## THERMAL SCIENCE

International Scientific Journal

### A NEW NOTION OF TRANSITIVE RELATIVE RETURN RATE AND ITS APPLICATIONS USING STOCHASTIC DIFFERENTIAL EQUATIONS

**ABSTRACT**

We introduce a new notion of transitive relative return rate and present its applications based on the stochastic differential equations. First, we define the notion of a relative return rate (RRR) and show how to construct the transitive relative return rate (TRRR) on it. Then, we state some propositions and theorems about RRR and TRRR and prove them. Moreover, we exhibit the theoretical framework of the generalization of TRRR for n≥3 cases and prove it, as well. Furthermore, we illustrate our approach with real data applications of daily relative return rates for Borsa Istanbul-30 (BIST-30) and Intel Corporation (INTC) indexes with respect to daily interest rate of Central Bank of the Republic of Turkey (CBRT) between 18.06.2003 and 17.06.2013. For this purpose, we perform simulations via Milstein method. We succeed to present usefulness of the relative return rate for the relevant real large data set using the numerical solution of the stochastic differential equations. The simulation results show that the proposed closely approximates the real data.

**KEYWORDS**

PAPER SUBMITTED: 2018-09-15

PAPER REVISED: 2018-10-31

PAPER ACCEPTED: 2018-11-16

PUBLISHED ONLINE: 2018-12-16

**THERMAL SCIENCE** YEAR

**2019**, VOLUME

**23**, ISSUE

**Supplement 1**, PAGES [S113 - S120]

- İzgi, B., and Bakkaloğlu, A., Invariant Approaches for the Analytic Solution of the Stochastic Black-Derman Toy Model, Thermal Science, 22 (2018), 1, pp. 265 - 275.
- Mao. X, Stochastic Differential Equations and Applications, second ed., WP, 2007.
- Baker, M., and Wurgler, J., Comovement and Predictability Relationships Between Bonds and the Cross-section of Stocks, Review of Asset Pricing Studies, 2 (2012), 1, pp. 57-87.
- Duran, A., and İzgi, B., Application of the Heston Stochastic Volatility Model for Borsa Istanbul Using Impression Matrix Norm, J. of Comp. and Appl. Math., 281 (2015), pp. 126-134.
- Geweke, J., Measurement of Linear Dependence and Feedback Between Multiple Time Series, J. of the American Stat. Assoc., 77 (1982), pp. 304-313.
- İzgi, B., Behavioral Classification of Stochastic Differential Equations in Mathematical Finance, Ph.D. thesis, Istanbul Technical University, 2015.
- Merton, R., Option Pricing when Underlying Stock Returns are Discontinuous, J. Financial Economics, 3 (1976), pp. 125-144.
- Milstein, G.N., Approximate Integration of Stochastic Differential Equations, Theor. Prob. Appl., 19, (1974), pp. 557-562.
- Kloeden, P.E., et al., Numerical Solution of SDE Through Computer Experiments, Springer, Berlin, 2003.
- Chen, H., et al., Comovement Revisited, J. of Financial Economics, 121 (2016), 3, pp. 624-644.
- Bunn, D. W., et al., Fundamental and Financial Influences on the Co-movement of Oil and Gas Prices, Energy Journal, 38 (2017), 2, pp. 201-228.