A squared binomial tree approach to discrete-time bond market modelling (Q2769691)

This paper deals with a discrete-time squared binomial bond market model \((\Omega,F,{\mathbf F},P,\{A_{n}, n=0,\ldots,N^{*}\}\), \(\{P(n,N), n=0,\ldots,N\}_{N\leq N^{*}})\), where \({\mathbf F}=\{F_{n}\}_{n=0}^{N^{*}},\;F_0\subset F_1\subset\ldots\subset F_{N^{*}}=F\); \(P\) is probability measure on \((\Omega,F)\); \(\{A_{n}\), \(n=0,\ldots,N^{*}\}\) is an \({\mathbf F}\)-predictable sequence of positive random prices of non-risky security; \(P(n,N), n=1,\ldots,N\) is a positive prices of zero-coupon bonds are traded at the moments \(n=1,\ldots,N^{*}\); \(P(N,N)=1, \forall N=0,\ldots, N^{*}\). Let us denote \(\overline\varepsilon=(\overline\varepsilon^1,\overline\varepsilon^2)= ((\varepsilon^1+1)/2,(\varepsilon^2+1)/2)\), \(\varepsilon=(\varepsilon^1,\varepsilon^2)\in\{-1;+1\}^2\), and let \(A_0>0, A_1,\ldots, A_{N^{*}}\) be the price of non-risky security given by the equality NEWLINE\[NEWLINEA_{n}=A_{n}(\varepsilon_1,\ldots,\varepsilon_{n-1})=X_1 X_2(\overline\varepsilon_1)\cdots X_{n}(\overline\varepsilon_1+\ldots+\overline\varepsilon_{n-1}),\quad n\geq 1,NEWLINE\]NEWLINE NEWLINE\[NEWLINEX_{n}:\{0,\ldots,n-1\}^2\to \mathbb{R}_{+},\quad n\geq 2,\;X_1>0,\;\varepsilon_{i}=(\varepsilon_{i}^1,\varepsilon_{i}^2).NEWLINE\]NEWLINE The bond price \(P(n,N)\) is given by equality NEWLINE\[NEWLINEP_{\varepsilon_1,\ldots,\varepsilon_{n}}(n,N)= P_{\varepsilon_1,\ldots,\varepsilon_{n-1}}(n-1,N) X_{n}(\overline\varepsilon_1+\ldots+\overline\varepsilon_{n-1})h(\varepsilon_{n};n,N),NEWLINE\]NEWLINE \(n=1,\ldots,N\), \(N=1,\ldots,N^{*}\), where \(P(0,N)>0\), \(P(N,N)=1\), \(h(\cdot ;n,N):\{1,-1\}^2\to R_{+}\) are random perturbation function such that \(h(\cdot ;N,N)=1\). The author define the functions \(X_{n}(\cdot)\) and \(h(\cdot ;\cdot ,\cdot)\) in such way that the squared binomial bond market model is arbitrage-free and path independent. The author demonstrates how the case of Ho-Lee model can be obtained. Equalities for forward rate and spot interest rate are established.NEWLINENEWLINEFor the entire collection see [Zbl 0968.00043].