1. Stability of Financial Models Anatoliy Swishchuk Mathematical and Computational Finance Laboratory Department of Mathematics and Statistics University of Calgary, Calgary, Alberta, Canada E-mail: aswish@math.ucalgary.caaswish@math.ucalgary.ca Web page: http://www.math.ucalgary.ca/~aswish/ Talk ‘Lunch at the Lab’ MS543, U of C 25th November, 2004

2. Outline Definitions of Stochastic Stability Stability of Black-Scholes Model Stability of Interest Rates: Vasicek, Cox- Ingersoll-Ross (CIR) Black-Scholes with Jumps: Stability Vasicek and CIR with Jumps: Stability

3. Why do we need the stability of financial models?

4. Definitions of Stochastic Stability 1) Almost Sure Asymptotical Stability of Zero State 2) Stability in the Mean of Zero State 3) Stability in the Mean Square of Zero State 4) p-Stability in the Mean of Zero State Remark: Lyapunov index is used for 1) ( and also for 2), 3) and 4)): Ifthen zero state is stable almost sure. Otherwise it is unstable.

5. Black-Scholes Model (1973) Bond Price Stock Price r>0-interest rate -appreciation rate >0-volatility Remark. Rendleman & Bartter (1980) used this equation to model interest rate

6. Ito Integral in Stochastic Term Difference between Ito calculus and classical (Newtonian calculus): 1) Quadratic variation of differentiable function on [0,T] equals to 0: 2) Quadratic variation of Brownian motion on [0,T] equals to T: In particular, the paths of Brownian motion are not differentiable.

7. Simulated Brownian Motion

8. Stability of Black-Scholes Model. I. Solution for Stock Price If, then S t =0 is almost sure stable Idea: and almost sure Otherwise it is unstable

9. Stability of Black-Scholes Model. II. p-Stability If then the S t =0 is p-stable Idea:

10. Stability of Black-Scholes Model. III. Stability of Discount Stock Price If then the X t =0 is almost sure stable Idea:

11. Black-Scholes with Jumps N t-Poisson process with intensity moments of jumps independent identically distributed r. v. in On the intervals At the moments Stock Price with Jumps The sigma-algebras generated by ( W t, t>=0), ( N t, t>=0) and ( U i; i>=1) are independent.

12. Simulated Poisson Process

13. Stability of Black-Scholes with Jumps. I. If, then S t=0 is almost sure stable Idea: Lyapunov index

14. Stability of Black-Scholes with Jumps. II. If, then S t =0 is p-stable. Idea: 1st step: 2nd step: 3d step:

15. Vasicek Model for Interest Rate (1977) Explicit Solution: Drawback: P ( r t 0, which is not satisfactory from a practical point of view.

16. Stability of Vasicek Model Mean Value: Variance: since

17. Vasicek Model with Jumps N t - Poisson process U i – size of ith jump

18. Stability of Vasicek Model with Jumps Mean Value: Variance: since

19. Cox-Ingersoll-Ross Model of Interest Rate (1985) Ifthen the process actually stays strictly positive. Explicit solution: b t is some Brownian motion, random time Otherwise, it is nonnegative

20. Stability of Cox-Ingersoll-Ross Model Mean Value: Variance: since

21. Cox-Ingersoll-Ross Model with Jumps N t is a Poisson process U i is size of ith jump

22. Stability of Cox-Ingersoll-Ross Model with Jumps Mean Value: Variance: since

23. Conclusions We considered Black-Scholes, Vasicek and Cox-Ingersoll-Ross models (including models with jumps) Stability of Black-Scholes Model without and with Jumps Stability of Vasicek Model without and with Jumps Stability Cox-Ingersoll-Ross Model without and with Jumps If we can keep parameters in these ways- the financial models and markets will be stable

24. Thank you for your attention!