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Stability of Financial Models
Anatoliy SwishchukMathematical and Computational Finance Laboratory
Department of Mathematics and StatisticsUniversity of Calgary, Calgary, Alberta, Canada
E-mail: [email protected] page: http://www.math.ucalgary.ca/~aswish/
Talk ‘Lunch at the Lab’
MS543, U of C25th 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 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.
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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 St=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 St=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 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.
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Simulated Poisson Process
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Stability of Black-Scholes with Jumps. I.
If , then St=0 is almost sure stable
Idea:
Lyapunov index
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Stability of Black-Scholes with Jumps. II.
If , then St=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)>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)
If then 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!