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 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 that defines fu- rates. However, when the interest rate model allows ture term structures. A term structure defines rates to grow superpolynomially, the said lattices gen- yields as a function of . 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 [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 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- . 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 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 Var of ln(St+∆t/St) are fusion process is derived from Eq. (2) thus:

2 dS µ ≡ r − 0.5σ ∆t, (5) = rdt + σdz, (1) S Var ≡ σ2∆t. (6) where r is the risk-free rate, σ is the volatility of the By matching the mean and the variance of stock price process, and the random variable dz is a ln(S /S ), the four parameters P , P , u, and d standard Brownian motion. Equation (1) has the so- t+∆t t u d in the CRR lattice is lution, √ (r−0.5σ2)t+σz(t) σ ∆t St = S0e . (2) u = e , (7) √ d = e−σ ∆t, (8) 2.2 The Stochastic Interest Rate er∆t − d P = , (9) Model u u − d r∆t In this paper, we use the Black-Derman-Toy (BDT) e − u Pd = . (10) model for the interest rate dimension (but the general d − u

2 3 rv4 4 μσˆ +Δ2 t r v2 33 α Pu 2 rv rv4 4 2 2 μ P β ˆ m μ r1 r33v P γ 0 rv d r 2 4 4 r μσˆ −Δ2 t 3 t Δt tt+Δ r4

Figure 3: The Trinomial Lattice. The stock price can move from node X to node A with probability Pu, node B with probability Pm, and node C with prob- Figure 2: The BDT Binomial Interest Rate Lat- ability Pd. Above, µ denotes the mean of the s(X)- tice. The distribution converges to the lognormal dis- log-price of St+∆t (recall Eq. (5)). ln(s(X)/s(X)) = 0, tribution. The rates r , r , r , r are called baseline 1 2 3 4 and ln(s(B)/s√(X)), ln(s(A)/s(√X), and ln(s(C)/s(X)) rates. areµ ˆ,µ ˆ + 2σ ∆t, andµ ˆ − 2σ ∆t, respectively. The numbers β, α, and γ are defined in Eqs. (17)–(19). The length of the time step for the trinomial lattice is The requirements 0 ≤ Pu,Pd ≤ 1 can be met by suit- ably increasing n [15]. Moreover, the lattice converges ∆t. to the continuous-time model as n increases. We re- mark that our results can generalize to any lattice with In the above relations, u and d are independent of the equal time step. prevailing interest rate r (recall Eqs. (7) and (8)). As the prevailing interest rate in the BDT model grows 3.2 The BDT Binomial Interest Rate superpolynomially (which will be proved in Section Lattice 5), eventually inequalities (13) will break. When this happens, the probabilities generated by the CRR lat- An array of yields of zero-coupon bonds for numerous tice will lie outside [0, 1]. maturities and an array of short-rate volatilities for On the other hand, suppose the correlation (ρ) be- the same bonds are the input of the BDT model. From tween the stock price and interest rate is not zero. The the term structure of yields, a procedure called cali- transition probabilities for the stock price dimension bration constructs a binomial lattice consistent with p will become Pu = p + p(1 − p)ρ and Pd = 1 − Pu, the term structure (see Fig. 2). In general, there are where p = (er∆t −d)/(u−d). Obviously, the transition j possible rates in period j: probabilities will be complex numbers if inequalities 2 j−1 (13) do not hold. rj, rjvj, rjvj , ··· , rjvj , (11) where √ 2σj ∆t vj = e (12) 4 Bivariate Lattice Construc- is the multiplicative ratio for the rates in period j, rj is tion called baseline rates, and σj is the annualized short- Our bivariate lattice has two dimensions: the stock rate volatility in period j. The subscript j in σj is meant to emphasize that the short rate volatility may price dimension and the interest rate dimension. For be time dependent. Each branch has a 50% chance the stock price dimension, the foundation of our lattice of occurring in a risk-neutralized economy (i.e., the is the CRR lattice mentioned in Section 3.1. To solve probability for each branch is 1/2). the invalid transition probability problem mentioned in Section 3.3, we use trinomial lattices to track the mean of the stock price. 3.3 The Invalid Transition Probability Let the stock price of node Z be s(Z) for conve- Problem nience. Define the S-log-price of the stock price S0 as 0 A few bivariate binomial lattices for pricing convert- ln(S /S) and the log-distance between stock prices S 0 0 ible bonds with the BDT interest rate model for the and S as | ln(S) − ln(S )|. Given a node X at time t interest rate component have been proposed [6, 12]. If and nodes at time t + ∆t. (The log-distance√ between there is no correlation between the stock and interest two adjacent nodes at time t + ∆t is 2σ ∆t, which rate, the transition probabilities for the stock price is the same as the CRR lattice). The mean (µ) and variance (Var) of the s(X)-log-price of S can be dimension are Pu and Pd given the prevailing inter- t+∆t est rate r (recall Eqs. (9) and (10)). The no-arbitrage obtained from Eqs. (5) and (6), respectively. requirements 0 ≤ Pu,Pd ≤ 1 is equivalent to In Fig. 3, the node B whose s(X)-log-price (ˆµ) is closest to µ among all the nodes at time t+∆t will be d < er∆t < u. (13) the destination of the middle branch of the trinomial

3 k 2 Stxt,rv++11 t S20 ,r3v3

S20 , r3v3

Sr20 , 3 2σ Δt 2 S21,r3v3

k+1 S21,r3v3 rtt++2v 2 β Sr21, 3 S(1)ty+ k rt++22vt S(ty++1)( 1) 2 S22 ,r3v3 S(1)(2)ty++ μ 2 S22 ,r3v3

Sr22 , 3 σ Δt Figure 4: The Concept of a Two-Dimensional S10 ,r2v2

Lattice. The node X at time step t has six branches Sr10 , 2 S ,r v2 to nodes A, B, C, D, E, and F at time step t+1. Let Sr, 23 3 3 00 1 S ,r v Sij denote the jth nodes at time step i. The stock 23 3 3 k price and the short rate at X are Stx and rt+1v , Sr23 , 3 t+1 S ,r v respectively. The stock price at nodes A and D is 11 2 2 Sr11 , 2 S(t+1)y, that at nodes C and E is S(t+1)(y+1), and 2 S24 ,r3v3 that at nodes D and F is S(t+1)(y+2). The interest S ,r v k+1 24 3 3 rate at node A, B, C is rt+2vt+2 (upward), and that Sr, k 24 3 at nodes D, E, F is rt+2vt+2 (downward). S12 ,r2v2

Sr12 , 2 σ Δt S25 ,r3v3 lattice (this is called meaning tracking). Note that S25 ,r3v3 the s(X)-log-price of the central node B will lie in the S , r √ √ Δ t 25 3 interval ( µ − σ ∆t, µ + σ ∆t ] [16]. We then select two nodes A and C, which are adjacent to node B, Figure 5: The Size of the Bivariate Lattice.√The and three transition probabilities Pu, Pm, and Pd for log-distance between two adjacent nodes is 2σ ∆t. the node X can be obtained by matching the mean 0 Sij denotes the jth nodes at time i, and µ2 = (r2v2 − and variance of the s(X)-log-price of St+∆t. 2 2 0 0.5σ )∆t = r2v2 − 0.5σ ∆t (recall Eq. (5)), where rj The transition probabilities for the node X (i.e., Pu, denotes the annualized baseline rate. Pm, Pd) can be derived by solving the following three equalities: k E, F is rt+2vt+2 (downward). In the above lattice construction, we simply assume that the interest rate Puα + Pmβ + Pdγ = 0, (14) process and the stock price process are independent, P (α)2 + P (β)2 + P (γ)2 = Var, (15) u m d so the joint probabilities for the six branches can be Pu + Pm + Pd = 1, (16) obtained by multiplying the transition probabilities of the BDT binomial lattice (i.e., 1/2) and the trino- where mial lattice (i.e., Pu, Pm, and Pd). If the correlation between two processes are not zero, the joint proba- β ≡ µˆ − µ, (17) √ √ bilities can be obtain by using the orthogonalization α ≡ µˆ + 2σ ∆t − µ = β + 2σ ∆t, (18) technique [9]. √ √ γ ≡ µˆ − 2σ ∆t − µ = β − 2σ ∆t, (19) µˆ ≡ ln (s(B)/s(X)). (20) 5 Complexity of the Bivariate √ √ (Note that −σ ∆t < β ≤ σ ∆t.) The above proce- Lattice dure will yield valid probabilities [16]. For the interest rate dimension, we first construct a The size of the proposed bivariate lattice is proved binomial lattice for the BDT model. The idea of our in this section. Figure 5 gives a three-period bi- bivariate lattice is illustrated in Fig. 4. The node X variate lattice as an example and illustrates the con- at time t has 6 branches, to nodes A, B, C, D, E, cept of the bivariate lattice. In the figure, µ2 = (r0 v − 0.5σ2)∆t = r v − 0.5σ2∆t (see Eq. (5)) and and F at time t + ∆t. Let Sij denote the jth nodes 2 2 2 2 at time step i. Assume that the stock price and the √ √ k −σ ∆t ≤ β < σ ∆t . (21) short rate at X are Stx and rt+1vt+1, respectively. The stock price at nodes A and D is S , that 0 (t+1)y (Note that rj denotes the annualized baseline rates.) at nodes C and E is S(t+1)(y+1), and that at nodes To investigate the complexity of the size of the bivari- D and F is S(t+1)(y+2). The interest rate at node ate lattice, we prove that the node count at each time k+1 A, B, C is rt+2vt+2 (upward), and that at nodes D, grows superpolynomially as n goes to infinity.

4 We fix the maturity T = 1 for simplicity. Let d(`) Then denote the number of nodes in the stock price dimen- j−1 ∗ j sion at time step ` and µj denote the log-mean of rjvj < rj vj (26) the stock price at time j with the biggest interest  j−1 j−1 2 2 j rate at time j − 1 (i.e., µj = rjvj − 0.5σ /n = = fjvj (27) √ 1 + vj r e2(j−1)σj / n − 0.5σ2/n) for j = 1, 2, ··· , n + 1. Fig- j  j−1 ure 5 shows that d(0) = 1, d(1) = 3, and 2vj = fjvj, (28) 1 + vj √1 √1 µ2 + β + 2σ n − σ n j−1 ∗ j−1 d(2) = d(1) + 1 + . rjvj > rj vj (29) 2σ √1 n  2 j−1 = f vj−1 (30) 1 1 j j √ √ 1 + vj By the fact −σ n < β ≤ σ n by inequalities (21),  j−1 we know that 2vj µ µ = fj. (31) d(1) + 1 + 2 ≤ d(2) ≤ d(1) + 1 + 2 + 1. 1 + vj √1 √1 2σ n 2σ n Let By induction, d(l+1) satisfies the following recurrence j j 2σ √1 ! relation:  2v  2e j+1 n A(n, j) = j+1 = . (32) 2σ √1 `+1 `+1 1 + vj+1 1 + e j+1 n X µj X µj d(1)+`+ ≤ d(`+1) ≤ d(1)+2`+ . 2σ √1 2σ √1 √ j=2 n j=2 n Because 1 + e2σj+1/ n > 2, (22)

Consequently, the total node count in the stock price 1 j 2σj+1 √ ! n j 2e 2σ √ dimension is j+1 n n+1 A(n, j) < = e . (33) X 2 d(j). (23) √ j=0 O(j/ n) Therefore, A(n, j) = e .√ On the other hand,√ for From Eqs. (22) and (23), we know that any bivariate 2σ 1 2σ 1 n sufficiently large, 2 × 2 j+1 n ≥ 1 + e j+1 n . So lattice method for stock and interest rate must grow superpolynomially if the interest rate (or µj) grows j 2σ √1 ! j j+1 n 2σ √ 2e  e  j+1 n superpolynomially (, to which we turn below). A(n, j) ≥ √ = . (34) 2σ 1 Next, we prove that the growth rate of µj is su- 2 × 2 j+1 n 2 perpolynomial in the BDT model. We first prove the √ superpolynomial growth rate of the biggest interest As a result, A(n, j) = (e/2)Ω(j/ n) for n sufficiently 2(j−1)σ √1 j−1 j n rate at time j − 1, rjvj = rje . Let fj large. denote the forward rate in period j. Build a binomial According to Eqs. (26), (29), (33), and (34), as n interest rate lattice with the baseline rates r∗ set by j−1 j goes to infinity, the upper and lower bounds of rjvj are  j−1 ∗ 2 rj = fj. (24) j−1 1 + vj A(n, j − 1)vj < rjvj < A(n, j − 1)fjvj, (35)

Theorem 5.1 The binomial interest rate lattice con- j−1 structed with Eq. (24) overestimates the prices of the From inequalities (35), we know that rjvj grows benchmark securities in the presence of volatilities. superpolynomially, and equivalently, µj grows super- j−1 2 The conclusion is independent of whether the volatility polynomially (recall µj = rjvj − 0.5σ /n). There- structure is matched. fore, the size of the bivariate lattice must grow super- polynomially in the BDT model. Proof: The details of the proof is presented in [15] . Theorem 5.2 The binomial interest rate lattice con- 6 Conclusions structed with Eq. (24) by using the forward rate vjfj instead of f underestimates the prices of the bench- j This paper presents the first bivariate lattice method mark securities in the presence of volatilities. The to solve the invalid transition probability problem by conclusion is independent of whether the volatility using the trinomial lattice and mean-tracking tech- structure is matched. niques. Furthermore, we prove that any bivariate lat- Proof: The details of the proof are available from the tice method for stock price and interest rate must grow authors upon request. superpolynomially if (1) the transition probabilities According to Theorems 5.1 and 5.2, we have are guaranteed to be valid and (2) the interest rate model allows rates to grow superpolynomially such as ∗ ∗ rj < rj < rj vj. (25) the BDT model.

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