Page 30 - CCFA Journal - Tenth Issue

数学建模 Math Modeling 加中金融

Jarrow-Yildirim Model

In the industry, a more commonly used inflation term structure model is the Jarrow-Yildirim (JY) model. The advantage of JY model
over the put-call parity is that it offers an implied interpolation that is smooth and natural.

Let ( , ) and ( , ) be the nominal and real discount bond prices at t respectively. Let ( ) = ( , ) be the inflation ratio over

[t,T].
Jarrow-Yildirim 模型

在行业中，更常用的通膨模型是 Jarrow-Yildirim（JY）模型。JY 模型相对于放空对冲平价法的优势在于它提供了一种平滑
自然的隐含插值。

设 ( , ) 和 ( , ) 分别为名义及实际折扣债券在 t 时刻的价格。又设 ( ) = ( , ) 为区间 [t,T]上的通膨率.

JY Model Setup
The Fisher equation is a well-known equation in macroeconomics that describes the relationship between nominal interest rates,
real interest rates, and expected inflation. It is named after economist Irving Fisher. The equation is stated as follows:

(1 + ) = (1 + )(1 + )

where:

r is the nominal interest rate
i is the real interest rate

is the expected rate of inflation

The Fisher equation can further be extended to

( , ) = ( ) ( , )

Let ( ) and ( ) be the nominal and real instantaneous forward rates respectively. The JY model assumes that, in the nominal

risk-neutral measure,
( ) = ( ) − ( ) + ( )

( ) = ( ) − ( ) − + ( )

,

( ) = ( )( ( ) − ( )) + ( ) ( )

where ( ), ( ), ( ) is a Brownian motion with constant correlation , , , and , , , , are constants, and
,

,

( ), ( ) are deterministic functions of t.

In JY model, both ( ) and ( ) follow the Hull-White model. It is well-known that, for ∈ { , },

( , ) = ( , ) ( , ) ( )

where
1 − ( )
( , ) =

Then it can be shown that the convexity adjustment over [ ， ] under the JY model can be given as

⎡ ( , , ( , ) ⎤

⎢ ) , ( ( , )) ⎥

= ( , ) ⎥
⎢

⎢ − , ( , ) ⎥

⎣ + ⎦

CCFA JOURNAL OF FINANCE March 2023
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