Vasicek model

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The Vasicek model in finance is a mathematical model describing the evolution of interest rates. It is a type of "one factor model" (Short rate model) as describes interest rate movements as driven by only one source of market risk. The model can be used in the valuation of interest rate derivatives. It was introduced in 1977 by Oldrich Vasicek.

The model specifies that the instantaneous interest rate follows the stochastic differential equation:

dr_t = a(b-r_t)\, dt + \sigma \, dW_t

where Wt is a Wiener process modelling the random market risk factor. The standard deviation parameter, σ, determines the volatility of the interest rate.

Vasicek's model was the first one to capture mean reversion, an essential characteristic of the interest rate that sets it apart from other financial prices. Thus, as opposed to stock prices for instance, interest rates can not rise indefinitely. This is because at very high levels they would hamper economic activity, prompting a decrease in interest rates. Similarly, interest rates can not decrease indefinitely. As a result, interest rates move in a limited range, showing a tendency to revert to a long run value.

The drift factor a(b − rt) represents the expected instantaneous change in the interest rate at time t. The parameter b represents the long run equilibrium value towards which the interest rate reverts. Indeed, in the absence of shocks (dWt = 0), the interest remains constant when rt = b. The parameter a, governing the speed of adjustment, needs to be positive to insure stability around the long term value. For example, when rt is bellow b, the drift term a(b − rt) becomes positive for positive a, generating a tendency for the interest rate to move upwards (toward equilibrium).

The main disadvantage is that, under Vasicek's model, it is theoretically possible for the interest rate to become negative, an undesirable feature. This shortcomming was fixed in the Cox-Ingersoll-Ross model. The Vasicek model was further extended in the Hull-White model

[edit] References

  • Hull, John C. (2003). Options, Futures and Other Derivatives. Upper Saddle River, NJ: Prentice Hall. ISBN 0130090565.
  • Vasicek, Oldrich (1977). "An Equilibrium Characterisation of the Term Structure". Journal of Financial Economics 5: 177-188.
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