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Unformatted text preview: IEOR E4731: Credit Risk and Credit Derivatives Lecture 05: Reduced form models. Notes originally written by Prof. Rama Cont Instructor: Xuedong He Spring, 2012 1 / 40 Reduced form models I Structural models have strong economic meaning but are not applicable I In practice, people use reduced form models I In reduced form models, people do not care about why defaults happen. They just model default times as random times. I Drawback of reduced form models: does not tell why defaults happen I Advantage of reduced form models: I Easy to implement I Easy to incorporate any market information I Can use the machinery of defaultfree termstructure modeling. I In view of riskneutral pricing, let us start from specifying information sets. 2 / 40 Information sets: modeled via filtrations I {H t } t ≥ : we observe default or no default only. Information at time t is: H t := σ { 1 τ>s , ≤ s ≤ t } I {F t } t ≥ : market information other than the default time (stock prices, interest rates, news,...) I {G t } t ≥ : full information, i.e., G t := F t W H t . I Example: Merton model. F t is the historical firm value up to time t . H t ⊂ F t , thus G t = F t . I Example: F t = {∅ , Ω } , i.e., no market information. In this case, G t = H t . 3 / 40 Risk Neutral Pricing I Starting point: the market value of a terminal cash flow H paid at T is given as Π t ( H ) = E Q h e R T t r ( u ) du H  G t i where Q is the risk neutral pricing measure (with e R t r ( u ) du ,t ≥ to be the denominator) I Example: for a zerocoupon zerorecovery bond that only defaults at maturity T , the payoff is H = 1 τ>T . I It is critical to compute the conditional expectation. 4 / 40 Conditional expectation I Formal definition of conditional expectation: let X be a random variable on a probability space (Ω , F , P ) and G be a sub σalgebra. Then, the conditional expectation Z := E [ X  G ] is a Gmeasurable random variable defined via E [ Z 1 A ] = E [ X 1 A ] , ∀ A ∈ G . I Example: let A 1 ,...,A n be a partition of Ω and G := { A 1 ,...,A n } , then E [ X  G ] = n X i =1 E [ X  A i ] 1 A i , where E [ X  A i ] := E [ X 1 A i ] P ( A i ) 5 / 40 Conditional expectation (cont’d) I For any B ∈ F , P [ B  G ] := E [ 1 B  G ] . I Example: if G := { A 1 ,...,A n } , then P [ B  G ] = n X i =1 P [ B  A i ] 1 A i where P ( B  A i ) := P ( B ∩ A i ) P ( A i ) I For any random variable X and Y , E [ X  Y ] := E [ X  σ ( Y )] where σ ( Y ) is the σalgebra generated by Y , i.e., the information one can obtain from Y . I Example: if Y = 1 A , then σ ( Y ) = { A,A c } . As a result, E [ X  1 A ] = E [ X  A ] 1 A + E [ X  A c ] 1 A c . 6 / 40 Compute Conditional Expection We start from the simplest information structure: no market information, i.e., F t = {∅ , Ω } . In this case, G t = H t ,t ≥ ....
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 Spring '12
 HeXuedong
 Probability theory

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