大浩浩的笔记课堂——FRM考试学习笔记086

Topic 15 The Art of Term Structure Models:Volatility and Distribution

1.Model 3:

time-dependent volatility:

⑴improvement:

It assigns a specific parameterization of time-dependent volatility.

⑵formula:

dr=λ(t)dt+σ(t)dw

⑶characteristics:

①They are very flexible and can incorporate increasing,decreasing and constant short-term rate volatilities between periods.

②This flexibility is useful for valuing interest rate caps and floors.

2.The Cox-Ingersoll-Ross(CIR) model:

volatility as a function of the short rate:

⑴formula:

⑵advantages:

①CIR model withe respect to not allowing negative rates.

②Basis-point volatility increases during periods of high inflation.

⑶disadvantage:

Option prices from the CIR distribution may differ significantly from lognormal or normal distribution.

3.Model 4:

lognormal model:

⑴assumption:

①Yield volatility,σ,is constant;

②But basis-point volatility,σr,increases with the level of the short-term rate.

③The natural log of the short-term rate follows a normal distribution.

⑵important models:

There are 2 lognormal models of importance:

①lognormal with deterministic drift:

Ho-Lee model based on the natural logarithm

②lognormal with mean reversion:

Black-Karasinski model

4.The Ho-Lee model:

a lognormal model with drift:

The Ho-Lee model based on the natural logarithm:

d[In(r)]=a(t)dt+σdw

The drifts terms are multiplicative.

5.The Black-Karasinski model:

a lognormal model with mean reversion:

⑴formula:

d[In(r)]=k(t)[Inθ(t)-In(r)]dt+σ(t)dw

①It is time-varying and mean reverting.

②It refers to Inθ(t)at a speed of k(t)with a volatility of σ(t).

⑵characteristics:

①Mean reversion controls the term structure of volatility.

②Time dependent volatility controls the future volatility of the short-term rate.

③The time intervals between interest rate changes are recalibrated to force the node to recombine.

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