On the Complexity of Bivariate Lattice with Stochastic Interest Rate Models
Chuan-Ju Wang
Department of Computer Science & Information Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., Taipei 106, Taiwan. E-mail: [email protected] Tel: 886-2-23625336 ext. 446
Yuh-Dauh Lyuu
Department of Finance and Department of Computer Science & Information Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., Taipei 106, Taiwan. E-mail: [email protected] Tel: 886-2-23625336 ext. 429 Fax: 886-2-23628167
Abstract stochastic stock prices and stochastic interest rates. In the traditional approach to valuing derivatives, Many complex financial instruments with multiple interest rates are often assumed to be constant. But state variables have no analytical formulas and thus the values of interest rate-sensitive securities such as must be priced by numerical methods like lattice. The callable bonds depend strongly on the interest rate, bivariate lattice is a numerical method that is widely which does not stay constant in the real world. Hence used to work with correlated state variables. Some it is critical for a general model to incorporate a research has focused on bivariate lattices for models stochastic interest rate component. with two state variables: stochastic underlying as- A stochastic interest rate model performs two tasks. set prices (e.g., stock prices) and stochastic interest First, it provides a stochastic process that defines fu- rates. However, when the interest rate model allows ture term structures. A term structure defines bond rates to grow superpolynomially, the said lattices gen- yields as a function of maturity. Second, the model erate invalid transition probabilities. With the trino- should be consistent with the observed term struc- mial lattice and the mean-tracking techniques, this pa- ture. There are two approaches on modeling inter- per presents the first bivariate lattice that guarantees est rates: the equilibrium model and the no-arbitrage valid probabilities. It also proves that any bivariate model. Equilibrium models usually start with the as- lattice for stock price and interest rate must grow su- sumption about economic variables and derive a pro- perpolynomially if the interest rate model allows rates cess for the risk-free rate. Two of the difficulties fac- to grow superpolynomially. ing equilibrium models are that (1) they usually re- Keywords: lattice, stochastic interest rate model, quire the estimation of the market price of risk and complexity (2) they cannot fit the observed market term struc- ture. Non-arbitrage models, in contrast, are designed 1 Introduction to be consistent with the market term structure [10]. This paper focuses on non-arbitrage lognormal inter- The pricing of financial instruments with two or more est rate models, such as Black-Derman-Toy, Black- state variables has been intensively studied. The Karasinski, and Dothan models [2, 3, 7]. This class added state variables to the underlying asset price can of interest rate models is assumed to follow the log- be stochastic volatility [8, 11] or stochastic interest normal interest rate process. In this paper, we adopt rate [4, 14]. (The underlying asset will be assumed to Black-Derman-Toy (BDT) model, which is extensively be stock for convenience.) Unfortunately, many com- used by practitioners, to explain the main idea of our plex financial instruments with multiple state vari- bivariate lattice [1, 5, 13]. The popularity of lognor- ables have no analytical formulas and thus must be mal models arises from the fact that negative interest priced by numerical methods like lattice. This paper rates are impossible. presents a bivariate lattice method for models with Using the BDT model, Hung and Wang propose a
1 Su3 bivariate binomial lattice for pricing convertible bonds 0
[12]. A convertible bond entitles the holder to convert 2 Su0 it into stocks. The lattice’s size has a cubic growth Su Su rate. Chambers and Lu extend Hung and Wang’s lat- 0 0 Pu tice by adding the correlation between stock and in- S0 S0 terest rate [6]. However, both approaches share the Pd Sd Sd same problem of invalid transition probabilities. This 0 0 is due to the fact that the BDT model allows the in- 2 Sd0 terest rate to grow superpolynomially, which makes Sd3 the stock prices on the lattice with high interest rates 0 unable to match the desired moments with valid prob- abilities. Figure 1: The CRR Lattice. The initial stock price This paper presents the first bivariate lattice for is S0. The upward and downward multiplicative fac- stock and interest rate that guarantees valid transition tors for the stock price are u and d, respectively. The probabilities. Our bivariate lattice has two dimen- transition probabilities are Pu and Pd = 1 − Pu. sions: the stock price dimension and the interest rate dimension. For the interest rate dimension, we con- struct a binomial interest rate lattice under the BDT conclusion applies to all lognormal interest rate mod- model. We then develop a trinomial lattice for the els). In the BDT model, the short rate r follows the stock price dimension with the help of mean-tracking stochastic process, techniques [16]. Furthermore, we prove that any bi- d ln r = θ(t)dt + σr(t)dz, (3) variate lattice method (such as ours) for stock price and interest rate must grow superpolynomially if (1) where θ(t) is a function of time calibrated to ensure the transition probabilities are guaranteed to be valid that the model fits the market term structure, σr(t) is and (2) the interest rate model allows rates to grow a function of time and denotes the instantaneous stan- superpolynomially such as the BDT model. dard deviation of the short rate, and dz is a standard Our paper is organized as follows: The mathemat- Brownian motion. ical models are introduced in Section 2. We review how to construct a binomial lattice and a BDT inter- est rate lattice in Section 3. Section 3 also introduces 3 Preliminaries the invalid transition probability problem. Section 4 describes the methodology for our proposed bivari- 3.1 The Binomial Lattice Model ate lattice. Section 5 proves the superpolynomial size A lattice partitions the time span from time 0 to time of any bivariate lattice method for stock and interest T into n equal time steps and specifies the stock price rate if the interest rate model grows superpolynomi- at each time step. The length of one time step ∆t ally. Section 6 concludes the paper. thus equals T/n. A 3-time-step Cox-Ross-Rubinstein (CRR) binomial lattice is illustrated in Fig. 1. At 2 Modeling and Definitions each time step, the stock price S can either make an up move to become Su with probability Pu or a down move to become Sd with probability Pd ≡ 1−Pu. The 2.1 Continuous-Time Stock Price Dy- relation namics ud = 1 (4)
Define St as the stock price at time t. The risk- is enforced by the CRR binomial lattice. neutralized version of the stock price lognormal dif- The mean (µ) and variance Var of ln(St+∆t/St) are fusion process is derived from Eq. (2) thus: