A simple trinomial lattice approach for the skew-extended CIR models

Abstract

In this paper, we introduce a simple and efficient trinomial lattice tree approach for the skew Cox-Ingersoll-Ross (CIR) model and the doubly skewed CIR model. Suffering from the terms of local times and non-constant volatility, we apply two transforms to the skew-extended CIR processes. Then we construct a modified trinomial tree for the transformed processes which are piecewise tractable diffusions with constant volatility. As a result, the tree for the original skew-extended CIR processes can be easily obtained by using the inverse transform. Results of applications to zero-coupon bonds, European and American options demonstrate that our simple tree approach is efficient and satisfactory.

Appendices

Appendix

Pricing the zero-coupon bond using the spectral expansion method

In this appendix, we apply the spectral expansion approach to value the zero-coupon bond when the term structure of interest rate satisfies the skew CIR process defined in Eq. (1). Following the idea in [34], we assume that

$$\begin{aligned} \beta =\frac{2k\theta }{\sigma ^{2}}\geqslant 1, \end{aligned}$$

which means the process $r_{t}$ defined in Eq. (1) satisfies the Feller condition and the state space of $r_{t}$ is $(0,\infty )$. Define

$$\begin{aligned} G(x)=\left\{ \begin{array}{l@{\quad }l} \displaystyle p(x-a)+a, &{} 0<x<a,\\ \displaystyle (1-p)(x-a)+a, &{} x\ge a, \end{array} \right. \end{aligned}$$

then the inverse function H(x) of G(x) satisfies

$$\begin{aligned} H(x)=\left\{ \begin{array}{l@{\quad }l} \displaystyle \frac{1}{p}(x-a)+a, &{} (1-p)a<x<a,\\ \displaystyle \frac{1}{1-p}(x-a)+a, &{} x\ge a. \end{array} \right. \end{aligned}$$

Applying the generalized Itó formula (see e.g., [32]) to $\eta _{t}=G(r_{t})$, we obtain that

$$\begin{aligned} {\mathrm {d}}\eta _{t}=\left\{ \begin{array}{ll} \displaystyle k(\theta _{2}-\eta _{t}){\mathrm {d}}t+\sigma _{2}\sqrt{\eta _{t}-l_{2}}{\mathrm {d}}W_{t}, &{}\quad l_{2}<\eta _{t}<a,\\ \displaystyle k(\theta _{1}-\eta _{t}){\mathrm {d}}t+\sigma _{1}\sqrt{\eta _{t}-l_{1}}{\mathrm {d}}W_{t}, &{}\quad \eta _{t}\ge a, \end{array} \right. \end{aligned}$$

(A.22)

where

$$\begin{aligned} \begin{array}{lll} \theta _{1} =(1-p)\theta +pa,&{}\quad \sigma _{1}=\sigma \sqrt{1-p},&{}\quad l_{1}=pa, \\ \theta _{2} = p\theta +(1-p)a,&{} \quad \sigma _{2}=\sigma \sqrt{p}, &{}\quad l_{2}=(1-p)a. \end{array} \end{aligned}$$

The process $\eta _{t}$ is a diffusion with discontinuous coefficients on $I^{\eta }=(l_{2},\infty )$. And its path behaves like the shifted CIR process when $\eta _{t}\ge a$ or $l_{2}<\eta _{t}<a$ with different coefficients. What’s more, through careful calculations (for more details please see [34]), we know that the boundary $l_{2}$ is entrance and $\infty$ is non-attracting natural. This indicates that $l_{2}$ and $\infty$ are both NONOSC⁷ natural boundaries for a shifted CIR process. The scale density $\mathfrak {s}_{\eta }(x)$ and the speed density $\mathfrak {m}_{\eta }(x)$ of the diffusion $\eta _{t}$ are

$$\begin{aligned}&\mathfrak {s}_{\eta }(x) = \left\{ \begin{array}{ll} \displaystyle (x-l_{2})^{-\beta }\mathrm {e}^{\frac{2k(x-l_{2})}{\sigma _{2}^{2}}}, &{}\quad l_{2}<x<a,\\ \displaystyle (a-l_{2})^{-\beta }\mathrm {e}^{\frac{2k(x-a)}{\sigma _{1}^{2}}+\frac{2k(a-l_{2})}{\sigma _{2}^{2}}}\left( \frac{x-l_{1}}{a-l_{1}}\right) ^{-\beta }, &{}\quad x\ge a, \end{array} \right. \nonumber \\&\mathfrak {m}_{\eta }(x) = \left\{ \begin{array}{ll} \displaystyle \frac{2}{\mathfrak {s}_{\eta }(x)\sigma _{2}^{2}(x-l_{2})}, &{}\quad l_{2}<x<a,\\ \displaystyle \frac{2}{\mathfrak {s}_{\eta }(x)\sigma _{1}^{2}(x-l_{1})}, &{}\quad x\ge a. \end{array} \right. \end{aligned}$$

Using the procedure similar to [10], we consider the risk-neutral expectation of the discounted payoff

$$\begin{aligned} V(t,x)=\mathbb {E}_{x}[\mathrm {e}^{-\int _{0}^{t}r_{u}{\mathrm {d}}u}f(r_{t})], \end{aligned}$$

where $\mathbb {E}_{x}[\cdot ]=\mathbb {E}[\cdot |r_{0}=x]$. According to the relation between $r_{t}$ and $\eta _{t}$, we obtain

$$\begin{aligned} V(t,x)=\mathbb {E}[\mathrm {e}^{-\int _{0}^{t}H(\eta _{u}){\mathrm {d}}u}g(\eta _{t})|\eta _{0}=G(r_{0})]=\mathbb {E}_{G(x)}[\mathrm {e}^{-\int _{0}^{t}H(\eta _{u}){\mathrm {d}}u}g(\eta _{t})], \end{aligned}$$

where $g(\cdot )=f(H(\cdot ))$. Define the pricing semigroup $(\mathcal {P}_{t}g)(u)=\mathbb {E}_{u}[\mathrm {e}^{-\int _{0}^{t}H(\eta _{s}){\mathrm {d}}s}g(\eta _{t})]$ as the strongly continuous semigroup of self-adjoint contractions in the Hilbert space $L^{2}(I^{\eta },\mathfrak {m}_{\eta })$ of real valued, square-integrable functions on $I^{\eta }$ with the inner product $(f,g)=\int _{I^{\eta }}f(x)g(x)\mathfrak {m}_{\eta }(x){\mathrm {d}}x$. Then the infinitesimal generator of the pricing semigroup $\{\mathcal {P}_{t}\}$ is self-adjoint operator in $L^{2}(I^{\eta },\mathfrak {m}_{\eta })$. In this special case, we can use the Spectral Theorem for self-adjoint operators in Hilbert space

$$\begin{aligned} (\mathcal {P}_{t}g)(u)=\mathbb {E}_{u}[\mathrm {e}^{-\int _{0}^{t}H(\eta _{s}){\mathrm {d}}s}g(\eta _{t})]=\int _{I^{\eta }}q(t;u,y)g(y){\mathrm {d}}y, \end{aligned}$$

where q(t; u, y) is called the state-price density. Since $l_{2}$ and $\infty$ are both NONOSC natural boundaries for $\eta _{t}$, the state-price density q(t; u, y) can be reduced to

$$\begin{aligned} q(t;u,y)=\mathfrak {m}_{\eta }(y)\sum _{n=0}^{\infty }\mathrm {e}^{-\lambda _{n}t}\varphi _{n}(u)\varphi _{n}(y), \end{aligned}$$

where $\varphi _{n}(x)$ is the normalized eigenfunction associated to $\lambda _{n}$ and $\mathfrak {m}_{\eta }(y)$ is the speed density of $\eta _{t}$. The infinitesimal generator with respect to the semigroup $(\mathcal {P}_{t}g)(u)$ is

$$\begin{aligned} ( \mathcal {G}g)(u)=\left\{ \begin{array}{ll} \displaystyle \frac{1}{2}\sigma _{2}^{2}(u-l_{2})g''(u)+k(\theta _{2}-u)g'(u)-\left( \frac{1}{p}(u-a)+a\right) g(u), &{} l_{2}<u<a,\\ \displaystyle \frac{1}{2}\sigma _{1}^{2}(u-l_{1})g''(u)+k(\theta _{1}-u)g'(u)-\left( \frac{1}{1-p}(u-a)+a\right) g(u), &{} u\ge a. \end{array} \right. \end{aligned}$$

(A.23)

The domain of $\mathcal {G}$ is:

$$\begin{aligned} D(\mathcal {G})=\,&\{g \in L^{2}(I^{\eta },\mathfrak {m}_{\eta }): g,g' \in AC_{loc}(I^{\eta }), \mathcal {G}g\in L^{2}(I^{\eta },\mathfrak {m}_{\eta }), \\&\quad \text {boundary conditions at} \ l_{2}\ \ \text {and} \ \infty \}, \end{aligned}$$

where $AC_{loc}(I^{\eta })$ is the space of absolutely continuous functions over each compact subinterval of $I^{\eta }$, and the appropriate boundary conditions at $l_{2}$ and $\infty$ are

$$\begin{aligned} \lim _{u\downarrow l_{2}}\frac{f'(u)}{\mathfrak {s}_{\eta }(u)}=0, \quad \lim _{u\uparrow +\infty }\frac{f'(u)}{\mathfrak {s}_{\eta }(u)}=0. \end{aligned}$$

Now we consider the zero-coupon bond

$$\begin{aligned} \mathbb {E}_{x}[\mathrm {e}^{-\int _{0}^{t}r_{u}{\mathrm {d}}u}]=\mathbb {E}_{G(x)}[\mathrm {e}^{-\int _{0}^{t}H(\eta _{u}){\mathrm {d}}u}]. \end{aligned}$$

Define

$$\begin{aligned} \begin{array}{lllll} \tilde{\sigma }_{1}=\sigma _{1}\sqrt{1-p}, &{} \tilde{k}_{1}=k(1-p), &{} \tilde{\lambda }_{1}=(1-p)\lambda +pa, &{} \gamma _{1}=\sqrt{ \tilde{k}_{1}^{2}+2\tilde{\sigma }_{1}^{2}}, &{} k_{1}=\frac{\tilde{\lambda }_{1}-l_{1}}{\gamma _{1}}+\frac{\beta \tilde{k}_{1}}{2\gamma _{1}}, \\ \tilde{\sigma }_{2}=\sigma _{1}\sqrt{p}, &{} \tilde{k}_{2}=kp, &{} \tilde{\lambda }_{2}=p\lambda +(1-p)a, &{} \gamma _{2}=\sqrt{ \tilde{k}_{2}^{2}+2\tilde{\sigma }_{2}^{2}}, &{} k_{2}=\frac{\tilde{\lambda }_{2}-l_{2}}{\gamma _{2}}+\frac{\beta \tilde{k}_{2}}{2\gamma _{2}},\\ \zeta _{1}(x)=\frac{2\gamma _{1}}{\tilde{\sigma }_{1}^{2}}(x-l_{1}), &{} \xi _{1}=\zeta _{1}(a), &{} \zeta _{2}(x)=\frac{2\gamma _{2}}{\tilde{\sigma }_{2}^{2}}(x-l_{2}), &{} \xi _{2}=\zeta _{2}(a),&{}m=\frac{\beta -1}{2}, \end{array} \end{aligned}$$

and

$$\begin{aligned} \displaystyle&\phi (x,\lambda )=(x-l_{2})^{-\frac{\beta }{2}}\mathrm {e}^{\frac{\tilde{k}_{2}(x-l_{2})}{\tilde{\sigma }_{2}^{2}}}M_{k_{2},m}(\zeta _{2}), \\ \displaystyle&\psi (x,\lambda )=(x-l_{1})^{-\frac{\beta }{2}}\mathrm {e}^{\frac{\tilde{k}_{1}(x-l_{1})}{\tilde{\sigma }_{1}^{2}}}W_{k_{1},m}(\zeta _{1}), \end{aligned}$$

where $M_{k,m}(z)$ and $W_{k,m}(z)$ are the Whittaker functions (See more details about the Whittaker functions in [30]). Then we have the following proposition about the value of zero-coupon bond.

Proposition A.1

The zero-coupon bond $\mathbb {E}_{x}[\mathrm {e}^{-\int _{0}^{t}r_{u}{\mathrm {d}}u}]$ has the following form

$$\begin{aligned} \mathbb {E}_{x}[\mathrm {e}^{-\int _{0}^{t}r_{u}{\mathrm {d}}u}]=\mathbb {E}_{G(x)}[\mathrm {e}^{-\int _{0}^{t}H(\eta _{u}){\mathrm {d}}u}]=\sum _{n=1}^{\infty }c_{n}\mathrm {e}^{\lambda _{n}t}\varphi _{n}(G(x)), \end{aligned}$$

where the eigenvalues $0\le \lambda _{1}<\lambda _{2}<\cdots <\lambda _{n}\rightarrow \infty$ as $n\uparrow \infty$ are the simple discrete zeros of the Wronskian equation

$$\begin{aligned} \omega (\lambda )=&\frac{1}{\mathfrak {s}_{\eta }(a)}(l_{1}l_{2})^{-\frac{\beta }{2}}\mathrm {e}^{\frac{2ka}{\sigma ^{2}}}\nonumber \\&\; \left\{ \left[ \frac{k}{\sigma _{1}^{2}}-\frac{k}{\sigma _{2}^{2}}-\frac{\beta }{2l_{2}}-\frac{\beta }{2l_{1}}+\frac{2\gamma _{1}}{\tilde{\sigma }_{1}^{2}}\left( \frac{1}{2}-\frac{k_{1}}{\xi _{1}}\right) -\frac{2\gamma _{2}}{\tilde{\sigma }_{2}^{2}}\left( \frac{1}{2}-\frac{k_{2}}{\xi _{2}}\right) \right] W_{k_{1},m}(\xi _{1})M_{k_{2},m}(\xi _{2})\right. \nonumber \\&\left. -\frac{2\gamma _{1}}{\tilde{\sigma }_{1}^{2}\xi _{1}}W_{k_{1}+1,m}(\xi _{1})M_{k_{2},m}(\xi _{2})-\left( \frac{1}{2}+m+k_{2}\right) \frac{2\gamma _{2}}{\tilde{\sigma }_{2}^{2}\xi _{2}}W_{k_{1},m}(\xi _{1})M_{k_{2}+1,m}(\xi _{2})\right\} , \end{aligned}$$

(A.24)

the normalized eigenfunction $\varphi _{n}(x)$ has the form

$$\begin{aligned} \varphi _{n}(x)=\left\{ \begin{array}{ll} \sqrt{\frac{\psi (a,\lambda _{n})}{\varDelta _{n}\phi (a,\lambda _{n})}}\phi (x,\lambda _{n}), &{}\quad l_{2}<x<a,\\ sign(\phi (a,\lambda _{n})\psi (a,\lambda _{n}))\sqrt{\frac{\phi (a,\lambda _{n})}{\varDelta _{n}\psi (a,\lambda _{n})}}\psi (x,\lambda _{n}), &{}\quad x\ge a, \end{array} \right. \end{aligned}$$

and the coefficients $c_{n}$ satisfy

$$\begin{aligned} c_{n}=\int _{I^{\eta }}\varphi _{n}(x)\mathfrak {m}_{\eta }{\mathrm {d}}x, \end{aligned}$$

where $\varDelta _{n}=\frac{d\omega (\lambda )}{d\lambda }|_{\lambda =\lambda _{n}}$.

Proof

This proof is similar to Proposition 3.3 and Proposition 5.1 in [16]. $\square$

After having this proposition, the value of a zero-coupon bond in spectral expansion form can be obtained. For simulations in this paper, we used $Mathematica^{\textregistered }$10.1.0.0 running on a PC (Intel Core i7-6820 @ 2.7GHz with 16GB memory). The eigenvalues $\lambda _{n}$ are determined by finding zeros of the function $\omega (\lambda )$ using the built-in function FindRoot. The special functions $M_{k,m}(z)$ and $W_{k,m}(z)$ in eigenfunctions $\varphi _{n}(x)$ are determined using the bulit-in functions WhittakerM[k,m,z] and WhittakerW[k,m,z] and the coefficients $c_{n}$ are computed numerically using the built-in numerical integration function NIntegrate.

Figure 4 shows the convergence of bond prices under the skew CIR model, using the spectral expansion method. It can be seen that the price converges smoothly and quickly to a steady level. With more than 30 roots, we can have quite accurate evaluations for bond prices.

Fig. 4

Convergence of bond prices computed by the spectral expansion method. The top panel shows the bond prices with respect to the number of roots under the skew CIR model when $p=0.5$. The bottom left panel and the bottom right panel depict the corresponding bond prices when $p=0.2$ and $p=0.8$, respectively. The other parameters are given as follows: $\theta = 8\%$; $\sigma =0.1$; $k=0.1$; $r_0=0.05$; $a=0.04$; $T=1$

In fact, [27] mentioned that the spectral approach requires that the instantaneous discount process should be non-negative. Generally, when modeling the interest rate with the CIR process or the skew CIR process, we can obtain bond or option prices using the spectral approach with some complex calculations. But for some frequently-used process, such as the Ornstein-Uhlenbeck (OU) process and the skew OU process, it is not workable to derive the prices of bond or option with the spectral method since they have negative values. While in these cases tree approach can do. In addition, when only discussing the spectral expansions for some certain models, it is very hard to obtain the spectral expansions for the geometric Brownian motion or stochastic volatility model since they have continuous spectrum. However, these models can be dealt with by tree approach easily.