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Technische Universit¨ at M¨ unchen Lehrstuhl f¨ ur Finanzmathematik Prof. Dr. Rudi Zagst PD Dr. Aleksey Min

Winter term 2019/2020 Sheet 3 Discussion in the week November 26, 2019

Fixed Income Markets Exercise Sheet 3 Exercise 3.1 ˜ (t) be a one dimensional Wiener process under the measure Q. ˜ Solve the stochastic Let W differential equation ˜ (t), dr (t) = [θ − ar (t)]dt + σdW

by considering d(eat r(t)). Note that the quantity eat is called the integration factor.

Exercise 3.2 Using the results of Exercise 3.1 show that the short rate in the Vasicek model (see Page ˜ and 67 of the lecture slides or Exercise 3.4) under the equivalent martingale measure Q given Ft0 is normally distributed with expectation and variance given by  θ E[r(t)|Ft0 ] = 1 − e−a(t−t0 ) + r(t0 )e−a(t−t0 ) ; a  σ2  1 − e−2a(t−t0 ) . V ar[r(t)|Ft0 ] = 2a Exercise 3.3 In the Cox-Ingersoll-Ross model the short rate r(t) is modeled by p dr (t) = [θ − ar (t)]dt + σ r(t)dW˜ (t),

˜ (t) is the Wiener process under the equivalent martingale measure Q. ˜ where W Show that the short rate r(t) in the Cox-Ingersoll-Ross model under the equivalent ˜ and given Ft0 has expectation and variance given by martingale measure Q  θ 1 − e−a(t−t0 ) + r(t0 )e−a(t−t0 ) ; E[r(t)|Ft0 ] = a   σ2 r(t0 ) e−a(t−t0 ) − e−2a(t−t0 ) V ar[r(t)|Ft0 ] = a  θσ 2  1 − 2e−a(t−t0 ) + e−2a(t−t0 ) . + 2 2a Hint: To compute the mean, first determine r(t) by using the method of Exercise 3.1. Note that the expectation of any Itˆo integral is equal to zero. To compute the variance, first determine r 2 (t) by considering d(e2atr 2 (t)). Then compute E[r 2 (t)|Ft0 ] by interchanging expectation and integration and using the computed expectation E[r(t)|Ft0 ]. Please turn the sheet 1

Exercise 3.4 In the Vasicek model the short rate r(t) is modeled by ˜ (t), dr (t) = [θ − ar (t)]dt + σdW ˜ (t) is the Wiener process under the equivalent martingale measure Q. ˜ where W Show that the Vasicek model is a short-rate model with affine term structure and ˜ and B from Definition 4.2 (s. Page 69 of the the corresponding deterministic function A ∗ lecture slides) for all t0 ≤ t ≤ T ≤ T are given by 2 2 2 ˜ T ) = [B(t, T ) − (T − t)](θa − 0.5σ ) − σ B (t, T ) ; A(t, 4a a2 −a(T −t) 1−e B(t, T ) = . a

Exercise 3.5(∗) In the Cox-Ingersoll-Ross model the short rate r(t) is modeled by p dr (t) = [θ − ar (t)]dt + σ r(t)dW˜ (t),

˜ (t) is the Wiener process under the equivalent martingale measure Q. ˜ where W Show that the Cox-Ingersoll-Ross model is a short-rate model with affine term structure and the corresponding deterministic function A˜ and B from Definition 4.2 (s. Page 69 of the lecture slides) for all t0 ≤ t ≤ T ≤ T ∗ are given by   2ce0.5(a+c)(T −t) 2θ ˜ ; A(t, T ) = 2 ln (a + c)(ec(T −t) − 1) + 2c σ √ 2[ec(T −t) − 1] B(t, T ) = , where c = a2 + 2σ 2 . (a + c)(ec(T −t) − 1) + 2c (∗)

May be time consuming and will be shown in the exercises. Hint: To compute B(t, T ), you have to solve the Riccati differential equation.

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