Stability of Financial Models

Anatoliy SwishchukMathematical and Computational Finance Laboratory

Department of Mathematics and StatisticsUniversity of Calgary, Calgary, Alberta, Canada

E-mail: aswish@math.ucalgary.caWeb page: http://www.math.ucalgary.ca/~aswish/

Talk ‘Lunch at the Lab’

MS543, U of C25th November, 2004

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

Why do we need the stability of financial models?

Definitions of Stochastic Stability1) 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.

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

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.

Simulated Brownian Motion

Stability of Black-Scholes Model. I.

Solution for Stock Price

If , then St=0 is almost sure stable

Idea:

and

almost sure

Otherwise it is unstable

Stability of Black-Scholes Model. II.

• p-Stability

If then the St=0 is p-stable

Idea:

Stability of Black-Scholes Model. III.

• Stability of Discount Stock Price

If then the X t=0 is almost sure stable

Idea:

Black-Scholes with JumpsN 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.

Simulated Poisson Process

Stability of Black-Scholes with Jumps. I.

If , then St=0 is almost sure stable

Idea:

Lyapunov index

Stability of Black-Scholes with Jumps. II.

If , then St=0 is p-stable.

Idea:

1st step:

2nd step:

3d step:

Vasicek Model for Interest Rate (1977)

Explicit Solution:

Drawback: P (r t<0)>0, which is not satisfactory from a practical point of view.

Stability of Vasicek Model

Mean Value:

Variance:

since

Vasicek Model with Jumps

N t - Poisson process

U i – size of ith jump

Stability of Vasicek Model with Jumps

Mean Value:

Variance:

since

Cox-Ingersoll-Ross Model of Interest Rate (1985)

If then the process actually stays strictly positive.

Explicit solution:

b t is some Brownian motion,

random time

Otherwise, it is nonnegative

Stability of Cox-Ingersoll-Ross Model

Mean Value:

Variance:

since

Cox-Ingersoll-Ross Model with Jumps

N t is a Poisson process

U i is size of ith jump

Stability of Cox-Ingersoll-Ross Model with Jumps

Mean Value:

Variance:

since

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

Thank you for your attention!