρdt.
42
Dynamics of rt and λt under Measure Qj
• The dynamics of rt and λt under the “Tj -forward” measure Qj are given by
drt =
κθ − ρσσZ(t, T ; κ)− σ2Z(t, T ;κ)− κrt
dt+ σdWjt ,
dλt =
κθ − ρσσZ(t, T ; κ)− σ2Z(t, T ; κ)− κλt
dt+ σdWjt .
• Namely, we have
rT = rte−κ(T−t) +
∫ T
t
e−κ(T−s)
κθ − ρσσZ(s, T ; κ)− σ2Z(s, T ; κ)
ds
+
∫ T
t
e−κ(T−s)σdW js ,
λT = λte−κ(T−t) +
∫ T
t
e−κ(T−s)
κθ − ρσσZ(s, T ;κ)− σ2Z(s, T ; κ)
ds
+
∫ T
t
e−κ(T−s)σdW js .
• Anyway, Qj(rTj−1∈ dr, λTj−1
∈ dλ|rt, λt) can be seen as two-dimensional normal
distribution.
43
Numerical Results - Parameters Setting• Default intensity λt :
dλt = κ(θ − λt)dt+ σdWt
where κ = 0.2; θ = 0.03; σ = 0.7;λ0 = 0.03
• Short rate rt :
drt = κ(θ − rt)dt+ σdWt
whereκ = 0.5; θ = 0.06;σ = 0.03; r0 = 0.06 . . .Ait-Sahalia (2002, Econometrica)
or κ = 0.261; θ = 0.07; σ = 0.022; r0 = 0.07 . . .Cheng and Chen (2009,Journal of
Econometrics)
• ρ : Correlation between rt and λt : ρ = 0.2, 0.4, 0.6, 0.8,−0.2,−0.4,−0.6,−0.8
• Tn = 5 or 7 or 10
• M = 5 (constant maturity)
44
Numerical Result - Base Case
Parameter Values - Base Case
t = 0 current time
T0 = t CMCDS starting date
Tn = T = 5 CMCDS maturity date
δ = 0.25 quarterly payemt
M = 5 constant maturity
ρ = 0.2 correlation between rt and λt
s (t;T0, Ti) = 150 bp = 1.5%, for all i flat initial spread curve
κ = 0.2; θ = 0.03; σ = 0.03;λ0 = 0.03 dλt = κ(θ − λt)dt+ σdWt
κ = 0.5; θ = 0.06;σ = 0.03; r0 = 0.06 drt = κ(θ − rt)dt+ σdWt
45
The Effect of Hazard Rate Volatility
Value of σ
0.01 0.02 0.03 0.04 0.05
η 0.8195 0.8149 0.8064 0.7929 0.7728
Θ(0, 1) 1.0291 1.0387 1.0563 1.0820 1.1203
Θ(0, 2) 1.1083 1.1228 1.1492 1.1921 1.2730
Θ(0, 3) 1.2420 1.2543 1.2797 1.3244 1.4098
Θ(0, 4) 1.4366 1.4398 1.4454 1.4487 1.4787
Θ(0, 5) 1.7047 1.6855 1.6499 1.5829 1.3754
46
The Effect of Short Rate Volatility
Value of σ
0.01 0.02 0.03 0.04 0.05
η 0.8077 0.8068 0.8066 0.8063 0.8055
Θ(0, 1) 1.0535 1.0540 1.0556 1.0570 1.0592
Θ(0, 2) 1.1449 1.1478 1.1482 1.1492 1.1526
Θ(0, 3) 1.2766 1.2772 1.2791 1.2794 1.2819
Θ(0, 4) 1.4482 1.4450 1.4469 1.4459 1.4415
Θ(0, 5) 1.6540 1.6558 1.6494 1.6446 1.6430
47
The Effect of Correlation between Hazard Rate and
Short Rate
Value of ρ
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8
η 0.8039 0.8052 0.8056 0.8066 0.8076 0.8084 0.8094 0.8104 0.8115
Θ(0, 1) 1.0675 1.0632 1.0588 1.0556 1.0519 1.0483 1.0448 1.0417 1.0373
Θ(0, 2) 1.1658 1.1589 1.1538 1.1485 1.1444 1.1390 1.1335 1.1307 1.1248
Θ(0, 3) 1.2887 1.2845 1.2820 1.2777 1.2754 1.2714 1.2694 1.2666 1.2633
Θ(0, 4) 1.4394 1.4381 1.4424 1.4436 1.4445 1.4475 1.4487 1.4504 1.4549
Θ(0, 5) 1.6090 1.6166 1.6348 1.6447 1.6610 1.6704 1.6819 1.6870 1.6979
48
The Effect of Initial Credit Spread Curve
Initial Credit Spread Curve s (t;T0, Ti)
flat curve upward-sloping curve downward-sloping curve
η 0.8063 0.5346 0.9713
Θ(0, 1) 1.0562 1.0562 1.0556
Θ(0, 2) 1.1492 1.1480 1.1483
Θ(0, 3) 1.2796 1.2785 1.2780
Θ(0, 4) 1.4446 1.4438 1.4439
Θ(0, 5) 1.6442 1.6479 1.6505
49
The Effect of Contract Maturity
Contract Maturity Tn
Tn = 5 Tn = 7 Tn = 10
η 0.8148 0.7083 0.5977
Θ(0, 1) 1.0387 1.0390 1.0388
Θ(0, 2) 1.1227 1.1228 1.1219
Θ(0, 3) 1.2544 1.2540 1.2543
Θ(0, 4) 1.4405 1.4395 1.4393
Θ(0, 5) 1.6882 1.6870 1.6850
50
Conclusion
• A broad class of exotic interest rate derivatives can be valued simply by
adjusting the forward interest rate. This adjustment is known in the market as
“convexity correction”. Various ad hoc rules are used to calculate the
convexity correction for different products.
• In this research, we examines the mechanism of a CMCDS contract and
develop a method of valuing the credit derivatives with constant maturity
features.
• We put convexity correction on a rigorous mathematical basis by showing that
it can be interpreted as the side-effect of a change of probability measure.
• The analysis here provides us with a theoretically consistent framework to
calculate convexity corrections.
51
