In this assignment we approximate Oldrich Vasicek's (1977) term structure model with a binomial approach and show that it is convenient to use a recombining binomial tree to value interest rate derivatives in the Vasicek model.
First, we illustrate that our applied binomial approximations converge to the dynamic continuous-time Vasicek model with an increasing number of time steps (subperiods). Furthermore, we apply the binomial approach to value a Discount Bond, Coupon Bond and a Futures Contract on both a Discount and Coupon Bond. The resulting approximations will be compared to the respective analytical solution, which we use as a benchmark.
Thirdly, we determine the fair value of both an European and American Call and Put on a Discount Bond and Coupon Bond, respectively. We demonstrate that our estimated binomial prices converge with an increasing number of time steps. Moreover, we analyze both the behaviour of a Sraddle on a Discount Bond and the Early Exercise Premium of the considered American Options as a function of spot interest rates.
We obtain all results shown in this report from the software "Matlab". Hence, the submitted "m.files" should be taken as a reference for a better understanding of the calculation procedures described in this report (Relevant Code is depicted in the Appendices).
Furthermore, to reduce computational effort and required time to run our code we apply a joint calculation of specific approximations rather than run a code individually for each Task. This is mainly because some specific securities and interest rate derivatives require the same underlying and identical matrices of the interest rates and transition probabilities from the binomial trees for the approximation procedure. This approach is suitable because we apply the identical number of subperiods for specific Tasks and, thus, for the respective securities and or derivatives.
Inhaltsverzeichnis (Table of Contents)
- Abstract
- A Binomial Approximation of Vasicek's Term Structure Model
- Valuation of Bonds and Derivatives
- Part I
- Part II
- Conclusion
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
The primary objective of this work is to approximate Oldrich Vasicek's term structure model for interest rates using a binomial approach. This method provides a convenient tool for valuing interest rate derivatives in the Vasicek model. The work will focus on demonstrating the convergence of binomial approximations to the continuous-time Vasicek model and applying this approach to value various financial instruments, including discount and coupon bonds, futures contracts, and European and American options on these bonds.
- Binomial approximation of Vasicek's term structure model
- Valuation of bonds and derivatives using the binomial approach
- Convergence of binomial approximations
- Comparison of binomial results with analytical solutions
- Analysis of early exercise premiums for American options
Zusammenfassung der Kapitel (Chapter Summaries)
- Abstract: The abstract provides a concise overview of the assignment's aims and key findings. It highlights the use of a binomial approach to approximate Vasicek's term structure model and its application to valuing interest rate derivatives.
- A Binomial Approximation of Vasicek's Term Structure Model: This chapter introduces Vasicek's term structure model and explains its application to pricing interest rate-contingent claims. It presents the mathematical formulation of the model and the properties of the spot interest rate process. The chapter then discusses the discretization of the time to maturity and the binomial approach to approximating the model.
- Valuation of Bonds and Derivatives: This chapter delves into the application of the binomial approximation to value various financial instruments. It focuses on the valuation of discount bonds, coupon bonds, and futures contracts on these bonds. The chapter also examines European and American call and put options on discount and coupon bonds, analyzing the early exercise premium for American options. This chapter demonstrates the convergence of binomial prices with an increasing number of time steps.
Schlüsselwörter (Keywords)
The core keywords and topics explored in this work include: Vasicek's term structure model, binomial approximation, interest rate derivatives, bond valuation, futures contracts, European and American options, early exercise premium, convergence analysis, and Matlab implementation.
- Citation du texte
- Alexander Esse (Auteur), 2017, Pricing Interest Rate Risk Derivatives Using Binomial Trees with MATLAB, Munich, GRIN Verlag, https://www.grin.com/document/429036
