Stability of Financial Models

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.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)): If then zero state is stable almost sure. Otherwise it is unstable.

5. Black-Scholes Model (1973) Bond Price r>0-interest rate Stock Price -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 , then St=0 is almost sure stable If Otherwise it is unstable Idea: and almost sure

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

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

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

15. Vasicek Model for Interest Rate (1977) Explicit Solution: Drawback: P (r t<0)>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 Nt - Poisson process Ui – size of ith jump

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

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

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

21. Cox-Ingersoll-Ross Model with Jumps Ntis a Poisson process U iis 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!