Credit Risk Modelling Additional Exercises
1. Let X ∼ Ga(α, β) be a gamma distributed random variable with density denot- ed f(x). Compute the exponential tilting density gt(x) = exp(tx)f(x)/MX(t) where MX (t) is the moment generating function.
Are there any constraints on the value of t? Can the mean of X be shifted to arbitrary values under the importance sampling density?
2. Let M ∼ Poi(μ) be Poisson distributed under the probability measure P. What is the distribution of M under the probability measure obtained by exponential tilting?
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Now suppose that Mi ∼ Poi(μi), i = 1,…,k, are independent Poisson vari- ables under P. We are interested in the random variable L = ki=1 eiMi where e1,…,ek are known deterministic exposures. Suppose we change the proba- bility measure to Qt by exponential tilting with the random variable L. What isthejointdistributionofM1,…,Mk underQt?
Suppose we want to estimate θ = P(L > c) for some large c which greatly exceeds E(L). What equation should we solve to determine t if we want to use importance sampling under Qt to estimate θ?
3. How are the preceding two questions relevant to the problem of using im- portance sampling to estimate tail probabilities of the loss distribution in CreditRisk+ ?
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