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16 December 2004
Claus M. Pedersen [email protected] +1 212 526 7775Saurav Sen [email protected] +44 20 7102 2940
Quantitative Credit ResearchLehman Brothers
Please see important analyst certification(s) on the back page of this report
Product Description
Constant-Maturity Default Swaps (CMDS)
� Evolved as a variant of standard credit default swaps (CDS)� Innovation is the introduction of a floating premium leg� Premium leg pays a floating coupon proportional to the observed market spread on
a standard CDS with a fixed time to maturity.
Standard CDS CMDSFixed premium leg cashflows Floating premium leg cashflows
Protection following credit event: 1-RProtection following credit event: 1-R
Time Time
Cap
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The CMDS Floating Premium Leg
� Floating payments are linked to the prevalent market spread of a standard CDS with constant time to maturity – this is called the tenor of the CMDS
� The constant maturity spread sets in avance and pays in arrears� The spread paid is a contractually fixed fraction of the reset spread – this is called
the participation rate and we denote it by α� A CMDS is quoted by a bid-offer on the participation rate.� The floating spread is capped to avoid circumstances in which the spread may
become unobservable if the credit goes to high yield.
Reset Date
5Y CDS Spread = 100bp
Participation Rate = 80%
Notional = 10m
Payment Date
Coupon
= 80% × 0.25Y × 10m × 100bp
= 20,000
Accrual period = 0.25Y
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Standardised Inter-Dealer CMDS Contract
� Standardization of the contract is in process but not complete.� The reset mechanism is probably the biggest hurdle in making CMDS a more
liquid product– Party receiving the floating premium makes a bid for CDS protection at a
spread that will become the reset spread.– The required notional to bid is determined so that the floating receiver has
incentive to bid at current market level.– If no bid is made and the reset spread will be determined by a dealer poll.
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Variations on CMDS
� Floating-for-Protection:– Reduced mark-to-market volatility– For investors concerned about volatility in their reported earnings.– For investors bullish about the credit fundamentals of an issuer but concerned
about current market valuations.� Fixed-for-Floating:
– Take a view on spread widening or narrowing – No recovery rate risk following a credit event
� Floating-for-Floating:– Take a view on the shape of the curve– No recovery rate risk
� CMDS can be structured into tranched products giving investors diversification and the option to choose leverage.
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Valuation
Valuation of the Floating Leg
� Forward spreads are the main determinant of reset spreads� A convexity adjustment is needed if spreads are volatile� Intuitively, this is because mark-to-market depends on the forward PV01, which
is a convex function of spreads� Mathematically, the PV of a future coupon is its discounted expected value� Expectation must be computed under the appropriate numeraire� The forward spread is the expected spread under the risky PV01 numeraire� A convexity correction is therefore required in order to use a risky discount bond
as numeraire
Reset Spread = Forward (Main) + Convexity Adjustment (Secondary)
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Stochastic Hazard Rate Models
� Assume, say, lognormal dynamics for the hazard rate� Implies almost lognormal spread dynamics� Can be implemented on a trinomial tree with jump-to-default at each node� Integrated pricing of embedded options� Lognormal is not perfect, but preferable to, say, a Gaussian model� Can incorporate features like mean-reverting hazard rates
Integrated framework for pricing embedded options
λu
λd
λm
λ
Jump to Default
Probability = λ dt
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Change of Numeraire Technique
� The “correct” expected reset spread has to be computed using a risky discount bond as numeraire
� The forward spread is the expected reset spread under the risky PV01 numeraire� The Black formula can be used to price embedded options� Change of numeraire:
Compute E[S] and use Black Model for pricing embedded options
Taylor Series
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Semi-Analytic Valuation
� Similar to affine term structure models in interest rates� Assume some dynamics for the hazard rate, e.g.:
� Leads to closed-form survival probabilities
� If the distribution of hazard rate is tractable, then expected future cashflows can be computed using numerical integration
� Model can be calibrated to match the initial term structure
Exponential-affine survival probabilities
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Volatility Modelling
� Embedded options are typically deep out of the money� For this reason, it is important to model the volatility smile carefully� To correctly price a strip of cashflows, we also need information about the term
structure of volatility� Models can incorporate volatility surfaces, but calibration data is still sparse� Single-name default swaptions are becoming more liquid� Otherwise, we have to rely on historical estimates
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Numerical Examples
Size of the Convexity Adjustment
Forwards are the main determinant of reset spreads
60
70
80
90
100
110
120
130
2005 2006 2007 2008 2009
Reset Time
Forward
E[S] (25% Vol)
E[S] (50% Vol)
E[S] (75% Vol)
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Participation Rate as a Function of Volatility
Moneyness of the embedded options affects the participation rate
60%
65%
70%
75%
80%
85%
0% 25% 50% 75% 100%
Hazard Rate Volatility (Lognormal)
Part
icip
atio
n R
ate
C = 250
C = 500
C = 1000
No Cap
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Sensitivity to Parallel Curve Movements
CMDS are much less sensitive to parallel curve shifts than CDS
-60,000
-40,000
-20,000
-
20,000
40,000
60,000
-10 -5 0 5 10
Parallel Spread Bump (bp)
Net
PV
5Y CMDS v Prot
5Y Fixed v Prot
5Y CMDS v 3Y CMDS
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Sensitivity to Curve Slope Changes
CMDS allow investors to take a view on the CDS curve slope
-100,000
-80,000
-60,000
-40,000
-20,000
-
20,000
40,000
60,000
80,000
100,000
-5 -3 -1 1 3 5
Slope Change (bp/Y) with 5Y Pivot
Net
PV
5Y CMDS v Prot
5Y Fixed v Prot
5Y CMDS v 3Y CMDS
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Effect of Volatility
-30,000
-20,000
-10,000
-
10,000
20,000
30,000
0% 20% 40% 60% 80% 100%
Hazard Rate Volatility
5Y CMDS v Prot (500bp Cap)5Y CMDS v Prot (No Cap)5Y CMDS v 3Y CMDS (500bp Cap)5Y CMDS v 3Y CMDS (No Cap)
Embedded caps can have a significant effect on the vega of CMDS
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Mark-to-Market
Mark-to-Market of CMDS
� MTM = Floating Premium Leg minus Protection Leg
� The value of the protection leg is:
� The breakeven participation ratio satisfies, by definition:
� MTM is given by:
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Example: Historical MTM Volatility
Historically, spreads tend to move more parallel and so CMDS have lower MTMvolatility (the example is for Ford Motor Credit)
-200,000
0
200,000
400,000
600,000
800,000
1,000,000CDS CMDS
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Hedging CMDS with CDS
-200
0
200
400
600
800
1,000
Maturity (Y)
Due to difference between premium and protection leg sensitivities
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Summary
� CMDSs represent an innovation in the credit derivatives market� Allow investors to take a view on the shape of the credit curve� Volatility becomes important when embedded options are near-the-money� Some outstanding issues concerning the reset mechanism� But of potential interest to a wide class of investors
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References
� Pedersen, C. M., and S. Sen: “Valuation of Constant-Maturity Default Swaps”, Lehman Brothers Quantitative Credit Research Quarterly, Vol. 2004-Q2.
� Naldi, M., D. O’Kane et. al.: The Lehman Brothers Guide to Exotic Credit Derivatives, special supplement to Risk, October 2003.
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The views expressed in this report accurately reflect the personal views of Claus M. Pedersen and Saurav Sen, the primary analyst(s) responsible for this report, about the subject securities or issuers referred to herein, and no part of such analyst(s)’ compensation was, is or will be directly or indirectly related to the specific recommendations or views expressed herein.Any reports referenced herein published after 14 April 2003 have been certified in accordance with Regulation AC. To obtain copies of these reports and their certifications, please contact Larry Pindyck ([email protected]; 212-526-6268) or Valerie Monchi ([email protected]; 44-(0)207-102-8035).Lehman Brothers Inc. and any affiliate may have a position in the instruments or the Company discussed in this report. The Firm’s interests may conflict with the interests of an investor in those instruments.The research analysts responsible for preparing this report receive compensation based upon various factors, including, among other things, the quality of their work, firm revenues, including trading, competitive factors and client feedback.
This material has been prepared and/or issued by Lehman Brothers Inc., member SIPC, and/or one of its affiliates (“Lehman Brothers”) and has been approved by Lehman Brothers International (Europe), authorised and regulated by the Financial Services Authority, in connection with its distribution in the European Economic Area. This material is distributed in Japan by Lehman Brothers Japan Inc., and in Hong Kong by Lehman Brothers Asia Limited. This material is distributed in Australia by Lehman Brothers Australia Pty Limited, and in Singapore by Lehman Brothers Inc., Singapore Branch. This material is distributed in Korea by Lehman Brothers International (Europe) Seoul Branch. This document is for information purposes only and it should not be regarded as an offer to sell or as a solicitation of an offer to buy the securities or other instruments mentioned in it. No part of this document may be reproduced in any manner without the written permission of Lehman Brothers. We do not represent that this information, including any third party information, is accurate or complete and it should not be relied upon as such. It is provided with the understanding that LehmanBrothers is not acting in a fiduciary capacity. Opinions expressed herein reflect the opinion of Lehman Brothers and are subject to change without notice. The products mentioned in this document may not be eligible for sale in some states or countries, and they may not be suitable for all types of investors. If an investor has any doubts about product suitability, he should consult his Lehman Brothers representative. The value of and the income produced by products may fluctuate, so that an investor may get back less than he invested. Value and income may be adversely affected by exchange rates, interest rates, or other factors. Past performance is not necessarily indicative of future results. If a product is income producing, part of the capital invested may be used to pay that income. Lehman Brothers may, from time to time, perform investment banking or other services for, or solicit investment banking or other business from any company mentioned in this document. © 2004 Lehman Brothers. All rights reserved. Additional information is available on request. Please contact a Lehman Brothers’ entity in your home jurisdiction.
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