C redit V alue A djustment by Expected Future Exposure M ethod M u - - PowerPoint PPT Presentation
C redit V alue A djustment by Expected Future Exposure M ethod M u M . Liu, CQF Agenda What is CVA and why it must be considered? Valuation M ethodology Current Exposure ( CE ) Expected Future Exposure (EFE)
by Expected Future Exposure M ethod
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Current Exposure (“ CE” )
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Expected Future Exposure (“EFE”)
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Interest rate simulation model
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Probability of default model
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Expected Positive Exposure (EPE)
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Expected Negative Exposure (ENE)
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Swaps (interest rate, currency, equity, commodity)
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Other OTC private contracts
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A Typical Exposure Profile for Interest Rate Swaps
exposure.
where 1 is the indicator function that can be interpreted as the risk-neutral probability of default when computed within an expectation; R is the recovery rate; is the discount factor; is the exposure at time . The exposure calculation varies depending on which method is used.
Probability of Default M odel Interest Rate M odel; PCA Spectral Analysis Spot rate models, HJM model, LIBOR market model, etc
{ }
1 (1 )
Q T
B CVA R E B
τ τ τ ≤
= −
0 (1
) ( , ) ( ) B CVA R PD t t t E t dt B
τ
δ
∞
= − +
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Things we need for CVA calculation: Principal Component Analysis
Spectral Decomposition (Attribution Analysis)
An Interest Rate M odel
Heath-Jarrow-M orton (HJM )
A PD Bootstrapping Algorithm
Iterative algo
A Programming Language
M atlab 6
Forward curve as of M ay 11, 2016. There are more than 30 market instruments that constitute the forward curve. Principal Component Analysis helps us answer the question: which are the most influential tenors?
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PCA is an approach to determine the dominant and independent factors of the evolution of a multivariate system as a whole. 1. Decorrelation: PCA factors are independent and their covariance is zero. 2. Reduced dimensionality 3. Data structure reveal As mentioned before, HJM model is usually implemented with multiple factors. The shape of a yield curve is very complex - in fact, there are many schools of thought about the relationships among the tenors on the yield curve. For instance, does the short-term tenor affect the shape of the long end of the yield curve, or vise versa? The yield curve can move in many different ways, such as parallel shift, concave and convex at different tenors. In determining which tenors have the most significant impact on the shape of the yield curve, we use Principle Component Analysis (PCA) to analyze the relationships among all the tenors based on the forward
analysis. It turns out that the dimensions of the HJM model are determined by the number of factors. It would be the modeler’s choice as to how many factors to model, but typically three factors would be sufficient to model the dynamics of a yield curve. Empirically, the first factor explains 70% of the yield curve variation; first two factors would explain a total of 87% of the variation; and the first three factors would explain about 95% - 97% of the total variation. Thus, modeling more than five factors would be unnecessary.
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The dynamics of yield curve movements are complex. Principal component analysis helps us to identify the key factors that drive the movements. PCA provides independent systematic factors that attribute to the overall movement of the yield curve. To get the components (a.k.s factors), we decompose the covariance matrix obtained from forward rates on all the tenors. A is a diagonal matrix with eigenvalues ranked in a descending order
respect to the orthogonal direction.
' V V = Α
(1) (2) ( )
, ...,
n
V e e e =
1 2
...
n
λ λ λ > >
1
...
N
λ λ Α =
(1) (2) (3) ( ) 1 2 3
...
n t t t n
f f e X e X e X e X
+
= + + + + +
V
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(mathematical definition)
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The volatility structure for HJ M model is implemented based on this relationship.
i i i
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Things we need for CVA calculation: Principal Component Analysis
Spectral Decomposition (Attribution Analysis)
An Interest Rate M odel
Heath-Jarrow-M orton (HJM )
A PD Bootstrapping Algorithm
Iterative algo 12
conjunction with the volatility structure. In other words, drift rate will be automatically determined when a volatility structure is specified. Challenges:
1 1
( , ) ( , ) ( , ) ( , ) ( , )
k k i i i i i
f t df t v t v t s ds dt v t dX
τ
τ τ τ τ τ
= =
∂ = + + ∂
( ; ) ( ; ) ( , ) ( ) df t T t T dt t T dW t α σ = + ( ; ) t T α
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M arket price of risk is, in fact, a stochastic process itself. Source: The M arket Price of Interest-rate Risk: M easuring and M odelling Fear and Greed in the Fixed-income M arkets. Riaz Ahmad and Paul
Wilmott.
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Differentiating our forward rate equation (1), we can obtain the following SDE: Under the risk-neutral measure, the SDEbecomes: Now applying the chain rule on the drift term, the HJM stochastic differential equation becomes: The term r(t) vanished because we’re differentiating with respect to T . ( ; ) log ( ; ) f t T Z t T T ∂ = − ∂
( ; )
( ; )
T t f t s ds
Z t T e
−∫
=
2
( , ) log ( ; ) 1 ( ) ( , ) ( , ) ( ) 2 df t T Z t T T r t t T dt t T dW t T T σ σ ∂ = − ∂ ∂ ∂ = − − − ∂ ∂
2
1 ( , ) ( , ) ( ) ( , ) ( ) 2
Q
df t T t T r t dt t T dW t T T σ σ ∂ ∂ = − − ∂ ∂ ( , ) ( , ) ( , ) ( , ) ( )
Q
df t T t T t T dt t T dW t T T σ σ σ ∂ ∂ = − ∂ ∂
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Let’s introduce a substitution Integrating both sides: Now our HJM SDE becomes: Or, Drift rate structure One of the unique features about HJM model is its non-M arkov feature. The drift rate in HJ M model is highly path-dependent. In a M arkov chain, only the present state of a variable determines the possible future states. M arkov process is a stochastic process without memory. ( , ) ( , ) v t T t T T σ ∂ = − ∂ ( , ) ( , )
T t
t T v t s ds σ = −∫ ( , ) ( , ) ( , ) ( , ) ( )
Q
df t T t T v t T dt v t T dW t σ = − + ( , ) ( , ) ( , ) ( , ) ( )
T Q t
df t T v t T v t s ds v t T dW t = +
∫
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Let be the forward curve as of today, then the spot rate is SDE of the forward rate
Differentiate r(t) with respect with t: Take a closer look at the drift rate in the SDEfor the spot rate process above, we will find that the drift rate is contingent upon the volatility from t=0 to a future time t as well as the dX. Also, this expression is under the real expectation. We need M PR adjustment in order to derive the risk- neutral version.
( , ) f t T
( ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( )
t t t t t t
r t f t t df s t ds f t t s t s t s t ds s T dX s t t t σ σ µ σ = + ∂ ∂ ∂ = + − − ∂ ∂ ∂
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2 2 2 2 2 2 2
( , ) ( , ) ( , ) ( , ) ( ) ( , ) ( , ) ( , ) ( ) ( , )
t t s t t t s t
s t s t s t s t dr t f t t t s s t ds dX s t t t t t t t s s dt dX σ σ µ σ µ σ σ
= =
∂ ∂ ∂ ∂ ∂ ∂ = − + + − − ∂ ∂ ∂ ∂ ∂ ∂ ∂ − ∂
Biggest Advantage: We don’t have to worry about M arket Price of Interest Rate Risk!
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Graph generated using M atLab.
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Things we need for CVA calculation: Principal Component Analysis
Spectral Decomposition (Attribution Analysis)
An Interest Rate M odel
Heath-Jarrow-M orton (HJM )
A PD Bootstrapping Algorithm
Iterative algo 21
Homogenous Poisson Process is generally used to model default events. The probability of default is characterized as follows: A default event is considered as the first jump of a Poisson process with intensity (a.k.a hazard rate). Hazard rate is closely connected to the likelihood of default (probability of default) and is an exogenous variable that can be calibrated from market credit spreads.
For each entity, a term structure of hazard rates was determined based on cumulative survival probability which was inferred from the input credit spreads. The relationship between survival probability and hazard rate is as follows: Or,
( ) 1 ( | ) ( ) 1 1 ( )
t h t n
h O h m P N n m N O h m h O h m λ λ
+ =
+ = = + = > − + =
λ
( | ) ( ) ( | ) lim .... log ( )
h
P t t h t h O h P t t h t h d S t dt τ τ λ τ τ λ
→
< ≤ + > = + < ≤ + > = = = −
( )
( )
t
s ds
S t e
λ −∫
=
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forward rates in fixed income analysis, is calculated based on . Below is the recursive algorithm that we implemented in M atlab for the bootstrapping procedure which requires an iterative solution
For the subsequent periods, where D is the discount factor and LGD is the loss given default. ( )
n
P T
1
( )
n
P T − 1 ( ) 1 1 1 2 (0, ){ ( ) ( )} ( ) 1 1 2 1 1 ( ) . 2 (0, )( ) 2 2 2 2 2 T LGD p T LGD t S T D T LGD LGD t S p T p T LGD p T D T LGD t S LGD t S = = +∆ = − +∆ = + +∆ +∆
1 1 1 1
(0, ){ ( ) ( ) ( )} ( ) ( ) , (0, )( )
N n n n N n T n N N n N N N
n N D T LGD p T LGD t S p T p T LGD p T D T LGD t S LGD t S
− − − =
= × − + ∆ = + + ∆ + ∆
∑
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Once we have those building blocks in place, we can perform a CVA calculation by the following steps: Step 1: Aggregate OTC derivatives by counterparty; Step 2: Perform yield curve attrition analysis; Step 3: Use HJM model to generate forward rates commensurate with the swap payment dates; Step 4: Use bootstrapping algorithm to calculate risk-neutral probability of default for each party by using the appropriate credit spread; Step 5: For each forward curve realization, calculate the aggregate net payment at each payment date. Step 6: Repeat Step 3 ~5 or many times and calculate the present value of discounted credit loss for each payment period.
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1. US and International Accounting standards suggest (require) the fair value of an Over-the-Counter (“OTC” ) derivative should reflect the credit quality of the derivative instrument. 2. The method presented barely scratches the surface of credit derivative valuation. 3. There are many other more advanced methods available, such as the S waption Approach. 4. Regardless of which method to use for CVA valuation, it must consider the modeling of counterparty default risk.
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