Filtration Shrinkage and Credit Risk Second Princeton Credit Risk - PowerPoint PPT Presentation

Filtration Shrinkage and Credit Risk Second Princeton Credit Risk Conference, May 2008 Philip Protter, Cornell University May 23, 2008

Credit Risk Credit risk investigates an entity (corporation, bank, individual) that borrows funds, promises to return them in a specified contractual manner, and who may not do so (default) Mathematical framework: (Ω , G , P , G ) are given, G = ( G t ) t ≥ 0 , 0 ≤ t ≤ T , usual hypotheses. For sake of this talk, let us consider a firm. The asset value process is dA t = A t α ( t , A t ) dt + A t σ ( t , A t ) dW t where α and σ are such that A exists, is well defined, and positive. Assume the liability structure of the firm is a single zero-coupon bond with maturity T and face value 1, and default occurs only at time T , and only if A T ≤ 1.

• The probability of default is P ( A T ≤ 1).

• The probability of default is P ( A T ≤ 1). • Time zero value of the firm’s debt is � T v (0 , T ) = E Q (( A T ∧ 1 exp( − r s ds )) . 0 This is the Black-Scholes-Merton model (from the early 1970s) viewed as a European call option on the firm’s assets, maturity T , and strike price equal to the value of the debt.

• The probability of default is P ( A T ≤ 1). • Time zero value of the firm’s debt is � T v (0 , T ) = E Q (( A T ∧ 1 exp( − r s ds )) . 0 This is the Black-Scholes-Merton model (from the early 1970s) viewed as a European call option on the firm’s assets, maturity T , and strike price equal to the value of the debt. • The Black-Scholes-Merton model has been extended to default before time T by considering a barrier L = ( L t ) t ≤ 0 . Augment the information to include the L information. Let H t = σ ( A s , L s ; s ≤ t ).

• Default time becomes a first passage time relative to the barrier L : τ = inf { t > 0 : A t ≤ L t } .

• Default time becomes a first passage time relative to the barrier L : τ = inf { t > 0 : A t ≤ L t } . • The value of the firm’s debt is � T �� � � v (0 , T ) = E 1 { t ≤ T } L τ + 1 { τ> T } 1 exp( − r s ds ) . 0

• Default time becomes a first passage time relative to the barrier L : τ = inf { t > 0 : A t ≤ L t } . • The value of the firm’s debt is � T �� � � v (0 , T ) = E 1 { t ≤ T } L τ + 1 { τ> T } 1 exp( − r s ds ) . 0 • The previous models are known as the structural approach to credit risk. The default time is predictable.

• Alternative: the reduced form approach of Jarrow–Turnbull, and Duffie–Singleton, of the 1990s. The observer sees only the filtration generated by the default time τ and a vector of state variables X t . F t = σ ( τ ∧ s , X s ; s ≤ t ) ⊂ G t

• Alternative: the reduced form approach of Jarrow–Turnbull, and Duffie–Singleton, of the 1990s. The observer sees only the filtration generated by the default time τ and a vector of state variables X t . F t = σ ( τ ∧ s , X s ; s ≤ t ) ⊂ G t • Assume a trading economy with a risky firm with outstanding debt as zero coupon bonds Assuming no arbitrage (but not completeness), there is an equivalent local martingale measure Q .

• Alternative: the reduced form approach of Jarrow–Turnbull, and Duffie–Singleton, of the 1990s. The observer sees only the filtration generated by the default time τ and a vector of state variables X t . F t = σ ( τ ∧ s , X s ; s ≤ t ) ⊂ G t • Assume a trading economy with a risky firm with outstanding debt as zero coupon bonds Assuming no arbitrage (but not completeness), there is an equivalent local martingale measure Q . • Let � t M t = 1 { t ≥ τ } − λ s ds = a Q F − martingale . 0 Recovery rate given by ( δ t ) 0 ≤ t ≤ T ; So change F : F t = σ ( τ ∧ s , X s , δ s ; s ≤ t ) ⊂ G t F X t = σ ( X s ; s ≤ t ).

Default prior to time T : � � E Q ( N T = 1 |F X ) Q ( τ ≤ T ) = E Q � T = E Q (exp( − λ s ds )) , and value of the firm’s debt is 0 � T � � v (0 , T ) = E [1 { τ ≤ T } δ τ + 1 { τ> T } 1] exp( − r s ds ) 0 The modeler does not see the process A = ( A t ) t ≥ 0 , but has instead only partial information. How does one model this partial information?

There are three main approaches, in general: 1. Duffie-Lando, Kusuoka: Observe A only at discrete intervals, and add independent noise 2. With Kusuoka, a twist is given by introduction of filtration expansion 3. Giesecke-Goldberg: The default barrier is a random curve; but A is still assumed to observed continuously 4. C ¸etin-Jarrow-Protter-Yildirim: Begin with a structural model under G , and then project onto smaller filtration F ; Use of cash flows.

• We are interested in the filtration shrinkage approach

• We are interested in the filtration shrinkage approach • Following C ¸etin-Jarrow-Protter-Yildiray, consider the cash balance of the firm.

• We are interested in the filtration shrinkage approach • Following C ¸etin-Jarrow-Protter-Yildiray, consider the cash balance of the firm. • X denotes the cash balance of the firm, normalized by the money market account dX t = σ dW t , X 0 = x , where x > 0 , σ > 0 Z = { t ∈ [0 , T ] : X t = 0 } g t = sup { s ≤ t : X s = 0; g t is the last time before t cash balance is zero τ α = inf { t > 0 : t − g t ≥ α 2 2 : X s < 0 , all s ∈ ( g t − , t ) } ; τ α represents the time of potential default τ is the time of default;

• We are interested in the filtration shrinkage approach • Following C ¸etin-Jarrow-Protter-Yildiray, consider the cash balance of the firm. • X denotes the cash balance of the firm, normalized by the money market account dX t = σ dW t , X 0 = x , where x > 0 , σ > 0 Z = { t ∈ [0 , T ] : X t = 0 } g t = sup { s ≤ t : X s = 0; g t is the last time before t cash balance is zero τ α = inf { t > 0 : t − g t ≥ α 2 2 : X s < 0 , all s ∈ ( g t − , t ) } ; τ α represents the time of potential default τ is the time of default; • Assume τ = inf { t > τ α : X t = 2 X τ α }

• We are interested in the filtration shrinkage approach • Following C ¸etin-Jarrow-Protter-Yildiray, consider the cash balance of the firm. • X denotes the cash balance of the firm, normalized by the money market account dX t = σ dW t , X 0 = x , where x > 0 , σ > 0 Z = { t ∈ [0 , T ] : X t = 0 } g t = sup { s ≤ t : X s = 0; g t is the last time before t cash balance is zero τ α = inf { t > 0 : t − g t ≥ α 2 2 : X s < 0 , all s ∈ ( g t − , t ) } ; τ α represents the time of potential default τ is the time of default; • Assume τ = inf { t > τ α : X t = 2 X τ α } • But, the investor does not see the entire cash balance process.

• In C ¸JPY the investor sees only when the balances are positive or negative, and whether or not the cash balances are above or below the default threshold.

• In C ¸JPY the investor sees only when the balances are positive or negative, and whether or not the cash balances are above or below the default threshold. • Default threshold: 2 X τ α

• In C ¸JPY the investor sees only when the balances are positive or negative, and whether or not the cash balances are above or below the default threshold. • Default threshold: 2 X τ α � X t for t < τ α • Y t = 2 X τ α − X t for t ≥ τ α Y defined this way is an F Brownian motion

• In C ¸JPY the investor sees only when the balances are positive or negative, and whether or not the cash balances are above or below the default threshold. • Default threshold: 2 X τ α � X t for t < τ α • Y t = 2 X τ α − X t for t ≥ τ α Y defined this way is an F Brownian motion � 1 if x > 0 • τ = inf { t ≥ τ α : Y t = 0 } . sign( x ) = − 1 if x ≤ 0

• In C ¸JPY the investor sees only when the balances are positive or negative, and whether or not the cash balances are above or below the default threshold. • Default threshold: 2 X τ α � X t for t < τ α • Y t = 2 X τ α − X t for t ≥ τ α Y defined this way is an F Brownian motion � 1 if x > 0 • τ = inf { t ≥ τ α : Y t = 0 } . sign( x ) = − 1 if x ≤ 0 • G is the filtration of sign( Y t )

• N t = 1 { t ≥ τ } with G compensator A

• N t = 1 { t ≥ τ } with G compensator A •

• N t = 1 { t ≥ τ } with G compensator A • Theorem � t ∧ τ 1 A t = λ s ds, and moreover λ t = 1 { t >τ g a } g t − ) , 0 ≤ t ≤ τ , 0 2( t − ˜ and where ˜ g t = sup { s ≤ t : Y s = 0 } .

• N t = 1 { t ≥ τ } with G compensator A • Theorem � t ∧ τ 1 A t = λ s ds, and moreover λ t = 1 { t >τ g a } g t − ) , 0 ≤ t ≤ τ , 0 2( t − ˜ and where ˜ g t = sup { s ≤ t : Y s = 0 } . • Nota Bene: We are able to calculate λ explicitly since we have a formula for the Az´ ema martingale.

• N t = 1 { t ≥ τ } with G compensator A • Theorem � t ∧ τ 1 A t = λ s ds, and moreover λ t = 1 { t >τ g a } g t − ) , 0 ≤ t ≤ τ , 0 2( t − ˜ and where ˜ g t = sup { s ≤ t : Y s = 0 } . • Nota Bene: We are able to calculate λ explicitly since we have a formula for the Az´ ema martingale. • Use knowledge of λ to calculate quantities of interest. Simple example: price of a risky zero coupon bond at time 0: √ � T � � �� α/ 2 S 0 = exp( − r u du ) 1 − Q ( τ α ≤ T ) − E ( 1 { τ α ≤ T } ) � T − ˜ g τ α 0