Abstract
In this chapter, we first present the concepts and the relationships between zero coupon bond, short rate and forward rate which are essential for interest rate term structure modelling.
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Notes
- 1.
We know that \( {d\left (\int _0^t f(t,u)du\right ) = f(t,t)dt + \int _0^t df(t,u)du}\) for any smooth function f(t, u). Actually, \( {d\left (\int _0^t f(t,u)du\right ) = \lim\limits _{\varDelta t\rightarrow 0} \left [ \int _0^{t+\varDelta t} f(t+\varDelta t,u)du - \int _0^t f(t,u)du \right ] = \lim\limits _{\varDelta t\rightarrow 0} \left [ \int _t^{t+\varDelta t}\right .}\) \( {\left . f(t+\varDelta t,u)du + \int _0^t f(t+\varDelta t,u)du - \int _0^t f(t,u)du \right ] = f(t,t)dt + \int _0^t df(t,u)du}\). Similarly, we have \( {d\left (\int _0^t f(t,u)dW_u\right ) = f(t,t)dW_t + \int _0^t df(t,u)dW_u}\).
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Chan, R.H., Guo, Y.Z., Lee, S.T., Li, X. (2019). Interest Rate Term Structure Modelling. In: Financial Mathematics, Derivatives and Structured Products. Springer, Singapore. https://doi.org/10.1007/978-981-13-3696-6_21
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DOI: https://doi.org/10.1007/978-981-13-3696-6_21
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