1. Monotone and Cash-Invariant Convex Functions and Hulls Damir Filipović (joint with Michael Kupper) University of Munich www.math.lmu.de/~filipo Advances in Mathematics of Finance Second General AMaMeF Conference and Banach Center Conference Bedlewo, Poland, 2 May 2007

2. 2 May 2007Damir Filipovic, LMU2 Overview Motivating examples Convex monotone cash-invariant functions Monotone and cash-invariant hulls Examples

3. 2 May 2007Damir Filipovic, LMU3

4. 2 May 2007Damir Filipovic, LMU4 021 1 t 1-p p

5. 2 May 2007Damir Filipovic, LMU5

6. 2 May 2007Damir Filipovic, LMU6 Program  Characterize convex monotone functions  Characterize convex cash-invariant functions  Construct monotone and cash-invariant hulls  What can we learn from it?  Construct new and rediscover old convex risk measures

7. 2 May 2007Damir Filipovic, LMU7

8. 2 May 2007Damir Filipovic, LMU8

9. 2 May 2007Damir Filipovic, LMU9

10. 2 May 2007Damir Filipovic, LMU10

11. 2 May 2007Damir Filipovic, LMU11

12. 2 May 2007Damir Filipovic, LMU12

13. 2 May 2007Damir Filipovic, LMU13

14. 2 May 2007Damir Filipovic, LMU14

15. 2 May 2007Damir Filipovic, LMU15

16. 2 May 2007Damir Filipovic, LMU16

17. 2 May 2007Damir Filipovic, LMU17

18. 2 May 2007Damir Filipovic, LMU18 Conclusion We found tractable ways to explicitly calculate monotone and cash-invariant hulls. → Fix defective risk measures → Construct new ones Thanks!