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![Page 1: Pricing and Evaluating a Bond Portfolio Using a Regime-Switching Markov Model Leela Mitra* Rogemar Mamon Gautam Mitra Centre for the Analysis of Risk and.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551ae71d5503465e7d8b4a57/html5/thumbnails/1.jpg)
Pricing and Evaluating a Bond Portfolio Using a Regime-Switching Markov Model
Leela Mitra* Rogemar Mamon Gautam Mitra
Centre for the Analysis of Risk and Optimisation Modelling
*sponsored by EPSRC and OptiRisk
![Page 2: Pricing and Evaluating a Bond Portfolio Using a Regime-Switching Markov Model Leela Mitra* Rogemar Mamon Gautam Mitra Centre for the Analysis of Risk and.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551ae71d5503465e7d8b4a57/html5/thumbnails/2.jpg)
Outline
1. Motivation & insights
2. Problem setting
3. Novel features
4. Asset model
5. Computational study
![Page 3: Pricing and Evaluating a Bond Portfolio Using a Regime-Switching Markov Model Leela Mitra* Rogemar Mamon Gautam Mitra Centre for the Analysis of Risk and.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551ae71d5503465e7d8b4a57/html5/thumbnails/3.jpg)
1. Motivation & Insights
• Credit risk – risk that a counterparty’s creditworthiness changes, in particular that of default on financial obligations.
• An important consideration for investors.
• An important distinctionRogers (1999) notes “the first and most important thing to realise about modelling credit risk is that we may be trying to answer questions of two different types”;Pricing credit risky assets and quantifying their risk exposure.
• Appropriate integration of significant risks.- Market risk in particular, interest rate risk.- Credit risk.
![Page 4: Pricing and Evaluating a Bond Portfolio Using a Regime-Switching Markov Model Leela Mitra* Rogemar Mamon Gautam Mitra Centre for the Analysis of Risk and.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551ae71d5503465e7d8b4a57/html5/thumbnails/4.jpg)
1. Motivation & Insights
• Two main categories credit risk model
- Structural models. Merton (1974), Option theoretic, Debt contingent claim on firm’s assets.
- Reduced form models. Jump processes.
![Page 5: Pricing and Evaluating a Bond Portfolio Using a Regime-Switching Markov Model Leela Mitra* Rogemar Mamon Gautam Mitra Centre for the Analysis of Risk and.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551ae71d5503465e7d8b4a57/html5/thumbnails/5.jpg)
2. Problem Setting - Jarrow, Lando & Turnbull (1997) (JLT)
• Credit migration process described by Markov chain.
• Fractional recovery on default; η of par value.
• Interest rate process.- Any process can be used to represent the default-free rate.
• Connecting risk neutral measure, P with physical world measure, leading to…
• Arbitrage-free pricing.- Prices of defaultable bonds are the expected values under P, discounted at the default-free rate.
![Page 6: Pricing and Evaluating a Bond Portfolio Using a Regime-Switching Markov Model Leela Mitra* Rogemar Mamon Gautam Mitra Centre for the Analysis of Risk and.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551ae71d5503465e7d8b4a57/html5/thumbnails/6.jpg)
2. Problem Setting - Thomas, Allen &
Morkel-Kingsbury (2002)
• Adapt JLT framework.
• Markov chain which represents the economy; “Regime-switching model”.
• Price defaultable bonds.
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3. Problem Setting – Our Work
• Pricing.
- Price a portfolio of defaultable bonds.
- Similar model to Thomas et al.
• Risk quantification.
- Extend the model to the physical measure to simulate the bond portfolio value.
- Calculate Value at Risk (VaR) and Conditional Value at Risk (CVaR) one year ahead.
![Page 8: Pricing and Evaluating a Bond Portfolio Using a Regime-Switching Markov Model Leela Mitra* Rogemar Mamon Gautam Mitra Centre for the Analysis of Risk and.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551ae71d5503465e7d8b4a57/html5/thumbnails/8.jpg)
3. Novel Features
• Yield Curve Modelling; no arbitrage across interest rates.
• Bond stripping: Discovering of yield and credit spreads using quadratic programming (QP1),
- constraints remove price anomalies.
• Calibration of the risk neutral credit migration process using quadratic programming (QP2),
- constraints remove negative probabilities.
![Page 9: Pricing and Evaluating a Bond Portfolio Using a Regime-Switching Markov Model Leela Mitra* Rogemar Mamon Gautam Mitra Centre for the Analysis of Risk and.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551ae71d5503465e7d8b4a57/html5/thumbnails/9.jpg)
4. Asset Model
EconomyProcess
CtE
Rating Migration
Process CtR
Interest Rate
Process CtI
0 - Treasury1 – Highest rated bondsM-1 Worst rated bonds
M Default - absorbing state.
![Page 10: Pricing and Evaluating a Bond Portfolio Using a Regime-Switching Markov Model Leela Mitra* Rogemar Mamon Gautam Mitra Centre for the Analysis of Risk and.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551ae71d5503465e7d8b4a57/html5/thumbnails/10.jpg)
4. Asset Models – Interest Rate Process
• Thomas et al. model spot rate without restrictions, on process for arbitrage.
• Whole yield curve should be modelled,
– Heath, Jarrow & Morton (1992).
• Interest Rates are functions of two time dimensions,- current time t,
- time to maturity T.
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4. Asset Models – Interest Rate Process
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4. Asset Models – Interest Rate Process• Model an auxiliary binomial process to describe
underlying interest rate process.
Figure 1: Possible evolution of state space tree over t = {0,1,2,3}
S0 ={0} S1 ={u, d} S2 ={uu, ud, du, dd} S3 ={uuu, …}
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4. Asset Models – Interest Rate Process• Forward Rate Process
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4. Asset Models – Interest Rate Process
• Treasury zero coupon bond prices process and forward rate process are equivalent.
• Zero coupon bond price process
• Absence of arbitrage can be understood as
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4. Asset Models – Interest Rate Process
• Continuously compounding forward rate
• Continuous time process described by SDE
•Defining,
the discrete process converges to the continuous.
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4. Asset Models – Interest Rate Process
• Assuming
• No arbitrage condition becomes
• Drift of process under the risk neutral measure is well defined, given the observed volatility.
![Page 17: Pricing and Evaluating a Bond Portfolio Using a Regime-Switching Markov Model Leela Mitra* Rogemar Mamon Gautam Mitra Centre for the Analysis of Risk and.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551ae71d5503465e7d8b4a57/html5/thumbnails/17.jpg)
4. Asset Models – Risky bond price
• The price of a credit risky bond is, discounted expected value under the risk neutral measure,
where,
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5. Computational Study
• Part 1 – Bond stripping; Discovering of yield and credit spreads.
• Part 2 – Model calibration.
• Part 3 – Simulation of bond portfolio value so determining its distribution.
![Page 19: Pricing and Evaluating a Bond Portfolio Using a Regime-Switching Markov Model Leela Mitra* Rogemar Mamon Gautam Mitra Centre for the Analysis of Risk and.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551ae71d5503465e7d8b4a57/html5/thumbnails/19.jpg)
5. Computational Study Bond Stripping - Motivation
• Bonds priced using zero coupon bond prices. - Any bond’s price is determined using zero coupon bonds prices as discount factors.- Models for term structure describe zero coupon bond prices.
• Market Data Available.- Mainly coupon bonds available in market.- Model calibration involves stripping coupons from coupon bonds to derive underlying zero coupon bond prices.
• Jarrow et al.- Bucket bonds by rating and maturity and find average prices and coupons for each bucket. - Solve system triangular equations to get zero coupon prices.- Can lead to mispricing.
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5. Computational Study Bond Stripping - Motivation
• Allen, Thomas and Zheng - Use linear programming to find zero coupon bond prices.- Minimise absolute pricing errors.- Constraints are used to remove anomalies =>mispricing.
• Our Work- Use same formulation but with quadratic programming so minimising squared pricing errors.- Equivalent to constrained regression.
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5. Computational Study Bond Stripping - Formulation
• Credit ratings {0, 1,…, M-1}
• Time set { 0,…,T}
• N coupon bonds - where bond b, 1 b N,
- current market price vb ,- rating r(b),- pays cash flow cb(t) at time t.
• Zero coupon bond prices- zk(t) price of a zero coupon bond, with rating k, which pays 1 at time t.
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5. Computational Study Bond Stripping - Formulation
for bond blet ob be the pricing error ‘over’
let ub be the pricing error ‘under’
Minimise b (ob +ub ) 2 (1)
subject to vb + ob = t cb(t) zr(b)(t) + ub b {1,…,N}
(2)
z 0(t) z 0(t+1) (1+ m(t)) t {0,…,T} (3)
z k(t+1) - z k+1(t+1) z k(t) - z k+1(t) t {0,…,T}
k {0,…,M-1} (4)m(t) – minimum discount rate over (t,t+1].
![Page 23: Pricing and Evaluating a Bond Portfolio Using a Regime-Switching Markov Model Leela Mitra* Rogemar Mamon Gautam Mitra Centre for the Analysis of Risk and.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551ae71d5503465e7d8b4a57/html5/thumbnails/23.jpg)
5. Computational Study Bond StrippingZero Coupon Bond Prices against Time to Maturity
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
t0 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15
Maturity Date
Zero
Co
up
on
Bo
nd
Pri
ce
rat0 rat1 rat2 rat3 rat4 rat5 rat6
rat7 rat8 rat9
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5. Computational Study Bond Stripping – Allowing Mispricing
• If we adapt (4) we allow some types of mispricing of zero coupon bonds.
• Minimise b (ob +ub ) 2 (1)
subject to vb + ob = t cb(t) zr(b)(t) + ub b {1,…,N} (2)
z 0(t) z 0(t+1) (1+ m(t)) t {0,…,T} k {0,…,M-1} (3)
z 0(t+1) - z k+1(t+1) z 0(t) - z k+1(t) t {0,…,T} k {0,…,M-1} (4)
![Page 25: Pricing and Evaluating a Bond Portfolio Using a Regime-Switching Markov Model Leela Mitra* Rogemar Mamon Gautam Mitra Centre for the Analysis of Risk and.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551ae71d5503465e7d8b4a57/html5/thumbnails/25.jpg)
5. Computational Study Bond Stripping – Allowing Mispricing
Zero Coupon Bond Prices against Time to Maturity
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
t0 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15
Maturity Date
Zero
Co
up
on
Bo
nd
Pri
ce
rat0 rat1 rat2 rat3 rat4 rat5 rat6
rat7 rat8 rat9
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5. Computational Study- Model Calibration
• Each chain
-Determine relevant states.
-Determine physical world and risk neutral measures.
• Economy
- Classify years as good or bad, using observed transitions to lower ratings and default.
- Physical probabilities as observed frequencies of transitions.
- Risk neutral measure assumed to be same as real world.
![Page 27: Pricing and Evaluating a Bond Portfolio Using a Regime-Switching Markov Model Leela Mitra* Rogemar Mamon Gautam Mitra Centre for the Analysis of Risk and.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551ae71d5503465e7d8b4a57/html5/thumbnails/27.jpg)
5. Computational Study - Model Calibration
• Interest Rate Process
- From observations of daily yield curve,
we derive observations of volatility. - Fit to functional form,
to give our volatility function. Vasicek model.
- Risk neutral drift is then well defined by no arbitrage condition.
![Page 28: Pricing and Evaluating a Bond Portfolio Using a Regime-Switching Markov Model Leela Mitra* Rogemar Mamon Gautam Mitra Centre for the Analysis of Risk and.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551ae71d5503465e7d8b4a57/html5/thumbnails/28.jpg)
5. Computational Study - Model Calibration
• Interest Rate Process
- The states of the processes are known, given Forward rates/ zero coupon bonds processes.
- Risk neutral probabilities are ½ by construction.
- Physical probabilities are derived from the observed drift of the yield curve.
![Page 29: Pricing and Evaluating a Bond Portfolio Using a Regime-Switching Markov Model Leela Mitra* Rogemar Mamon Gautam Mitra Centre for the Analysis of Risk and.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551ae71d5503465e7d8b4a57/html5/thumbnails/29.jpg)
5. Computational Study– Model Calibration
• Credit Rating Migrations
- States are well defined; ratings classes and default state.
- Historical Standard & Poors default frequency data for physical world measure.
- Risk neutral measure backed out from zero coupon bond prices.
![Page 30: Pricing and Evaluating a Bond Portfolio Using a Regime-Switching Markov Model Leela Mitra* Rogemar Mamon Gautam Mitra Centre for the Analysis of Risk and.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551ae71d5503465e7d8b4a57/html5/thumbnails/30.jpg)
5. Computational Study– Model Calibration
• Credit Rating Migrations
- physical world measure
- risk neutral / pricing measure
- Similar expressions for when the economy is in state B
![Page 31: Pricing and Evaluating a Bond Portfolio Using a Regime-Switching Markov Model Leela Mitra* Rogemar Mamon Gautam Mitra Centre for the Analysis of Risk and.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551ae71d5503465e7d8b4a57/html5/thumbnails/31.jpg)
5. Computational Study– Model Calibration
• Assuming the economy does not change, pricing equation is
. • So given the zero coupon bond prices we are able to derive the implied probabilities of default under the risk neutral measure,
,
that is risk neutral probabilities over (0, T] are known.
Credit Rating Migrations
![Page 32: Pricing and Evaluating a Bond Portfolio Using a Regime-Switching Markov Model Leela Mitra* Rogemar Mamon Gautam Mitra Centre for the Analysis of Risk and.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551ae71d5503465e7d8b4a57/html5/thumbnails/32.jpg)
5. Computational Study– Model Calibration
• Risk neutral probabilities over (0, T0] can be found from zero coupon bond prices for bond maturing at time T0.
• Risk neutral probabilities over (0, Tn] can be found from zero coupon bond prices for bond maturing at time Tn .
•As process is Markov, if probabilities over (0, T n -1] and probabilities over (0, T n] are known we can derive probabilities over (T n -1, T n].
•We are able to derive risk neutral probabilities over all timeperiods, recursively.
Credit Rating Migrations
![Page 33: Pricing and Evaluating a Bond Portfolio Using a Regime-Switching Markov Model Leela Mitra* Rogemar Mamon Gautam Mitra Centre for the Analysis of Risk and.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551ae71d5503465e7d8b4a57/html5/thumbnails/33.jpg)
5. Computational Study– Model Calibration
•Negative values for the probabilities possible.-Nothing in structure of recursive equations solved that ensure probabilities take sensible values.
•Solve set of recursive QPs with restrictions on values of probabilities. - Initial Credit Fit – derives probabilities over (0,T0] given zero
coupon bond prices maturing at T0 - Recursive Credit Fit – derives probabilities over (Tn-1,Tn],given zero coupon bond prices maturing at Tn and probabilities over (0,Tn-1]
•Use prices at more than one date.- Incorporates more information into risk neutral measure that is derived.
Credit Rating Migrations
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5. Computational Study– Model Calibration
•Initial Credit Fit – derives probabilities over (0,T0]
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5. Computational Study– Model Calibration
•Recursive Credit Fit – derives probabilities over (Tn-1,Tn]
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5. Computational Study- Simulation of VaR & CVaR
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5. Computational Study- Simulation of VaR & CVaR
Rating MaturityExpected
Return95% VaR
Loss95% CVaR
Loss
Treasury 7 years 4.29% 1.00% 2.77%
AA 7 years 4.52% 2.12% 4.31%
BBB 7 years 6.95% 14.16% 27.77%
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Questions ?
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5. QP vs LP Efficient FrontierAllen, Thomas and Zheng – LP Formulation
- Objective: Minimise b (ob +ub )
Efficient Frontier of absolute errors against squared errors
684 7528000
8500
9000
9500
10000
680 700 720 740 760
Absolute Errors
Sq
ua
red
Err
ors