Methods of PC realization of the stochastic models of stock and bond values
This article deals with algorithms for PC realization of some well-known stochastic models for stock and bond values. For the bond value \(B_{t}(T)\) the following stochastic differential equation is considered:NEWLINE\[NEWLINEB_{t}(T)=B_0+\int_{t}^{T}B_{s}(T)\beta_{s}(T) ds+ \int_{t}^{T}B_{s}(T)\sigma_{s}(T) dw_{s},NEWLINE\]NEWLINE where \(T\) is the maturity time for the bond, \(w_{s}\) is the standard Wiener process. Let \(B_{k}\equiv B_{t_{k}}(T)\), \(\beta_{k}(T)\equiv \beta_{t_{k}}(T)\), \(\sigma_{k}(T)\equiv \sigma_{t_{k}}(T)\), \(k=0,\ldots,n\), \(0=t_0<t_2< \dots<t_{n}=T\), \(t_{k}=kh\), \(0<h\ll 1\), then as an approximate value for \(B_{i}\) the author proposes NEWLINE\[NEWLINE\widetilde B_{i}^{(T)}= \widetilde B_{i+1}^{(T)}/(1+\beta_{k}(T)h+\sigma_{k}(T)\Delta w_{i}), \quad i=n-1,\dots,0,NEWLINE\]NEWLINE \(\Delta w_{i}=w_{t_{i+1}}-w_{t_{i}}=\sqrt{T/2n}\) \(\eta_{i}\), \(i=1,2,\dots,n\), \(\eta_{i}\) are independent Gaussian random variables with \(E\eta_{i}=0\), \(E\eta_{i}^2=1\), \(i=1,\ldots,n\).
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