仿射期限結構模型中的“仿射”一詞是什麼意思？

我是數學金融領域的新手，想了解仿射期限結構模型中“仿射”一詞的直覺、物理和數學含義。任何文獻參考也將不勝感激。

另一方面，它是否與各種隨機過程的多尺度和自相似特性有關？

在仿射期限結構模型中，零息債券價格可以寫為 $ P\left(t, T\right) = e^{A\left(t, T\right) - B\left(t, T\right) r_t} $ . 零票面利率 $ R\left(t, T\right) = -\frac{\ln \left(P\left(t, T\right) \right)}{T - t} $ 因此是短期內的仿射函式 $ r_t $ .

許多教科書對這些模型都有一些專門的段落；如果你想要一本詳盡的利率模型專著，我推薦 Brigo 和 Mercurio 的利率模型。

根據莫妮卡皮亞澤西的說法：

The word “affine term structure model” is often used in different ways. I will use the
word to describe any arbitrage-free model in which [zero coupon] bond yields are affine 
(constant plus-linear) functions of some state vector x.

Affine models are thus a special class of term structure models, which write the yield 
y(τ) of a τ-period bond as
                                  y(τ) = A(τ) + B(τ) x
for coefficients A(τ) and B(τ) that depend on maturity τ. The functions A(τ) and B(τ)
make these yield equations consistent with each other for different values of τ. The
functions also make the yield equations consistent with the state dynamics.

The main advantage of affine models is tractability. Having tractable solutions for
bond yields is useful because otherwise yields need to be computed with Monte Carlo
methods or solution methods for PDEs. Both approaches are computationally costly,
[...].

The literature on bond pricing starting with Vasicek (1977) and Cox et al. (1985), 
therefore has focused on closed-form solutions.The riskless rate in these early
setups was the only state variable in the economy so that all bond yields were 
perfectly correlated. A number of extensions of these setups followed both in terms
of the number of state variables and the data-generating processes for these 
variables. Duffie and Kan (1996) finally provided a more complete characterization of
models with affine bond yields.

資料來源：https ://web.stanford.edu/~piazzesi/s.pdf

這個定義比上面的更一般，因為 $ x_t $ （“狀態變數”）可以是向量而不是標量 $ r_t $ （通常代表瞬時無風險利率）在最早的模型中使用。

引用自：https://quant.stackexchange.com/questions/54344