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Stability of Financial Models Anatoliy Swishchuk Mathematical and Computational Finance Laboratory Department of Mathematics and Statistics University of Calgary, Calgary, Alberta, Canada Web page: Talk ‘Lunch at the Lab’ MS543, U of C 25th November, 2004

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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

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Why do we need the stability of financial models?

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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.

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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

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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.

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Simulated Brownian Motion

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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

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

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

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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.

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Simulated Poisson Process

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

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

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

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Stability of Vasicek Model Mean Value: Variance: since

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

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

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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

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

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

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

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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

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Thank you for your attention!

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