lecture6 - CREDIT RISK: Lecture IV Lina El-Jahel 2006 1...

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CREDIT RISK: Lecture IV Lina El-Jahel 2006 1 Introduction to Jump Processes Why Poisson Processes? 1. Ultimate goal: A mathematical model of defaults that is realistic and tractable and useful for pricing and hedging. 2. Defaults are (a) sudden, usually unexpected (b) rare (hopefully :-) (c) cause large, discontinuous price changes. 3. Require from the mathematical model the same properties. 4. Furthermore: The probability of default in a short time interval is approximately proportional to the length of the interval. 2 What is a Poisson Process? N ( t ) = value of the process at time t. 1. Starts at zero: N (0) = 0 2. Integer-valued: N ( t ) = 0 , 1 , 2 ,... 3. Increasing or constant 4. Main use: marking points in time T 1 ,T 2 ,... the jump times of N 5. Here Default: time of the first jump of N , τ = T 1 6. Jump probability over small intervals proportional to that interval. 7. Proportionality factor = λ 1
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3 Discrete-time approximation: 1. divide [0 ,T ] in n intervals of equal length Δ t = T/n 2. Make the jump probability in each interval [ t i ,t i + Δ t ] proportional to Δ t : p := P [ N ( t i + Δ t ) - N ( t i ) = 1] = λ Δ t (1) 3. more exact approximation: p = 1 - e - λ Δ t 4. Let n → ∞ or Δ t 0 4 Important Properties Homogeneous Poisson process with intensity λ . Jump Probabilities over interval [ t,T ]: 1. No jump: P [ N ( T ) = N ( t )] = exp {- ( T - t ) λ } (2) 2. n jumps: P [ N ( T ) = N ( t ) + n ] = 1 n ! ( T - t ) n λ n e - ( T - t ) λ (3) 3. Inter-arrival times P [( T n +1 - T n ) tdt ] = λe - λt dt 4. Expectation (locally) E [ dN ] = λdt 5 Distribution of the Time of the first Jump T 1 time of first jump. Distribution:
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lecture6 - CREDIT RISK: Lecture IV Lina El-Jahel 2006 1...

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