Transcript Applications of Arbitrage-free Models

Applications of Arbitrage-free Models:
New Frontiers in Interest Rate, Credit and
Energy Risks
Third Annual Bloomberg Lecture in Finance
THOMAS S. Y. HO PhD
PRESIDENT
THC
OCTOBER 26, 2009
[email protected]
Arbitrage-free Term Structure Models
2
 Valuation models

Derivative pricing (relative valuation) under interest rate, credit
and other risk drivers
 Applications

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
Trading
Portfolio management
Enterprise risk management
 Impacts on the markets


Price discovery process
Regulatory policies in the financial markets
Introduction
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Questions Addressed
3
 What are the model’s economic principles that make
the model popular and fundamental?
 What are the frontiers of applications of the model in
going forward?
 What are my cautionary notes on the use of the model?
Detail discussions are available in the references
Introduction
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References
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Amin, Kaushik I., and Andrew J. Morton, 1994, “ Implied Volatility Functions in Arbitrage-free Term Structure Models, “ Journal of
Financial Economics, 35 (2), 141-180
Benth, Fred Espen, Lars Ekeland, Ragner Hauger and Bjorn Fredrik Nielsen 2003 “A Note on Arbitrage-free Pricing of Forward
Contracts in Energy Market” Applied Mathematical Finance 10, 325-336
Eydeland, Alexander and Krzysztof Wolyniec 2003 Energy and Power Risk Management, Wiley Finance
Harrison, J Michael, and David M. Kreps, 1979 “Martingales and Arbitrage in Multiperiod Securities Markets<” Journey of Economic
Theory, 20(3), 381-408
Ho, Thomas S. Y. 1992 “Key rate durations: measures of interest rate risks” Journal of Fixed-Income, 2(2), 19-44
Ho, Thomas S. Y. and Sang-Bin Lee 2003, The Oxford Guide to Financial Modeling, Oxford University Press
Ho, Thomas S. Y. and Sang-Bin Lee 1986, “Term Structure Movements and the Pricing of Interest Rate Contingent Claims,” Journal of
Finance, 41 (5), 1011-1029
Ho, Thomas S. Y. and Sang Bin Lee,2009 “ Valuation of Credit Contingent Claims: An Arbitrage-free Credit Model” Journal of
Investment Management vol 7 No 5
Ho, Thomas S. Y. Ho and Sang Bin Lee, 2009 ”A Unified Credit and Interest Rate Arbitrage-Free Contingent Claim Model” Journal of
Fixed-Income
Ho, Thomas S. Y. and Blessing Mudavanhu,2007 “Stochastic Movement of the Implied Volatility Function” Journal of Investment
Management 4th quarter
Ho, Thomas S. Y. and Sang Bin Lee, 2007 ““Generalized Ho-Lee Model: A Multi-factor State-Time Dependent Implied Volatility
Function Approach” Journal of Fixed Income 4th quarter
Ho, Thomas S. Y. 2007 “Managing Interest Rate Volatility Risk: Key Rate Vega” Journal of Fixed Income 4th quarter
Ho, Thomas S. Y. and Sang-Bin Lee 2009 “ A Unified Model: Arbitrage-free Term Structure Movements of Flow Risks”
Ho, Thomas S. Y. and Sang Bin Lee “ Pricing of Contingent Claims on Natural Gas” working paper
Nawalkha, Sanjay K., Natalia A. Beliaeva and Gloria M Soto 2007 Dynamic Term Structure Modeling Wiley Finance
References
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Outline
5
 Salient features of the model
 A perspective of arbitrage-free term structure models
 A framework to explore new frontiers in applications
 Apply the framework to …
 Interest rate risk
 Credit risk
 Energy risk
 Going forward: Managing model risks
 Edwards Deming approach to risk management
Introduction
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The Black-Scholes Model
Components of an Arbitrage-free Model
6
 Valuation component
C( S, t) Specify the contingent claim
 dS = r(t) Sdt + σ (t) Sdz


Specify the underlying risk process
 Application component


Delta: dynamic replication
Calibration to determine the implied volatility
Arbitrage-free Models
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Arbitrage-free Term Structure Model
Valuation Component of the Model
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 C = C( r , p(t, T), t)
 Contingent claims on the discount function
 dr = F( p(t, T), t)dt + σ (r, t) dw
 Short rate model
 Forward rate model
 Market model
 X(n-1,i) = 0.5 p(n,i)(B(n,i) + B(n, i+1))
 Rolling back adjusted by the discount rate
B(n-1, i) = max ( X(n-1, i), K)
Arbitrage-free models
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Key Rate Duration – Dynamic Replication
Application Component of the Model
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 Callable bond
 Maturity 2020-10-15
 SA fixed coupon rate
5.65%
 Bermudan callable at par
 Used for hedging, risk
management, and
investment
Arbitrage-free Models
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Implied Volatility Function
Calibration: Application Component of the Model
9
Arbitrage-free Models
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Term Structure Models and the Black-Scholes
Model: a Comparison
10
 The term structure model: Time dimension, rate, a
“flow concept”

From the economic modeling perspective, the Black
Scholes model is not a special case of a term structure
model – hence “term structure”
 Correlations of the securities of the term structure:
Principal component methods
Ho, Thomas S. Y. and Sang-Bin Lee 2009 “ A Unified Model: Arbitrage-free Term Structure Movements of
Flow Risks”
Arbitrage-free models
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Proposed Perspective of Term Structure Models
 “Stock” versus “Flow”
11
Arbitrage-free models have two components
 Valuation component (some examples)
 Multi-factor models
 Time and state dependent implied volatility function
 Unspanned stochastic volatility function
 Application component
 Effectiveness of dynamic hedging and implications of the
implied volatilities

Arbitrage-free Models
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Credit Term Structure
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 Valuation of fixed-income instruments with credit risk
Reduced form and structural models
 Credit default swap (CDS)
 Survival function vs discount function
 State and time dependent survival rate s(n,i)

Ho, Thomas S. Y. and Sang Bin Lee,2009 “ Valuation of Credit Contingent Claims: An
Arbitrage-free Credit Model” Journal of Investment Management vol 7 No 5
Applications: Credit Risk Modeling
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Valuing Credit Contingent Claims
Valuation Component of the Model
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 Make-whole Option
 X(n-1,i) = 0.5 p(n) s(n,i) (B(n,i) + B(n, i+1))
Rolling back adjusted by the survival rate
 B(n-1, i) = max ( X(n-1, i), K)
 Boundary and terminal conditions
 p(n) one period time value discount factor
 K strike price (an example), present value of the yield adjusted
Treasury bonds

Applications: Credit Risk Modeling
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Valuation of Embedded Credit Options
14
Performance Profile
108
prices
106
Straight Bond
Make-Whole
Callable
104
102
100
98
0.000 0.004 0.008 0.012 0.016 0.020
hazard rate
Applications: Credit Rsk Modeling
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Implications of the Credit Term Structure
Application Component of the Model
15
 Determine the credit key rate durations for credit
hedging

Specify the precise dollar credit exposure in the term
structure
 Identify the implied credit volatilities

Use of the structural models
 Interest rate and credit risk relationship

Applications to a callable instruments
Applications: Credit Risk Modeling
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Applications of the Term Structure Credit Model
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 Dynamic movements of the term structure of credit
 Specify the embedded make whole option in
commercial mortgages

Impact of the correlation to the interest rate level
 Relating a reduced form model to the structural model
 Multi-factor credit model
Applications: Credit Risk Modeling
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Natural Gas Futures Term Structure
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 Basic economics of natural gas: Henry Hub data
 Well head cost, gathering and processing costs
 Storage and cost to carry
 Demand: Weather affects heating; power generation
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Injection season: April - October
Withdrawal season: November – March
Ho, Thomas S. Y. and Sang Bin Lee “ Pricing of Contingent Claims on Natural Gas”
working paper
Applications: Energy Risk Modeling
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Importance of Modeling NG Term Structure
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 Deregulation of power industry
 Supply: Horizontal rigs
 Power prices
Depending on the bid stack and power demand
 Bid stack depends on the fuel price and the outage
 Use of derivatives to manage energy risk and capital
investments

Applications: Energy Risk Modeling
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Modeling of the Natural Gas Price Process
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 Abadic, Luis M and Jose Chamorra (2006)
 2 factor model with the stochastic fuel price mean reverting to a
stochastic long term price
 The stochastic long term price mean reverting to a constant price
 Eydeland, Alexander and Krzysztof Wolyniec (2003)
and Benth et al (2003)

Use arbitrage-free models
 Ho and Lee (2009)
 Identify the term structure “flow risk” and the “stock risk”
Applications: Energy Risk Modeling
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Data and Methodology
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 Futures and spot daily prices from 1/3/2006 -
12/27/2006
 Futures delivery dates: monthly from 1/1/2007 and
1/1/2010
 Implied cost of carry c(t, T) = (1/(T-t))ln F(t,T)/S(t)
 Use the principal component approach to specify the
movements
Data Source: Logical Information Machines (LIM)
Applications: Energy Risk Modeling
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2005/12/12
2006/1/12
2006/2/12
2006/3/12
2006/4/12
2006/5/12
2006/6/12
2006/7/12
2006/8/12
2006/9/12
2006/10/12
2006/11/12
2006/12/12
2007/1/12
2007/2/12
2007/3/12
2007/4/12
2007/5/12
2007/6/12
2007/7/12
2007/8/12
2007/9/12
2007/10/12
2007/11/12
2007/12/12
2008/1/12
2008/2/12
2008/3/12
2008/4/12
2008/5/12
2008/6/12
2008/7/12
2008/8/12
2008/9/12
2008/10/12
2008/11/12
2008/12/12
2009/1/12
2009/2/12
2009/3/12
2009/4/12
2009/5/12
2009/6/12
2009/7/12
Henry Hub $ MMBtu (12/12/05-8/7/09)
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18
16
14
12
10
8
6
4
2
0
Applications: Energy Risk Modeling
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Applications: Energy Risk Model
Sep/14
Jul/14
May/14
Mar/14
Jan/14
Nov/13
Sep/13
Jul/13
May/13
Mar/13
Jan/13
Nov/12
Sep/12
Jul/12
May/12
Mar/12
Jan/12
Nov/11
8
Sep/11
Jul/11
May/11
Mar/11
Jan/11
Nov/10
Sep/10
Jul/10
May/10
Mar/10
Jan/10
Nov/09
Term Structure of Henry Hub Futures Prices
10/16/2009
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Futures Prices
7.5
7
6.5
6
5.5
5
4.5
4
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Applications: Energy Risk Modeling
2010/1/1
2009/12/1
2009/11/1
2009/10/1
2009/9/1
2009/8/1
2009/7/1
2009/6/1
2009/5/1
2009/4/1
2009/3/1
2009/2/1
2009/1/1
2008/12/1
2008/11/1
2008/10/1
2008/9/1
2008/8/1
2008/7/1
2008/6/1
2008/5/1
2008/4/1
2008/3/1
2008/2/1
2008/1/1
2007/12/1
2007/11/1
2007/10/1
2007/9/1
2007/8/1
2007/7/1
2007/6/1
2007/5/1
2007/4/1
2007/3/1
2007/2/1
2007/1/1
Natural Gas Futures Term Structure
Movements
23
13
1/3/2006
12
1/4/2006
1/5/2006
11
1/6/2006
1/9/2006
10
1/10/2006
1/11/2006
9
1/12/2006
1/13/2006
1/17/2006
8
1/18/2006
1/19/2006
7
1/20/2006
1/23/2006
1/24/2006
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Arbitrage-free Natural Gas Model
Valuation Component of the Model
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 Contingent claims on the NG spot and futures prices
 Implied cost to carry: The term structure
 Futures contracts determining the implied cost of carry
 The one period cost to carry is equivalent to the survival rate
 Dynamics of the term structure: The term structure of
cost to carry and the spot rates


dS = c(t) S dt + σ(t) S dz
dc(t) = F( c(t, T), t) dt + σ* dw
Applications: Energy Risk Modeling
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Preliminary Results on Dynamic Hedging
Application Component of the Model
25
 The futures term structure movements have two
factors: “Price” (85%), “Cost to Carry” (14%), 3rd vector
(0.3%)
 The cost of carry movements has one factor: level
movement (99.5%) and 2nd factor (0.4%)
 Correlation of the spot price and cost to carry: low
 The use of key rate durations on the cost to carry and
the spot price for hedging
Applications: Energy Risk Modeling
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Applications: Energy Risk Modeling
2010/01/01
2009/12/01
2009/11/01
2009/10/01
2009/09/01
2009/08/01
2009/07/01
2009/06/01
2009/05/01
2009/04/01
2009/03/01
2009/02/01
2009/01/01
2008/12/01
2008/11/01
2008/10/01
2008/09/01
2008/08/01
2008/07/01
2008/06/01
2008/05/01
2008/04/01
2008/03/01
2008/02/01
2008/01/01
2007/12/01
2007/11/01
2007/10/01
2007/09/01
2007/08/01
2007/07/01
2007/06/01
2007/05/01
2007/04/01
2007/03/01
2007/02/01
2007/01/01
Rate Shift
1st Principal Movement of the Cost to Carry
26
Price
-0.157
-0.158
-0.159
-0.16
-0.161
-0.162
-0.163
-0.164
-0.165
-0.166
Maturity
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Applications: Energy Risk Modeling
2010/01/01
2009/12/01
2009/11/01
2009/10/01
2009/09/01
2009/08/01
2009/07/01
2009/06/01
2009/05/01
2009/04/01
2009/03/01
2009/02/01
2009/01/01
2008/12/01
2008/11/01
2008/10/01
2008/09/01
2008/08/01
2008/07/01
2008/06/01
2008/05/01
2008/04/01
2008/03/01
2008/02/01
2008/01/01
2007/12/01
2007/11/01
2007/10/01
2007/09/01
2007/08/01
2007/07/01
2007/06/01
2007/05/01
2007/04/01
2007/03/01
2007/02/01
2007/01/01
Rate Shift
2nd Principal Movement of the Cost to Carry
27
Slope
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
Maturity
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Calibrate the Model to the Implied Volatilities
Application component of the Model
28
60
2006-01-03
2006-01-04
50
2006-01-05
2006-01-06
2006-01-09
40
2006-01-10
2006-01-11
30
2006-01-12
2006-01-13
2006-01-17
20
2006-01-18
2006-01-19
10
2006-01-20
2006-01-23
0
2006-01-24
1
2
3
4
5
6
7
Applications: Energy Risk Modeling
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
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Implications to Energy Trading
29
 Quantitative analysis of NG contracts and the changing
shape of the implied cost to carry curve
 Relate the NG futures option prices and to other
derivatives
 Applications to the calendar basis trades

Use of the multi-factor model to determine the correlations of the
cycles
 Correlation of interest rates with the cost to carry curve
 Power stack function: relation to coal and crude oil
Applications: Natural Gas Contracts
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Going Forward
30
 A lesson learnt from the financial crisis
 Mispricing and hence misallocation of resources
 Justification for the dynamic replication and volatility
calibration


Revisiting the application component of an arbitrage-free model
Internal consistency of the model
Manage Model Risks
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Managing Model Risk
31
 Example of mis-valuation: The CDO copula model
 The use of implied correlations
 How to manage model risk?
Managing Model Risks
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A Solution to Managing Model Risk (Deming)
32
 Deming: Statistical approach to quality control
 Defects are often traced directly to the cause. But…
 Statistical approach is more objective
 Catching the defects when they are small
 Return attributions of Treasury futures
 Cheapest to deliver
 Delivery options, timing options, end of month options
Managing Model Risks
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Return Attribution and Risk Management
33
Data source: BGCantor Market Data
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Analysis of the Model Risk
34
 Explanatory power of the model?
 Mean reversion behavior of the residuals?
 Effectiveness of the dynamic replication?
 Detect “defects” in the time series in relation to events
Manage Model Risks
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Analysis of the Residuals over the Roll Month
35
 September and December 5 year futures over the
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

month of September
Market price = Mid quotes
10 minute intervals from 7:00am till 5:30 pm
Both the explanatory power and mean reversion rate
decline by mid month
This behavior varies across the contracts
Manage Model Risks
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Data source: BGCantor Market Data
10/16/2009
10/15/2009
10/14/2009
10/13/2009
10/9/2009
10/7/2009
9/30/2009
9/29/2009
9/28/2009
9/25/2009
9/24/2009
9/23/2009
9/22/2009
9/21/2009
9/18/2009
9/17/2009
9/16/2009
9/15/2009
9/14/2009
5yr-Sep09
9/10/2009
9/9/2009
9/8/2009
9/4/2009
9/3/2009
9/2/2009
9/1/2009
8/31/2009
8/25/2009
8/24/2009
8/21/2009
8/20/2009
8/18/2009
8/13/2009
8/12/2009
8/11/2009
8/10/2009
8/7/2009
8/6/2009
8/5/2009
8/4/2009
Explanatory Power Metric : R squared
p(market)= a + b p(model) + e
36
5yr-Dec09
120.00%
100.00%
80.00%
60.00%
40.00%
20.00%
0.00%
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Data source: BGCantor Market Data
10/16/2009
10/15/2009
10/14/2009
10/13/2009
10/9/2009
10/7/2009
9/30/2009
9/29/2009
9/28/2009
9/25/2009
9/24/2009
9/23/2009
9/22/2009
9/21/2009
9/18/2009
9/17/2009
9/16/2009
9/15/2009
9/14/2009
10yr-Sep09
9/10/2009
9/9/2009
9/8/2009
9/4/2009
9/3/2009
9/2/2009
9/1/2009
8/31/2009
8/25/2009
8/24/2009
8/21/2009
8/20/2009
8/18/2009
8/13/2009
8/12/2009
8/11/2009
8/10/2009
8/7/2009
8/6/2009
8/5/2009
8/4/2009
Explanatory Power Metric : R squared
p(market)= a + b p(model) + e
37
10yr-Dec09
120.00%
100.00%
80.00%
60.00%
40.00%
20.00%
0.00%
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Data source: BGCantor Market Data
10/16/2009
10/15/2009
10/14/2009
10/13/2009
10/9/2009
10/7/2009
9/30/2009
9/29/2009
9/28/2009
9/25/2009
9/24/2009
9/23/2009
9/22/2009
9/21/2009
9/18/2009
9/17/2009
9/16/2009
9/15/2009
9/14/2009
5yr-Sep09
9/10/2009
9/9/2009
9/8/2009
9/4/2009
9/3/2009
9/2/2009
9/1/2009
8/31/2009
8/25/2009
8/24/2009
8/21/2009
8/20/2009
8/18/2009
8/13/2009
8/12/2009
8/11/2009
8/10/2009
8/7/2009
8/6/2009
8/5/2009
8/4/2009
Mean Reversion Metric: ( 1 -b )
ch/rh(n) = a + b ch/rh(n-1) + e
38
5yr-Dec09
120.00%
100.00%
80.00%
60.00%
40.00%
20.00%
0.00%
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Data source: BGCantor Market Data
10/16/2009
10/15/2009
10/14/2009
10/13/2009
10/9/2009
10/7/2009
9/30/2009
9/29/2009
9/28/2009
9/25/2009
9/24/2009
9/23/2009
9/22/2009
9/21/2009
9/18/2009
9/17/2009
9/16/2009
9/15/2009
9/14/2009
10yr-Sep09
9/10/2009
9/9/2009
9/8/2009
9/4/2009
9/3/2009
9/2/2009
9/1/2009
8/31/2009
8/25/2009
8/24/2009
8/21/2009
8/20/2009
8/18/2009
8/13/2009
8/12/2009
8/11/2009
8/10/2009
8/7/2009
8/6/2009
8/5/2009
8/4/2009
Mean Reversion Metric: ( 1 -b )
ch/rh(n) = a + b ch/rh(n-1) + e
39
10yr-Dec09
120.00%
100.00%
80.00%
60.00%
40.00%
20.00%
0.00%
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Replication Metric : R squared
p(model) = a + b TSY returns + e
40
Date
5yr-Dec09
10yr-Dec09
10yr-Mar10
30yr-Dec09
30yr-Mar10
10/7/2009
0.9099
0.9908
0.9612
0.9724
0.7282
10/9/2009
0.8300
0.9894
0.9767
0.9580
0.9755
10/13/2009
0.9009
0.9926
0.9812
0.9864
0.9893
10/14/2009
0.7084
0.8854
0.7809
0.7777
0.8822
10/15/2009
0.8597
0.9647
0.9376
0.8452
0.9215
10/16/2009
0.8288
0.9931
0.9700
0.7679
0.8971
Data source: BGCantor Market Data
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Summary: Proposed Perspective
41
 Term structure models deal with “rates”, flows ( interest rate,
default rate, cost to carry, inflation rate )
 Contrast to the Black-Scholes
 Valuation and application components to the model
 Similar to the Black-Scholes
 Manage model risk: Statistical approach (Deming)
 Examples of term structure
 Interest rate, credit, energy
 Natural gas shows the use of both price and rate models
Conclusions
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Conclusions and Implications
42
 Trading

Return attributions on performance measure
 Model risk

Statistical Approach: explanatory, mean reversion, replication
 Securities valuation


Interest rate, default rate, inflation rate, liquidity, cost to carry
Hybrid models
 Identifies the price formation process


From the basic valuation building blocks to exotic structures
Regulatory policy on market transparency and the role of exchanges
Conclusions
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References
43
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Amin, Kaushik I., and Andrew J. Morton, 1994, “ Implied Volatility Functions in Arbitrage-free Term Structure Models, “ Journal of
Financial Economics, 35 (2), 141-180
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