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.
Table of Contents
1 Abstract
2 A Binomial Approximation of Vasicek’s Term Structure Model
3 Valuation of Bonds and Derivatives
3.1 Part I
3.2 Part II
4 Conclusion
Objectives and Topics
This assignment aims to approximate Oldrich Vasicek’s (1977) term structure model using a recombining binomial tree approach implemented in MATLAB. The work evaluates the convergence of these binomial approximations toward the continuous-time Vasicek model and demonstrates the practical valuation of various interest rate derivatives, including bonds, futures, and European or American options.
- Binomial approximation of the Vasicek term structure model.
- Numerical valuation of Discount Bonds, Coupon Bonds, and Futures Contracts.
- Pricing European and American Call and Put options.
- Analysis of Put-Call-Parity and Early Exercise Premiums.
- Convergence analysis of numerical approximations vs. analytical benchmarks.
Excerpt from the Book
A Binomial Approximation of Vasicek’s Term Structure Model
To price a contingent claim, we first need to specify the driving process of the respective underlying source of uncertainty. Hence, in our case of interest rate contingent claims we need a dynamic model for the term strucuture of interest spot rates. In the following, we will apply Oldrich Vasicek (1977) dynamic model. In this model the spot interest rate r(t) follows the so-called Ornstein-Uhlenbeck (or mean-reverting Wiener model) process
dr(t) = κ(μ − r(t))dt + σdw(t) (1)
with the constant parameters κ (mean reversion factor) μ (long-run average of the interest rate) and σ (interest rate volatility). Basically κ reflects the adjustment speed of the interest rate towards μ. The speed of adjustment is greater for higher values of κ and vice versa. If r>μ then the drift is negative, and when r<μ is present, the drift is positive (Ritchken (1996), p.543).
For an interest rate at time t0 (r(0) = r0), the spot interest rate at time T (r(T)) is normally distributed with mean (Vasicek (1977), p. 185)
E[r(t) | r0] = μ + (r0 − μ)e−κ(t−t0) (2)
and variance
V ar[r(t) | r0] = σ2 / 2κ (1 − e−2κ(t−t0)). (3)
As the interest rates are normally distributed, we could obtain negative interest rates with this process. However, when t0 → −∞, the distribution becomes stationary with E[r(t)] = μ and V ar[r(t)] = σ2/2κ. As a result, the probability that interest rates become negative is largely mitigated (Ritchken (1996), p. 543).
In the following, we want to approximate the term structure model of Vasicek (1977) with a binomial approach. First, we discretize the time to maturity (T − t0) into n subperiods with width Δt = (T − t0) / n.
Summary of Chapters
Abstract: Provides an overview of approximating the Vasicek model with a binomial approach, including the valuation of various securities and the use of MATLAB for computational results.
A Binomial Approximation of Vasicek’s Term Structure Model: Details the mathematical foundation of the Ornstein-Uhlenbeck process and explains the discretization of time and interest rates into a recombining binomial tree.
Valuation of Bonds and Derivatives: Divided into two parts, this section covers the numerical valuation of bonds, futures, and options, including the study of Put-Call-Parity and American option features.
Part I: Focuses on the approximation of Discount Bonds, Coupon Bonds, Forwards, and Futures, alongside European options and Straddles.
Part II: Discusses the valuation of European options on Coupon Bonds and extends the methodology to price American-style options.
Conclusion: Summarizes the effectiveness of the binomial tree method in replicating the Vasicek model and highlights the observed convergence properties for various derivatives.
Keywords
Vasicek Model, Binomial Tree, Interest Rate Derivatives, Discount Bond, Coupon Bond, MATLAB, Numerical Approximation, European Options, American Options, Put-Call-Parity, Early Exercise Premium, Ornstein-Uhlenbeck Process, Term Structure, Backward Induction, Financial Engineering.
Frequently Asked Questions
What is the primary focus of this work?
This work focuses on using a binomial tree approximation to value interest rate derivatives within the framework of the Vasicek term structure model.
What are the central thematic fields?
The work covers financial mathematics, specifically interest rate modeling, numerical analysis using MATLAB, and the pricing of bonds and options.
What is the research goal?
The goal is to demonstrate that a binomial approach is a convenient and effective way to value complex interest rate derivatives and to analyze how these approximations converge to analytical solutions.
Which scientific method is applied?
The author applies a numerical binomial approach, utilizing backward induction and discretization of the continuous-time Vasicek stochastic process.
What does the main body treat?
The main body treats the discretization of the term structure model, the valuation procedures for bonds and various types of options, and a comparative analysis between numerical approximations and analytical benchmarks.
Which keywords define this paper?
Key terms include Vasicek Model, Binomial Tree, Numerical Approximation, Bond Valuation, and American/European Options.
How is the computational effort managed in the code?
The author uses a joint calculation approach for specific approximations and leverages MATLAB's matrix capabilities to avoid redundant computations, especially when evaluating multiple derivatives with the same underlying parameters.
Why is the "recombining" nature of the binomial tree significant?
Recombining trees are computationally efficient because they keep the number of nodes at each time step manageable, which is crucial for achieving high accuracy as the number of subperiods (n) increases.
How is the "Early Exercise Premium" analyzed for American options?
It is calculated as the difference between the American option price and the price of its European counterpart, helping to determine the value of the option-holder's right to exercise before the expiration date.
What is the role of the MATLAB m-files provided?
They serve as the practical implementation of the numerical procedures described, allowing the reader to reproduce the results and gain a deeper understanding of the calculation logic.
- Arbeit zitieren
- Alexander Esse (Autor:in), 2017, Pricing Interest Rate Risk Derivatives Using Binomial Trees with MATLAB, München, GRIN Verlag, https://www.grin.com/document/429036
