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2) we expect that the IEE scheme performs less accurate for the approximation of ”non-normal” shaped cdf‘s. 2) for the approximation of 2 -cdf. g. 5). 34 4 The Integrated Edgeworth Expansion 2 -cdf running an IEE Fig. 1 Approximation of a χ100 Nevertheless, again the IEE performs very accurately in the midsection (∆rel ≈ ∆abs ≈ 10−10 − 10−8 ) and the left end of the distribution (∆rel ≈ 10−4 − 10−2 and ∆abs ≈ 10−6 − 10−4 ), whereas the error increases significantly at the right end (∆abs ≈ 10−4 − 10−2 ).

1) the price of a call option on a discount bond is given by Q Q ZBO1 (t, T0 , T1 ) = Πt,1 [k] − K · Πt,0 [k] . The price of the equivalent put option can be easily found via Q Q ZBO−1 (t, T0 , T1 ) = K · 1 − Πt,0 [k] − 1 − Πt,1 [k] . Then the risk-neutral probabilities Q [k] Πt,a 1 1 = + 2 π ∞ Θt (a + iφ )e−iφ k dφ iφ Re 0 can be derived, by performing a Fourier inversion of the transform Θt (z) = EtQ e− T0 t r(s)ds+zX(T0 ,T1 ) , for a ∈ {0, 1}. 2 Pricing of zero-coupon bond options 45 ϒt (z) = EtT0 ezX(T0 ,T1 ) .

This can be easily shown by deriving the integrated forward rate T ε(t, T ) = f(t, y)dy t in differential form N dε(t, T ) = µ(t, T )dt + ∑ T ∗ σ i (t, y)dydwQ i (t) − f(t,t)dt. 3) together with P(ε) = e−ε(t,T ) . 3) directly leads to ⎛ ⎛ ⎞ ⎞ 1 N dP(t, T ) ⎜ = ⎝r(t) − µ(t, T ) + ∑ ⎝ P(t, T ) 2 i=1 T 2 ⎟ σ (t, y)dy⎠ ⎠ dt i∗ t N − ∑ σ i (t, T )dwQ i (t) . i=1 Now, demanding the absence of arbitrage opportunities requires that the drift term equals the risk-free interest rate, which implies ⎛ ⎞2 T N ∗ 1 µ(t, T ) = ∑ ⎝ σ i (t, y)dy⎠ 2 i=1 t 5 Multi-Factor HJM models 41 or correspondingly µ ∗ (t, T ) = = = ∂ µ (t, T ) ∂T T N ∑ i=1 t N ∑σ T ∗ σ i (t, y) dy · ∂y i∗ ∗ σ i (t, y)dy t (t, T )σ i (t, T ).

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A Bayesian procedure for the sequential estimation of the mean of a negative-binomial distribution by Marcus R.


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