The Heath–Jarrow–Morton (HJM) framework is a general framework to model the evolution of interest rate curves – instantaneous forward rate curves in particular (as opposed to simple forward rates).

Heath–Jarrow–Morton（HJM）框架由 David Heath， Robert Jarrow 和Andrew Morton提出，它是一个对利率曲线的运动进行建模的一般框架，尤其是 瞬时远期利率 （instantaneous forward rate）曲线。 若将零息债券（zero-coupon bond）在时间 t 的价格表示为 P (t,T) ，根据 风险中性定价，到期时间为 T 的零息债券在时间 t 的价格为 P (t,T)=\mathrm {E}^Q\left [e^ {-\int_t^T r_u\ du}\bigg|\mathcal {F}_t\right]\\ 其中 Q 是风险中性测度（risk-neutral measure）。

HJM (Heath-Jarrow-Morton) model is a very general framework used for pricing interest rates and credit derivatives. Big banks trade hundreds, sometimes even thousands, of different types of

HJM model under the physical measure P: dB(t;T) = B(t;T) R(t) (t;T) + 1 2 (˙(t;T))2 dt B(t;T)˙(t;T)dW(t): This is a geometric Brownian motion model with drift term (t;T) := R(t) (t;T) + 1 2 (˙(t;T))2 and volatility ˙(t;T). Thus, the market price of risk equation for this particular bond with expiry T is: to nd an adapted process (t) such that

2018年6月5日 · HJM generalizes the BSM model in two important ways. One, the BSM model assumes deterministic interest rates. Two, the BSM has only one traded risky asset. The HJM model has stochastic interest rates and a continuum of traded risky assets.

Morton (HJM) pricing paradigm. The equivalence between the forward rate and the conditional expectation of the short rate under the forward measure is discussed. Elaborating on the work of Geman, El Karoui and Rochet (1995) using the bond price as a num´eraire, the short rate’s dy-namics is obtained under the forward measure. Consequently ...

所以在本文的HJM模型，我们将从 f(t,T) 出发得到债券价格的表达式。与之前类似的，我们假设对于所有T有： df(t,T)=\alpha(t,T)dt + \sigma(t,T)dW_t \; \forall T \geq t\\ 注意到 \alpha(t,T), \sigma(t,T) 都是 \mathcal{F_t} 可测的，且 f(0,T) 在当前已知。

HJM模型主要是用来描述 利率期限结构模型 的，有如下重要的概念。 Main Concept: 1、Instantaneous Forward Rate. 若是Forward Rate则需要用 f(t,T_1,T_2) 来表示。 当 T_1\rightarrow T_2 时即为Instantaneous Forward Rate，用 f(t,T) 来表示。 2、Short rate

HJM 模型的主要方法是无套利分析法，即在n个因子风险模型下，可以通过一个无风险资产和n个风险资产的组合构造资产市场上的所有资产。 给定 债券 波动率的期限结构，就可以得到债券定价的全部信息，它是无套利模型的基准模型。

赫斯–加罗–莫顿模型（HeathJarrow-Morton model; HJM model ）是2016年公布的管理科学技术名词。 应用无套利分析法，即在n个风险因子模型下，可以通过一个无风险资产和n个风险资产的组合构成资产市场上所有的资产。 《管理科学技术名词》第一版。 [1] 赫斯–加罗–莫顿模型（HeathJarrow-Morton model; HJM model ）是2016年公布的管理科学技术名词。