Unified Model Arbitrage-Free Term Structure of Flow Risks

Abstract

Interest movement models are important to financial modeling because they can be used for valuing any financial instruments whose values are affected by interest rate movements. Specifically, we can classify the interest rate movement models into two categories: equilibrium models and no-arbitrage models. The equilibrium models emphasize the equilibrium concept. However, the no-arbitrage models argue that the term structure movements should satisfy the no-arbitrage condition. The arbitrage-free interest rate model is an extension of the Black-Scholes model to value interest rate derivatives. The model valuation is assured to be consistent with the observed yield curve in valuing interest rate derivatives and providing accurate pricing of interest rate contingent claims. Therefore, it is widely used for portfolio management and other capital market activities.

The authors would like to thank the participants at Financial Innovation Conference, Owen School, Vanderbilt University 2008 for their comments. We would also like to thank Eric A Anderson and Charles Fenner for their insights into energy contracts. We are responsible for any errors in the chapter.

Appendices

Appendix A: A General Form of the Arrow-Debreu Primitive Securities

The Arrow-Debreu primitive securities at time n are equal to a vector with (n + 1) elements, which we can calculate by multiplying the following n matrixes.

$$ {\displaystyle \begin{array}{c}\ \left[\frac{e^{-\rho (0)}}{1+{\delta}^0}\kern0.5em \frac{e^{-\rho (0)}}{1+{\delta}^0}\right]\times \left[\begin{array}{ccc}\frac{e^{-\rho (1)}\delta }{1+{\delta}^1}& \frac{e^{-\rho (1)}\delta }{1+{\delta}^1}& 0\\ {}0& \frac{e^{-\rho (1)}}{1+{\delta}^1}& \frac{e^{-\rho (1)}}{1+{\delta}^1}\end{array}\right]\times \left[\begin{array}{cccc}\frac{e^{-\rho (2)}{\delta}^2}{1+{\delta}^2}& \frac{e^{-\rho (2)}{\delta}^2}{1+{\delta}^2}& 0& 0\\ {}0& \frac{e^{-\rho (2)}\delta }{1+{\delta}^2}& \frac{e^{-\rho (2)}\delta }{1+{\delta}^2}& 0\\ {}0& 0& \frac{e^{-\rho (2)}}{1+{\delta}^2}& \frac{e^{-\rho (2)}}{1+{\delta}^2}\end{array}\right]\\ {}\kern3.5em 1\times 2\kern6.75em 2\times 3\kern12.25em 3\times 4\\ {}\\ {}\times \left[\begin{array}{ccccc}\frac{e^{-\rho (3)}{\delta}^3}{1+{\delta}^3}& \frac{e^{-\rho (3)}{\delta}^3}{1+{\delta}^3}& 0& 0& 0\\ {}0& \frac{e^{-\rho (3)}{\delta}^2}{1+{\delta}^3}& \frac{e^{-\rho (3)}{\delta}^2}{1+{\delta}^3}& 0& 0\\ {}0& 0& \frac{e^{-\rho (3)}{\delta}^1}{1+{\delta}^3}& \frac{e^{-\rho (3)}{\delta}^1}{1+{\delta}^3}& 0\\ {}0& 0& 0& \frac{e^{-\rho (3)}}{1+{\delta}^3}& \frac{e^{-\rho (3)}}{1+{\delta}^3}\end{array}\right]\kern0.75em \cdots \cdots \cdots \\ {}\kern9em 4\times 5\\ {}\\ {}\cdots \cdots \cdots \kern0.75em \times \kern0.5em \left[\begin{array}{cccccc}\frac{e^{-\rho \left(n-1\right)}{\delta}^{n-1}}{1+{\delta}^{n-1}}& \frac{e^{-\rho \left(n-1\right)}{\delta}^{n-1}}{1+{\delta}^{n-1}}& 0& \cdots & 0& 0\\ {}0& \frac{e^{-\rho \left(n-1\right)}{\delta}^{n-2}}{1+{\delta}^{n-1}}& \frac{e^{-\rho \left(n-1\right)}{\delta}^{n-1}}{1+{\delta}^{n-1}}& \cdots & 0& 0\\ {}\vdots & \vdots & \frac{e^{-\rho \left(n-1\right)}{\delta}^{n-3}}{1+{\delta}^{n-1}}& \frac{e^{-\rho \left(n-1\right)}{\delta}^{n-3}}{1+{\delta}^{n-1}}& \vdots & \vdots \\ {}0& 0& \cdots & \ddots & \ddots & 0\\ {}0& 0& \cdots & 0& \frac{e^{-\rho \left(n-1\right)}}{1+{\delta}^{n-1}}& \frac{e^{-\rho \left(n-1\right)}}{1+{\delta}^{n-1}}\end{array}\right]\\ {}\kern15.25em n\times \left(n+1\right)\end{array}} $$

For example, when n is equal to 1, the Arrow-Debreu primitive securities at time 1 is the first matrix such that A(1, 1) is equal to $\frac{e^{-\rho (0)}}{1+{\delta}^0}$ and A(1, 0) is equal to $\frac{e^{-\rho (0)}}{1+{\delta}^0}$. When n is equal to 2, we can generate the Arrow-Debreu primitive securities at time 2 by multiplying the first two matrixes such that

$$ {\displaystyle \begin{array}{l}\left[\frac{e^{-\rho (0)}}{1+{\delta}^0}\kern0.5em \frac{e^{-\rho (0)}}{1+{\delta}^0}\right]\times \left[\begin{array}{ccc}\frac{e^{-\rho (1)}\delta }{1+{\delta}^1}& \frac{e^{-\rho (1)}\delta }{1+{\delta}^1}& 0\\ {}0& \frac{e^{-\rho (1)}}{1+{\delta}^1}& \frac{e^{-\rho (1)}}{1+{\delta}^1}\end{array}\right]\\ {}\kern0.5em =\kern0.5em \left[\frac{e^{\sum \limits_{t=0}^1-\rho (t)}\delta }{\prod \limits_{t=1}^11+{\delta}^t}\kern1em \frac{e^{\sum \limits_{t=0}^1-\rho (t)}\left(1+\delta \right)}{\prod \limits_{t=1}^11+{\delta}^t}\kern1em \frac{e^{\sum \limits_{t=0}^1-\rho (t)}1}{\prod \limits_{t=1}^11+{\delta}^t}\right]\\ {}\kern2.25em 1\times 2\kern8em 2\times 3\kern11.75em 1\times 3\kern2em \end{array}} $$

The first element of the vector is A(2, 2), the second element is A(2, 1), and the last element is A(2, 0). Similarly, we can generate the Arrow-Debreu primitive securities at the following periods and determine them accordingly. The intuition behind the general form of the Arrow-Debreu primitive securities is the recursive relationship among A(n, i), A(n − 1, i), and A(n − 1, i − 1) for any n and i such as $A\left(n,i\right)=\frac{e^{-r\left(n-1\right)}{\delta}^i}{\left(1+{\delta}^{n-1}\right)}A\left(n-1,i\right)+\frac{e^{-r\left(n-1\right)}{\delta}^{i-1}}{\left(1+{\delta}^{n-1}\right)}A\left(n-1,i-1\right)$.

Appendix B: Three-Factor Unified Model

The model is provided in Ho-Lee (2007). For completeness, we specify the model in this appendix. Let ${B}_{i,j,l}^n(T)$ be the T year bond price at time n, at state (i, j, k). Then ${B}_{i,j,l}^n(T)$ is specified by combining three one-factor models. Specifically, we have

$$ {\displaystyle \begin{array}{l}{B}_{i,j,l}^n(T)=\frac{P\left(n+T\right)}{P(n)}\prod \limits_{k=1}^n\left(\frac{1+{\delta}_{0,r}^{k-1}\left(n-k\right)}{1+{\delta}_{0,r}^{k-1}\left(n-k+T\right)}\right)\prod \limits_{k=0}^{i-1}{\delta}_{k,r}^{n-1}(T)\\ {}\kern3.25em \times \frac{S\left(n-1+T\right)}{S\left(n-1\right)}\prod \limits_{k=1}^n\left(\frac{1+{\delta}_{0,h}^{k-1}\left(n-k\right)}{1+{\delta}_{0,h}^{k-1}\left(n-k+T\right)}\right)\prod \limits_{k=0}^{j-1}{\delta}_{k,h}^{n-1}(T)\\ {}\kern3.25em \times \frac{L\left(n+T\right)}{L(n)}\prod \limits_{k=1}^n\left(\frac{1+{\delta}_{0,l}^{k-1}\left(n-k\right)}{1+{\delta}_{0,l}^{k-1}\left(n-k+T\right)}\right)\prod \limits_{k=0}^{l-1}{\delta}_{k,l}^{n-1}(T)\end{array}} $$

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where

$$ {\displaystyle \begin{array}{c}{\delta}_{i,r}^n(T)={\delta}_{i,r}^n{\delta}_{i,r}^{n+1}\left(T-1\right)\left(\frac{1+{\delta}_{i+1,r}^{n+1}\left(T-1\right)}{1+{\delta}_{i,r}^{n+1}\left(T-1\right)}\right)\\ {}{\delta}_{i,h}^n(T)={\delta}_{i,h}^n{\delta}_{i,h}^{n+1}\left(T-1\right)\left(\frac{1+{\delta}_{i+1,h}^{n+1}\left(T-1\right)}{1+{\delta}_{i,h}^{n+1}\left(T-1\right)}\right)\\ {}{\delta}_{i,l}^n(T)={\delta}_{i,l}^n{\delta}_{i,l}^{n+1}\left(T-1\right)\left(\frac{1+{\delta}_{i+1,l}^{n+1}\left(T-1\right)}{1+{\delta}_{i,l}^{n+1}\left(T-1\right)}\right)\end{array}} $$

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and the one-period forward volatilities are given by definition,

$$ {\displaystyle \begin{array}{c}{\delta}_{i,r}^m(1)={\delta}_{i,r}^m=\exp \left(-2\cdot {\sigma}_r(m)\min \left({h}_{i,r}^m,{H}_r\right)\Delta {t}^{3/2}\right)\\ {}{\delta}_{i,h}^m(1)={\delta}_{i,h}^m=\exp \left(-2\cdot {\sigma}_h(m)\min \left({h}_{i,h}^m,{H}_h\right)\Delta {t}^{3/2}\right)\\ {}{\delta}_{i,l}^m(1)={\delta}_{i,l}^m=\exp \left(-2\cdot {\sigma}_l(m)\min \left({h}_{i,l}^m,{H}_l\right)\Delta {t}^{3/2}\right)\end{array}} $$

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Where the functions σ_m(n) = (a + bn) exp (−cn) + d where m = r, h, l is specified by the parameters a, b, c, and d, which can be obtained from the calibration to the market price of an option on CDS, etc. This specification of the implied volatility function allows for a broad range of shapes including downward sloping or dumped shapes.

Using the direct extension, we can specify the one-period hazard rates for the two-factor model for any future period m and state i, and ${h}_{i,1}^m$ and ${h}_{i,2}^m$ are defined by

$$ {\displaystyle \begin{array}{l}{h}_{i,r}^m\Delta t=-\log \left(\frac{P\left(m+1\right)}{P(m)}\right)-\sum \limits_{k=1}^m\log \left(\frac{1+{\delta}_{0,r}^{k-1}\left(m-k\right)}{1+{\delta}_{0,r}^{k-1}\left(m-k+1\right)}\right)-\sum \limits_{k=0}^{i-1}\log \left({\delta}_{k,r}^{m-1}(1)\right)\\ {}{h}_{i,h}^m\Delta t=-\log \left(\frac{S(m)}{S\left(m-1\right)}\right)-\sum \limits_{k=1}^m\log \left(\frac{1+{\delta}_{0,h}^{k-1}\left(m-k\right)}{1+{\delta}_{0,h}^{k-1}\left(m-k+1\right)}\right)-\sum \limits_{k=0}^{i-1}\log \left({\delta}_{k,h}^{m-1}(1)\right)\\ {}{h}_{i,l}^m\Delta t=-\log \left(\frac{L(m)}{L\left(m-1\right)}\right)-\sum \limits_{k=1}^m\log \left(\frac{1+{\delta}_{0,l}^{k-1}\left(m-k\right)}{1+{\delta}_{0,l}^{k-1}\left(m-k+1\right)}\right)-\sum \limits_{k=0}^{i-1}\log \left({\delta}_{k,l}^{m-1}(1)\right)\end{array}} $$

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