DOI: https://doi.org/10.30970/jps.25.2801
APPLICATION OF THE PATH INTEGRAL METHOD TO SOME STOCHASTIC MODELS OF FINANCIAL ENGINEERING
V. Yanishevsky 
National University ‟Lvivska Polytechnika”,
12, Bandery St., Lviv, UA–79013, Ukraine
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Journal of Physical Studies 25(2), Article 2801 [10 pages] (2021)
DOI: https://doi.org/10.30970/jps.25.2801 APPLICATION OF THE PATH INTEGRAL METHOD TO SOME STOCHASTIC MODELS OF FINANCIAL ENGINEERING
V. Yanishevsky
National University ‟Lvivska Polytechnika”,
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The path integral method is applied to investigation of time dependence of interest rate of Merton and Vasicek stochastic models. The path integral is based on Wiener integral by means of variable substitution based on stochastic equations of the models. The results were obtained for term structure of interest rates in Merton and Vasicek models.
The path integral method is applied to the investigation of the time dependence of the interest rate in the Merton and Vasicek stochastic models. The path integral is based on the Wiener integral by means of variable substitution based on stochastic equations of the models. The results were obtained for the term structure of interest rates in the Merton and Vasicek models.
As is known, the drawback of those models is that they allow for negative values of the interest rate. Hence, one should introduce boundary conditions, which limit the stochastic variable domain to positive values. During the construction of the path integral for respective mean values, one needs to use a transition probability that satisfies those boundary conditions. However, the construction of such path integral for average values, which contains an exponent dependent on some functional of integration variables, is in general a very complicated task.
In the work, an easier approach is used that consists in limiting the integral domain in path integrals for the Merton and Vasicek models. In particular, in path integrals the integral domain is limited using condition $ınt_{t_0}^{t}r(τ)dτ>0$, which is given by the Heaviside step function $θ(x)$. It is shown that limiting the integral domain in such a way allows us to get rid of negative values of the term structure of interest rates.
The path integrals for the limited integral domain are calculated using the transformation of the Heaviside step function into a complex plane. During the calculation of mean values, a normalization of the measure is applied to a selected domain. As a result, in the models where the interest rate takes only positive values, equations for the term structure of interest rates were obtained. Positive values of interest rates are visually demonstrated by means of asymptotic analysis of their time dependence.
The obtained functional forms of the interest rates dependencies can be used to calibrate models with the help of respective optimization procedures, which minimize deviation of theoretical and experimental values. Expressions found in the work allow one to accomplish the above procedures using numeric methods.
Key words: stochastic model, Merton model, Vasicek model, interest rate, functional integral