# 利率模型 Interest Rate Models MATH97114

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Integrating both sides and taking exponentials of both sides we get the following expression for the unit zero coupon bond:
$$P_{t}(x)=e^{-\int_{0}^{x} f_{t}(s) d s} .$$
If we rewrite using the notation $x=T-t$, we get:
$$P_{t}(x)=e^{-x r_{t}(x)}$$

for the relationship between the discount unit bond and the spot rate. In terms of the forward rates it reads:
$$r_{t}(x)=\frac{1}{x} \int_{0}^{x} f_{t}(s) d s .$$
This relation can be inverted to express the forward rates as function of the spot rate:
$$f_{t}(x)=r_{t}(x)+x \frac{\partial}{\partial r} r_{t}(x) .$$

## MATH97114COURSE NOTES ：

If $X^{(1)}, X^{(2)} \ldots$ form a sequence of random vectors, each with the same law in $\mathbb{R}^{m}$, then the strong law of large numbers tells us that the estimators
$$\mu_{N}=\frac{1}{N} \sum_{i=1}^{N} X^{(i)}$$
and
$$Q_{N}=\frac{1}{N-1} \sum_{i=1}^{N}\left(X^{(i)}-\mu_{N}\right) \otimes\left(X^{(i)}-\mu_{N}\right)$$