PAK Study Manual Volatility Modeling Structured Finance Interest Rate Models

PAK Study Manual
Quantitative Finance and Investment Advanced (QFIA) Exam
Fall 2014 Edition
CDS
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Derivatives
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Interest Rate Models
Performance Measurement
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QFI Exam 2013-2014
PAK Study Manual
IRM-Theory-and-Practice-4
IRM-Theory-and-Practice-4
Two-Factor Short Rate Models
Background
The chapter introduces one major two factor models:

The Two-Additive-Factor Gaussian Model (G2++)
We introduce the dynamics of the model, the pricing of zero coupon bonds/European options on zero coupon
bonds/Caps/Floors/Swaptions, the correlation structures in the model, construction of a tree and calibration of
this model to market data.
The previous chapter (IRM-Theory-and-Practice-3) was devoted to one factor model, r(t). Under such model, the
knowledge of the short rate dynamics (dr(t)) leads to the knowledge of bond prices and of the whole zero coupon
interest rate curve at time t via the formula:
𝑇
𝑃(𝑡, 𝑇) = 𝐸𝑡 �𝑒𝑥𝑝 �− � 𝑟𝑠 𝑑𝑠��

𝑡
Choosing a poor model for the evolution of the short rate r(t) will result in a poor evolution of the yield
curve
Consider the one factor Vasicek model. We have seen that since this model is an ATS (Affine Term Structure, as
explained in IRM-Theory-and-Practice-3), we have:
𝑃(𝑡, 𝑇) = 𝐴(𝑡, 𝑇) × 𝑒 𝑟(𝑡)×𝐵(𝑡,𝑇)
We also saw that this implies that the continuously compounded spot rate at time t is of the form:
𝑅(𝑡, 𝑇) = −
𝑙𝑛[𝐴(𝑡, 𝑇)] 𝐵(𝑡, 𝑇)
+
𝑟 = 𝑎(𝑡, 𝑇) + 𝑏(𝑡, 𝑇)𝑟𝑡
𝑇−𝑡
𝑇−𝑡 𝑡
Consider a payoff depending on the joint distribution of two such rates at time t, say:

𝑅(𝑡, 𝑡 + 1) 𝑎𝑛𝑑 𝑅(𝑡, 𝑡 + 10)
The correlation between the two rates plays a crucial role in calculating this payoff.
Let us calculate this correlation:
Thus:
𝐶𝑜𝑟𝑟[𝑅(𝑡, 𝑇1 ), 𝑅(𝑡, 𝑇2 )] = 𝑐𝑜𝑟𝑟[𝑎(𝑡, 𝑇1 ) + 𝑏(𝑡, 𝑇1 )𝑟𝑡 ; 𝑎(𝑡, 𝑇2 ) + 𝑏(𝑡, 𝑇2 )𝑟𝑡 ] = 1

A weakness of the one factor model: At every time instant, rates for all maturities in the curve are
perfectly correlated.

Truly, interest rates are known to exhibit some decorrelation, so that a more satisfactory model of curve
evolution has to be defined.
1
QFI Exam 2013-2014
PAK Study Manual
IRM-Theory-and-Practice-4
Nonetheless, one factor models such as HW, BK, CIR++, EEV may still prove useful when the product to be priced
does not depend on the correlations of different rates but depends at every instant on a single rate of the whole
interest rate curve.
Whenever the correlation plays a major role, or whenever higher precision is needed, we need to move to a
model allowing for more realistic correlation pattern. This is achieved with multi factor models like the G2++ we
are about to introduce.
Replace the Vasicek model with the hypothetical Gaussian 2 factor models as follows:
𝑟𝑡 = 𝑥𝑡 + 𝑦𝑡
𝑑𝑥𝑡 = 𝑘𝑥 × (𝜃𝑥 − 𝑥𝑡 )𝑑𝑡 + 𝜎𝑥 𝑑𝑊1 (𝑡)
And
With:
𝑑𝑦𝑡 = 𝑘𝑦 × �𝜃𝑦 − 𝑦𝑡 �𝑑𝑡 + 𝜎𝑦 𝑑𝑊2 (𝑡)
𝐶𝑜𝑟𝑟(𝑊1 , 𝑊2 ) = 𝜌𝑑𝑡
For this kind of model, the bond price is an affine function of two variables:
𝑃(𝑡, 𝑇) = 𝐴(𝑡, 𝑇)𝑒𝑥𝑝[−𝐵 𝑥 (𝑡, 𝑇) × (𝑥𝑡 ) − 𝐵 𝑦 (𝑡, 𝑇) × (𝑦𝑡 )]
Now, computing the correlation between two points of the curve:
𝐶𝑜𝑟𝑟[𝑅(𝑡, 𝑇1 ), 𝑅(𝑡, 𝑇2 )] = 𝑐𝑜𝑟𝑟[𝑏 𝑥 (𝑡, 𝑇1 )𝑥𝑡 + 𝑏 𝑦 (𝑡, 𝑇1 )𝑦𝑡 ; 𝑏 𝑥 (𝑡, 𝑇2 )𝑥𝑡 + 𝑏 𝑦 (𝑡, 𝑇2 )𝑦𝑡 ]
This is no longer always equal to one. It depends on the correlation between the two factors x and y.
This methodology can be extended to more than two factors. But, remember that the choice of the number of
factors is a compromise between:
1.
2.
Numerical efficiency and
Capability of the model to represent realistic correlation patterns and fitting of enough market data.
Historical analysis of the yield curve, based on PCA analysis (Tsay-9, in the syllabus) suggests that two components
can explain more than 85% of the variation of the entire yield curve.
The syllabus sticks to the two factors model (G2++):
𝑟𝑡 = 𝑥𝑡 + 𝑦𝑡
𝑑𝑥𝑡 = 𝑘𝑥 × (𝜃𝑥 − 𝑥𝑡 )𝑑𝑡 + 𝜎𝑥 𝑑𝑊1 (𝑡)
And
𝑑𝑦𝑡 = 𝑘𝑦 × �𝜃𝑦 − 𝑦𝑡 �𝑑𝑡 + 𝜎𝑦 𝑑𝑊2 (𝑡)
The chapter answers some interesting questions:


Is the two factor model like the G2++ flexible enough to be calibrated to a large set of swaptions, or even
to caps and swaptions at the same time?
How many swaptions can be calibrated in a sufficiently satisfactory way?
2
QFI Exam 2013-2014



PAK Study Manual
IRM-Theory-and-Practice-4
What is the evolution of the term structure of volatilities as implied by the calibrated model?
How can one implement trees for models such as G2++?
Is Monte Carlo feasible?
Can the model be used for Quanto-like products and products depending on more than an interest rate?
The Two-Additive-Factor Gaussian Model (G2++)
First:
“The dynamics of the G2++ are on the fall 2013 formula sheet”
The short rate dynamics under the risk adjusted measure Q of the G2++ model is given by:
𝑟(𝑡) = 𝑥(𝑡) + 𝑦(𝑡) + 𝜑(𝑡);
Where
𝑑𝑥(𝑡) = −𝑎𝑥(𝑡)𝑑𝑡 + 𝜎𝑑𝑊1 (𝑡),
And
𝑟(0) = 𝑟0
𝑥(0) = 0
𝑑𝑦(𝑡) = −𝑏𝑦(𝑡)𝑑𝑡 + 𝜂𝑑𝑊2 (𝑡),
𝑦(0) = 0
𝑊1 𝑎𝑛𝑑 𝑊2 𝑎𝑟𝑒 𝑡𝑤𝑜 𝐵𝑟𝑜𝑤𝑛𝑖𝑎𝑛 𝑚𝑜𝑡𝑖𝑜𝑛 𝑤𝑖𝑡ℎ 𝒊𝒏𝒔𝒕𝒂𝒏𝒕𝒂𝒏𝒆𝒐𝒖𝒔 𝒄𝒐𝒓𝒓𝒆𝒍𝒂𝒕𝒊𝒐𝒏 𝝆
𝑟0 , 𝑎, 𝜎, 𝑏, 𝜂 𝑎𝑟𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠
The function (𝜑(𝑡)) is deterministic and well defined (for all time t from 0 to typically 10, 30 or 50 years). It is
chosen so as to fit the initial term structure (just like in IRM-Theory-and-Practice-4). Also:
𝝋(𝟎) = 𝒓(𝟎) = 𝒓𝟎
Second: some comments about the G2++ model:





The G2++ model is useful in practice, despite the feature of theoretically possible negative interest rates.
The model is analytically tractable: Possible to price plain vanilla instruments and derive spot and forward
rates/curves at any point in time.
De facto: an “almost” negative perfect correlation of factors x(t) and y(t) is introduced, resulting in a
more precise calibration to correlation-based products like caps and European swaptions.
Useful model when pricing out-of-the money exotic instruments after calibration to at the money plain
vanilla products.
Related to the Hull White two factors model, but has less complicated formula than the HW, and is easier
to implement in practice than the HW.
Third: We use the G2++ to do some real things:
Integration of expressions dx(t) and dy(t) implies:
𝑟(𝑡) = 𝑥(𝑠)𝑒
Therefore:
−𝑎(𝑡−𝑠)
+ 𝑦(𝑠)𝑒
−𝑏(𝑡−𝑠)
𝑡
+ 𝜎�𝑒
𝑠
−𝑎(𝑡−𝑢)
𝑡
𝑑𝑊1 (𝑢) + 𝜂 � 𝑒 −𝑏(𝑡−𝑢) 𝑑𝑊2 (𝑢) + 𝜑(𝑡)
𝑠
“Note that the solution to the SDE is not on the formula sheet”
3
QFI Exam 2013-2014
PAK Study Manual
IRM-Theory-and-Practice-4
“The condition mean and variance of the G2++ are on the formula sheet”
Conditional on the information up to time s (𝐹𝑠 ), the short rate under the G2++ model is normally distributed with:
And
In particular:
𝑉𝑎𝑟�𝑟(𝑡)� =
𝐸�𝑟(𝑡)� = 𝑥(𝑠)𝑒 −𝑎(𝑡−𝑠) + 𝑦(𝑠)𝑒 −𝑏(𝑡−𝑠) + 𝜑(𝑡)
𝜎2
𝜂2
𝜎𝜂
�1 − 𝑒 −2𝑎×(𝑡−𝑠) � + �1 − 𝑒 −2𝑏×(𝑡−𝑠) � + 2𝜌
�1 − 𝑒 −(𝑎+𝑏)×(𝑡−𝑠) �
2𝑎
2𝑏
𝑎+𝑏
𝒕
𝒕
𝒓(𝒕) = 𝝈 � 𝒆−𝒂(𝒕−𝒖) 𝒅𝑾𝟏 (𝒖) + 𝜼 � 𝒆−𝒃(𝒕−𝒖) 𝒅𝑾𝟐 (𝒖) + 𝝋(𝒕)
𝟎
𝟎
“Exam Question: Be ready to answer questions like finding out the probability that the model G2++ results in
negative rates at time t”
Pricing of a zero coupon bond
In order to compute it, we need the following Lemma:
For each t, T, the random variable:
𝑇
𝐼(𝑡, 𝑇) = �[𝑥(𝑢) + 𝑦(𝑢)] 𝑑𝑢
𝑡
Is normally distributed with mean M(t,T) and variance V(t,T) respectively given by:
𝑀(𝑡, 𝑇) =
And
𝑉(𝑡, 𝑇) =
1 − 𝑒 −𝑎(𝑇−𝑡)
1 − 𝑒 −𝑏(𝑇−𝑡)
𝑥(𝑡) +
𝑦(𝑡)
𝑎
𝑏
𝜎2
2 −𝑎(𝑇−𝑡)
1 −2𝑎(𝑇−𝑡)
3
𝜂2
2
1
3
×
�𝑇
−
𝑡
+
𝑒
−
𝑒
−
�
+
× �𝑇 − 𝑡 + 𝑒 −𝑏(𝑇−𝑡) − 𝑒 −2𝑏(𝑇−𝑡) − �
𝑎2
𝑏2
𝑎
2𝑎
2𝑎
𝑏
2𝑏
2𝑏
+2𝜌
𝜎𝜂
𝑒 −𝑎(𝑇−𝑡) − 1 𝑒 −𝑏(𝑇−𝑡) − 1 𝑒 −(𝑎+𝑏)(𝑇−𝑡) − 1
�𝑇 − 𝑡 +
+
−
�
𝑎
𝑏
𝑎+𝑏
𝑎𝑏
“These M and V are on the formula sheet”
Now we get state the price of a zero coupon bond under the model
4
QFI Exam 2013-2014
PAK Study Manual
IRM-Theory-and-Practice-4
Under the G2++ model, the price a zero coupon bond at time t, with maturity at time T is:
𝑇
1
𝑃(𝑡, 𝑇) = 𝑒𝑥𝑝 �− � 𝜑(𝑢) 𝑑𝑢 − 𝑀(𝑡, 𝑇) + 𝑉(𝑡, 𝑇)�
2
𝑡
“The price under G2++ is also on the formula sheet”
Denote:
𝑃𝑀 (0, 𝑇) = 𝑇ℎ𝑒 𝑡𝑒𝑟𝑚 𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒 𝑜𝑓 𝑚𝑎𝑟𝑘𝑒𝑡 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑑𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑓𝑎𝑐𝑡𝑜𝑟𝑠
Thus:
𝑓 𝑀 (0, 𝑇) = −
Thus:
𝜕𝑙𝑛[𝑃𝑀 (0, 𝑇)]
= 𝑇ℎ𝑒 𝑚𝑎𝑟𝑘𝑒𝑡 𝑖𝑚𝑝𝑙𝑖𝑒𝑑 𝑖𝑛𝑠𝑡𝑎𝑛𝑡𝑎𝑛𝑒𝑜𝑢𝑠 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑟𝑎𝑡𝑒 𝑎𝑡 𝑡𝑖𝑚𝑒 0 𝑓𝑜𝑟 𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑇
𝜕𝑇
The G2++ model fits the currently observed term structure of discount factors if and only if, for each T:
𝜑(𝑇) = 𝑓 𝑀 (0, 𝑇) +
That is, if and only if:
𝜂2
𝜎𝜂
𝜎2
−𝑎𝑇 ]2
[1
×
−
𝑒
+
× [1 − 𝑒 −𝑏𝑇 ]2 + 𝜌 (1 − 𝑒 −𝑎𝑇 )(1 − 𝑒 −𝑏𝑇 )
2
2
2𝑎
2𝑏
𝑎𝑏
𝑇
𝑒𝑥𝑝 �− � 𝜑(𝑢)𝑑𝑢� =
𝑡
“Both equations are on the formula sheet”
Be able to restate the price of the zero coupon bond P(t,T) with this fitting condition:

𝑷(𝒕, 𝑻) =
Where:

𝐴(𝑡, 𝑇) =
2

𝑷𝑴 (𝟎, 𝑻)
𝒆𝒙𝒑�𝑨(𝒕, 𝑻)�
𝑷𝑴 (𝟎, 𝒕)
1
× [𝑉(𝑡, 𝑇) − 𝑉(0, 𝑇) + 𝑉(0, 𝑡)] − 𝑀(𝑡, 𝑇)
2
By fitting the currently observed term structure of discount factors, the expected instantaneous short rate
at time t becomes:
𝐸�𝑟(𝑡)� = 𝜇𝑟 (𝑡) = 𝑓 𝑀 (0, 𝑡) +
And
𝑃𝑀 (0, 𝑇)
1
𝑒𝑥𝑝 �− × [𝑉(0, 𝑇) − 𝑉(0, 𝑡)]�
𝑃𝑀 (0, 𝑡)
2
𝜂2
𝜎𝜂
𝜎2
−𝑎𝑡 ]2
[1
×
−
𝑒
+
× [1 − 𝑒 −𝑏𝑡 ]2 + 𝜌 (1 − 𝑒 −𝑎𝑡 )(1 − 𝑒 −𝑏𝑡 )
2
2
2𝑎
2𝑏
𝑎𝑏
𝑉𝑎𝑟�𝑟(𝑡)� = �𝜎𝑡 (𝑡)� =
𝜎2
𝜂2
𝜎𝜂
× (1 − 𝑒 −2𝑎𝑡 ) +
× (1 − 𝑒 −2𝑏𝑡 ) + 2𝜌 �1 − 𝑒 −(𝑎+𝑏)𝑡 �
2𝑎
2𝑏
𝑎𝑏
This implies that the risk neutral probability of negative rates at time t is:
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𝑸(𝒓(𝒕) < 0) = ∅ �−
IRM-Theory-and-Practice-4
𝝁𝒓 (𝒕)
�
𝝈𝒕 (𝒕)
“Notice how we are simply using the unconditional mean and variance of the short rate process”

Furthermore, as t tends to infinity, the limiting Gaussian distribution has finite mean and variance.
Volatility and Correlation Structures in two factor models
We derive the volatility structure of the forward rate under the G2++ model. We follow the HJM framework (IRMTheory-and-Practice-5 as background reading). Define:
𝐴(𝑡, 𝑇) =
And
𝑃𝑀 (0, 𝑇)
1
𝑒𝑥𝑝 � × [𝑉(𝑡, 𝑇) − 𝑉(0, 𝑇) + 𝑉(0, 𝑡)]�
𝑀
𝑃 (0, 𝑡)
2
𝐵(𝑧, 𝑡, 𝑇) =
We can write:
1 − 𝑒 −𝑧(𝑇−𝑡)
𝑧
𝑃(𝑡, 𝑇) = 𝐴(𝑡, 𝑇) × 𝑒𝑥𝑝[−𝐵(𝑎, 𝑡, 𝑇)𝑥(𝑡) − 𝐵(𝑏, 𝑡, 𝑇)𝑦(𝑡)]
The instantaneous forward rate at time t for maturity T is:
Then compute the quantity:
You find:
𝑓(𝑡, 𝑇) = −
𝜕𝑙𝑛[𝑃(𝑡, 𝑇)]
𝜕𝑇
𝑉𝑎𝑟�𝑑𝑓(𝑡, 𝑇)�
𝑑𝑡
𝑉𝑎𝑟�𝑑𝑓(𝑡, 𝑇)�
= 𝜎 2 𝑒 −2𝑎(𝑇−𝑡) + 𝜂 2 𝑒 −2𝑏(𝑇−𝑡) + 2𝜌𝜎𝜂𝑒 −(𝑎+𝑏)(𝑇−𝑡)
𝑑𝑡
Therefore:
“The absolute volatility of the instantaneous forward rate is documented in the formula sheet”
Under the G2++ model, the absolute volatility of the instantaneous forward rate f(t,T) is given by:
𝜎𝑓 (𝑡, 𝑇) = �𝜎 2 𝑒 −2𝑎(𝑇−𝑡) + 𝜂 2 𝑒 −2𝑏(𝑇−𝑡) + 2𝜌𝜎𝜂𝑒 −(𝑎+𝑏)(𝑇−𝑡)

If the correlation between the two factors is negative (𝜌 < 0), then we observe a hump volatility structure
similar to what is currently observed in the market.

If the correlation is positive, no hump is observed.
In reading IRM-Theory-and-Practice-3 (Around Page 24 of the study manual), we discussed:
1.
2.
How cap and caplet volatilities are defined in the market,
How the term structure of the volatility is obtained by those
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3.
4.
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How to transliterate the market definition of caplet volatility above to short rate models (model-implicit
caplet volatility) and saw what went wrong, and finally
How to modify the definition so that things work again: model-implied caplet/cap volatilities, so that
things work again.
More precisely (“this is an opportunity to learn again”):
“I believe you are responsible for this formula, since it is not documented on the formula sheet Fall 2013”
𝑇
𝑻𝒉𝒆 𝑻 𝒆𝒙𝒑𝒊𝒓𝒚 𝒗𝒐𝒍𝒂𝒕𝒊𝒍𝒊𝒕𝒚 𝒄𝒂𝒑𝒍𝒆𝒕 = 𝑣𝑇2 𝑐𝑎𝑝𝑙𝑒𝑡
𝑇
1
1
= × � �𝑑𝑙𝑛�𝐹(𝑡; 𝑇, 𝑇 + 𝜏)�� × �𝑑𝑙𝑛�𝐹(𝑡; 𝑇, 𝑇 + 𝜏)�� = × � 𝜎(𝑡; 𝑇, 𝑇 + 𝜏)2 𝑑𝑡
𝑇
𝑇
0
0
𝜎(𝑡; 𝑇, 𝑇 + 𝜏) = 𝑇ℎ𝑒 𝑖𝑛𝑠𝑡𝑎𝑛𝑡𝑎𝑛𝑒𝑜𝑢𝑠 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑖𝑚𝑝𝑙𝑦 𝑐𝑜𝑚𝑝𝑜𝑢𝑛𝑑𝑒𝑑 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑟𝑎𝑡𝑒
𝐹(𝑡; 𝑇, 𝑇 + 𝜏) 𝑡ℎ𝑎𝑡 𝑢𝑛𝑑𝑒𝑟𝑙𝑖𝑒𝑠 𝑡ℎ𝑒 𝑇 𝑒𝑥𝑝𝑖𝑟𝑦 𝑐𝑎𝑝𝑙𝑒𝑡.
The quantities (𝜎(𝑡; 𝑇, 𝑇 + 𝜏)) are determined in Black’s model for the cap market, so that the caplet volatility
(𝑣𝑇2 𝑐𝑎𝑝𝑙𝑒𝑡 ) is also deterministic.
The curve:
𝑇 → 𝑣𝑇2 𝑐𝑎𝑝𝑙𝑒𝑡
Is called the term structure of caplet volatilities at time 0.
In a model like the G2++, the integrals of the form:
𝑇
Is not deterministic, it is rather stochastic.
1
× � 𝜎(𝑡; 𝑇, 𝑇 + 𝜏)2 𝑑𝑡
𝑇
0
Therefore, the term structure of caplet volatilities at time 0 is also stochastic.
“I believe you are responsible for this formula, since it is not documented on the formula sheet Fall 2013”
The model intrinsic T-caplet volatility at time 0 is defined as the random variable:
𝑇
1
� × � 𝜎(𝑡; 𝑇, 𝑇 + 𝜏)2 𝑑𝑡
𝑇
0

This is just not the way caplet volatilities shall be defined for models such as G2++.
So:
Modify the Definition of caplet volatility for Short Rate Models so that things Work again
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The procedure consists of:
1- Compute the model price of the ATM T-caplet at time 0:
𝑪𝒑𝒍�0, 𝑇, 𝑇 + 𝜏, 𝐹(0; 𝑇, 𝑇 + 𝜏)�
2- Invert the T-expiry Black market formula for caplets to find the percentage Black volatility that once
plugged into the formula for the caplet (in step 1) gives the model price:
𝒗𝑮𝟐++
𝑻 𝒄𝒂𝒑𝒍𝒆𝒕
More precisely, solve the following equation for: 𝑣 𝑇𝐺2++
𝑐𝑎𝑝𝑙𝑒𝑡 ”
“I believe you are responsible for this formula, since it is not documented on the formula sheet Fall 2013”


𝑷(𝟎, 𝑻 + 𝝉) × 𝝉 × 𝑭(𝟎; 𝑻, 𝑻 + 𝝉) × �𝟐∅ �
𝒗𝑮𝟐++
𝑻 𝒄𝒂𝒑𝒍𝒆𝒕 × √𝑻
𝟐
� − 𝟏� = 𝑪𝒑𝒍�𝟎, 𝑻, 𝑻 + 𝝉, 𝑭(𝟎; 𝑻, 𝑻 + 𝝉)�
The left hand side is the Market’s Black formula for a T-expiry T+τ maturity at the money caplet
The right hand side is the corresponding G2++ model formula.
Do this for all the expiry T and plot the function:

𝑇 → 𝑣𝐺2++
𝑇 𝑐𝑎𝑝𝑙𝑒𝑡
This is the term structure of caplet volatility implied the G2++ model.
Implied cap volatilities shall be defined the same way.
We also saw the following important points:
1.
2.

No humps in (𝑻 → 𝝈𝒇 (𝒕, 𝑻)) implies only that small humps for (𝑻 → 𝒗𝑴𝒐𝒅𝒆𝒍
𝑻 𝒄𝒂𝒑𝒍𝒆𝒕 ) are possible.
Humps in (𝑻 → 𝝈𝒇 (𝒕, 𝑻)) implies large humps for (𝑻 → 𝒗𝑴𝒐𝒅𝒆𝒍
𝑻 𝒄𝒂𝒑𝒍𝒆𝒕 ) are possible.
In the example of the CIR++ model, we established that, (𝑻 → 𝝈𝒇 (𝒕, 𝑻)) is monotonically decreasing,
thus usually implying small humps in the model caplet/cap volatilities.
We now turn our attention to the instantaneous covariance between two forward rates.
Let two forward rates 𝑓(𝑡, 𝑇1 ) and 𝑓(𝑡, 𝑇2 ), we are interested in the quantity:
𝐶𝑜𝑣[𝑑𝑓(𝑡, 𝑇1 ), 𝑑𝑓(𝑡, 𝑇2 )]
𝑑𝑡
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Under the G2++ model,
𝐶𝑜𝑣[𝑑𝑓(𝑡, 𝑇1 ), 𝑑𝑓(𝑡, 𝑇2 )]
= 𝜎 2 𝑒 −𝑎(𝑇1+𝑇2 −2𝑡) + 𝜂 2 𝑒 −𝑏(𝑇1+𝑇2−2𝑡) + 𝜌𝜎𝜂�𝑒 −𝑎𝑇1 −𝑏𝑇2 +(𝑎+𝑏)𝑡 + 𝑒 −𝑏𝑇1−𝑎𝑇2+(𝑎+𝑏)𝑡 �
𝑑𝑡
And the instantaneous correlation between two forward rates is:
Where:
𝐶𝑜𝑟𝑟�𝑑𝑓(𝑡, 𝑇1 ), 𝑑𝑓(𝑡, 𝑇2 )� =
𝐶𝑜𝑣[𝑑𝑓(𝑡, 𝑇1 ), 𝑑𝑓(𝑡, 𝑇2 )]
𝜎𝑓 (𝑡, 𝑇1 )𝜎𝑓 (𝑡, 𝑇2 )
𝜎𝑓 (𝑡, 𝑇) = �𝜎 2 𝑒 −2𝑎(𝑇−𝑡) + 𝜂 2 𝑒 −2𝑏(𝑇−𝑡) + 2𝜌𝜎𝜂𝑒 −(𝑎+𝑏)(𝑇−𝑡)
“The formula for the covariance above is on the formula sheet”
The continuously compounded forward rate at time t between times 𝑇1 and 𝑇2 is:
𝑓(𝑡, 𝑇1 , 𝑇2 ) =
𝑙𝑛[𝑃(𝑡, 𝑇1 )] − 𝑙𝑛[𝑃(𝑡, 𝑇2 )]
𝑇2 − 𝑇1
Under the G2++ model, the absolute volatility of the forward rate 𝑓(𝑡, 𝑇1 , 𝑇2 ) is:
Where:
𝜎𝑓 (𝑡, 𝑇1 , 𝑇2 ) = �𝜎 2 𝛽(𝑎, 𝑡, 𝑇1 , 𝑇2 )2 + 𝜂 2 𝛽(𝑏, 𝑡, 𝑇1 , 𝑇2 )2 + 2𝜂𝜎𝜌 × 𝛽(𝑎, 𝑡, 𝑇1 , 𝑇2 ) × 𝛽(𝑏, 𝑡, 𝑇1 , 𝑇2 )
𝛽(𝑧, 𝑡, 𝑇1 , 𝑇2 ) =
𝐵(𝑧, 𝑡, 𝑇2 ) − 𝐵(𝑧, 𝑡, 𝑇1 )
𝑇2 − 𝑇1
And the instantaneous covariance per unit time between the two forward rates 𝑓(𝑡, 𝑇1 , 𝑇2 ) and 𝑓(𝑡, 𝑇3 , 𝑇4 ) is:
𝐶𝑜𝑣[𝑑𝑓(𝑡, 𝑇1 , 𝑇2 ); 𝑑𝑓(𝑡, 𝑇3 , 𝑇4 )]
𝐵(𝑎, 𝑡, 𝑇2 ) − 𝐵(𝑎, 𝑡, 𝑇1 ) 𝐵(𝑎, 𝑡, 𝑇4 ) − 𝐵(𝑎, 𝑡, 𝑇3 )
= 𝜎2
×
+
𝑑𝑡
𝑇2 − 𝑇1
𝑇4 − 𝑇3
𝜂2
𝐵(𝑏, 𝑡, 𝑇2 ) − 𝐵(𝑏, 𝑡, 𝑇1 ) 𝐵(𝑏, 𝑡, 𝑇4 ) − 𝐵(𝑏, 𝑡, 𝑇3 )
×
𝑇2 − 𝑇1
𝑇4 − 𝑇3
𝐵(𝑎, 𝑡, 𝑇2 ) − 𝐵(𝑎, 𝑡, 𝑇1 ) 𝐵(𝑏, 𝑡, 𝑇4 ) − 𝐵(𝑏, 𝑡, 𝑇3 ) 𝐵(𝑎, 𝑡, 𝑇4 ) − 𝐵(𝑎, 𝑡, 𝑇3 ) 𝐵(𝑏, 𝑡, 𝑇2 ) − 𝐵(𝑏, 𝑡, 𝑇1 )
+𝜂𝜎𝜌 × �
×
+
×
�
𝑇2 − 𝑇1
𝑇4 − 𝑇3
𝑇4 − 𝑇3
𝑇2 − 𝑇1
“The formula above are also on the formula sheet”
Using the G2++ model to price a European Option on a zero-coupon bond
As seen in IRM-Theory-and-Practice-2, the price a time t of a European call option with maturity T and strike K,
written on a zero-coupon bond with unit face value and maturity τ is:
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𝑻
𝒁𝑩𝑪(𝑡, 𝑇, 𝜏, 𝐾) = 𝑬 �𝒆𝒙𝒑 �− � 𝒓(𝒔)𝒅𝒔� × (𝑷(𝑻, 𝛕) − 𝑲)+ �
𝒕
“Note: The book has a typo here. The maturity of the bond τ shall be in the value of ZBC not the letter S”
How to derive this expression?
For every maturity T, define the forward measure as (𝑄 𝑇 ).
The change of measure is:
𝑇
𝑑𝑄 𝑇
1
= 𝑒𝑥𝑝 �− 𝑉(0, 𝑇) − �[𝑥(𝑢) + 𝑦(𝑢)]𝑑𝑢�
𝑑𝑄
2
0
Get the dynamics of the process x and y under the T-forward measure
Then compute the expected value and variance of the process r(t) in the T-forward measure
Under the G2++ model, Given that we are at time s, the distribution of the short rate r(t) in the T-forward measure
is normal with mean and variance:
And
𝐸�𝑟(𝑡)� = 𝑥(𝑠)𝑒 −𝑎(𝑡−𝑠) − 𝑴𝑻𝒙 (𝒔, 𝒕) + 𝑦(𝑠)𝑒 −𝑏(𝑡−𝑠) − 𝑴𝑻𝒚 (𝒔, 𝒕) + 𝜑(𝑡)
𝑉𝑎𝑟�𝑟(𝑡)� =
𝜎2
𝜂2
𝜎𝜂
�1 − 𝑒 −2𝑎×(𝑡−𝑠) � + �1 − 𝑒 −2𝑏×(𝑡−𝑠) � + 2𝜌
�1 − 𝑒 −(𝑎+𝑏)×(𝑡−𝑠) �
2𝑎
2𝑏
𝑎+𝑏
Note that the variance of r(t) is the same as in the risk neutral world, while the expectation is reduced by the
quantities in bold.
Where:
𝑀𝑥𝑇 (𝑠, 𝑡) = �
And
𝜎𝜂
𝜎2
𝜎2
−𝑎(𝑡−𝑠)
+
𝜌
�
�1
−
𝑒
�
−
× �𝑒 −𝑎(𝑇−𝑡) − 𝑒 −𝑎(𝑇+𝑡−2𝑠) �
𝑎2
2𝑎2
𝑎𝑏
𝜎𝜂
−𝜌
�𝑒 −𝑏(𝑇−𝑡) − 𝑒 −𝑏𝑇−𝑎𝑡+(𝑎+𝑏)𝑠 �
𝑏(𝑎 + 𝑏)
𝑀𝑦𝑇 (𝑠, 𝑡) = �
𝜎𝜂
𝜂2
𝜂2
−𝑏(𝑡−𝑠)
+
𝜌
�
�1
−
𝑒
�
−
× �𝑒 −𝑏(𝑇−𝑡) − 𝑒 −𝑏(𝑇+𝑡−2𝑠) �
𝑏2
2𝑏 2
𝑎𝑏
𝜎𝜂
−𝜌
�𝑒 −𝑎(𝑇−𝑡) − 𝑒 −𝑎𝑇−𝑏𝑡+(𝑎+𝑏)𝑠 �
𝑎(𝑎 + 𝑏)
We are ready to state the value of the European bond option.
“Please don’t be offended by the coloring. I am doing this on purpose, so that you don’t get confused but you
understand precisely how the formulas and the input parameters interact together”
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Under the G2++ model, the value of a European call option with maturity T and strike K, written on a zero-coupon
bond with unit face value and maturity S is given by:
𝑃(𝑡, 𝑆)
𝑃(𝑡, 𝑆)
𝑙𝑛 �
𝑙𝑛 �
� 𝛴(𝑡, 𝑇, 𝑆)
� 𝛴(𝑡, 𝑇, 𝑆)
𝐾 × 𝑃(𝑡, 𝑇)
𝐾 × 𝑃(𝑡, 𝑇)
𝒁𝑩𝑪(𝑡, 𝑇, 𝑆, 𝐾) = 𝑃(𝑡, 𝑆)∅ �
+
� − 𝑃(𝑡, 𝑇)𝐾∅ �
−
�
𝛴(𝑡, 𝑇, 𝑆)
𝛴(𝑡, 𝑇, 𝑆)
2
2
Where:
“It looks a bit like the standard BSM model, hopefully you see that too”
𝛴(𝑡, 𝑇, 𝑆)2 =
𝜎2
𝜂2
2
−𝑎(𝑆−𝑇) 2
−2𝑎(𝑇−𝑡)
�1
−
𝑒
�
�1
−
𝑒
�
+
�1 − 𝑒 −𝑏(𝑆−𝑇) � �1 − 𝑒 −2𝑏(𝑇−𝑡) �
2𝑎3
2𝑏 3
𝜎𝜂
+ 2𝜌
�1 − 𝑒 −𝑎(𝑆−𝑇) ��1 − 𝑒 −𝑏(𝑆−𝑇) ��1 − 𝑒 −(𝑎+𝑏)(𝑇−𝑡) �
𝑎𝑏(𝑎 + 𝑏)
Now, it is “easy” to get the following prices:

The price at time t of a European put option with maturity T and strike K, written on a zero coupon bond
with unit face value and maturity S:
𝐾 × 𝑃(𝑡, 𝑇)
𝐾 × 𝑃(𝑡, 𝑇)
𝑙𝑛 �
𝑙𝑛 �
� 𝛴(𝑡, 𝑇, 𝑆)
� 𝛴(𝑡, 𝑇, 𝑆)
𝑃(𝑡, 𝑆)
𝑃(𝑡, 𝑆)
𝒁𝑩𝑷(𝑡, 𝑇, 𝑆, 𝐾) = −𝑃(𝑡, 𝑆)∅ �
−
� + 𝑃(𝑡, 𝑇)𝐾∅ �
+
�
𝛴(𝑡, 𝑇, 𝑆)
𝛴(𝑡, 𝑇, 𝑆)
2
2

The price at time t of a European call option with maturity T and strike K, written on a zero coupon bond
with face value N and maturity S:
𝑵𝑃(𝑡, 𝑆)
𝑵𝑃(𝑡, 𝑆)
𝑙𝑛 �
𝑙𝑛 �
� 𝛴(𝑡, 𝑇, 𝑆)
� 𝛴(𝑡, 𝑇, 𝑆)
𝐾 × 𝑃(𝑡, 𝑇)
𝐾 × 𝑃(𝑡, 𝑇)
𝒁𝑩𝑪(𝑡, 𝑇, 𝑆, 𝑁, 𝐾) = 𝑵𝑃(𝑡, 𝑆)∅ �
+
� − 𝑃(𝑡, 𝑇)𝐾∅ �
−
�
𝛴(𝑡, 𝑇, 𝑆)
𝛴(𝑡, 𝑇, 𝑆)
2
2

The price at time t of a European put option with maturity T and strike K, written on a zero coupon bond
with face value N and maturity S:
𝐾 × 𝑃(𝑡, 𝑇)
𝐾 × 𝑃(𝑡, 𝑇)
𝑙𝑛 �
𝑙𝑛 �
� 𝛴(𝑡, 𝑇, 𝑆)
� 𝛴(𝑡, 𝑇, 𝑆)
𝑵𝑃(𝑡, 𝑆)
𝑵𝑃(𝑡, 𝑆)
𝒁𝑩𝑷(𝑡, 𝑇, 𝑆, 𝑁, 𝐾) = −𝑵𝑃(𝑡, 𝑆)∅ �
−
� + 𝑃(𝑡, 𝑇)𝐾∅ �
+
�
𝛴(𝑡, 𝑇, 𝑆)
𝛴(𝑡, 𝑇, 𝑆)
2
2
Let us continue to get the pricing of caps and floors
The Pricing of caplets and floorlets
Given the current t, and two future times, an in-arrears caplet pays off the excess of the LIBOR rate over the cap
rate as graphed below.
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Where:
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𝐿(𝑇1 , 𝑇2 ) = 𝐿𝑖𝑏𝑜𝑟 𝑟𝑎𝑡𝑒 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑇1 𝑎𝑛𝑑 𝑚𝑎𝑡𝑢𝑟𝑖𝑛𝑔 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑇2
“Ooppss…The time fraction function and the notional amount shall be factored in the payoff as well!!!”
Thus, the caplet pays off:
𝑁 × 𝛼(𝑇1 , 𝑇2 ) × 𝑚𝑎𝑥(𝐿(𝑇1 , 𝑇2 ) − 𝑋, 0)
From what we saw in IRM-Theory-and-Practice-1, the LIBOR rates are simply-compounded market rate and are
linked to zero coupon bond prices. Thus:
𝐿(𝑇1 , 𝑇2 ) =
Let us restate the payoff of the in-arrears cap:
1
1
�
− 1�
𝑃(𝑇1 , 𝑇2 )
𝛼(𝑇1 , 𝑇2 )
𝑁 × 𝛼(𝑇1 , 𝑇2 ) × 𝑚𝑎𝑥 �
This is the same expression as:
The same as:
1
1
�
− 1� − 𝑋, 0�
𝑃(𝑇1 , 𝑇2 )
𝛼(𝑇1 , 𝑇2 )
1
𝑁 × 𝑚𝑎𝑥 ��
− 1� − 𝛼(𝑇1 , 𝑇2 )𝑋, 0�
𝑃(𝑇1 , 𝑇2 )
1
𝑁 × 𝑚𝑎𝑥 ��
� − (1 + 𝛼(𝑇1 , 𝑇2 )𝑋), 0�
𝑃(𝑇1 , 𝑇2 )
So, to value this caplet, we just need to make an appropriate change of variable for the notional amount and use
the value of ZBPs
Thus:
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Under the G2++ model, the value of an in-arrears caplet is given by:
More precisely:
𝑪𝒑𝒍(𝑡, 𝑇1 , 𝑇2 , 𝑁, 𝑋) = 𝑁 ′ 𝒁𝑩𝑷(𝑡, 𝑇1 , 𝑇2 , 𝑋 ′ ) = 𝑍𝐵𝑃(𝑡, 𝑇1 , 𝑇2 , 𝑁 ′ , 𝑁)
𝑁𝑃(𝑡, 𝑇1 )
𝑁𝑃(𝑡, 𝑇1 )
� 𝛴(𝑡, 𝑇 , 𝑇 )
� 𝛴(𝑡, 𝑇 , 𝑇 )
𝑙𝑛 � ′
𝑙𝑛 � ′
)
𝑁
𝑁
×
𝑃(𝑡,
𝑇
× 𝑃(𝑡, 𝑇2 )
1
2
1 2
2
𝑪𝒑𝒍(𝑡, 𝑇1 , 𝑇2 , 𝑁, 𝑋) = −𝑁 ′ 𝑃(𝑡, 𝑇2 )∅ �
−
� + 𝑃(𝑡, 𝑇1 )𝑁∅ �
+
�
𝛴(𝑡, 𝑇1 , 𝑇2 )
𝛴(𝑡, 𝑇1 , 𝑇2 )
2
2
Where:
𝑋′ =
And
1
(1 + 𝛼(𝑇1 , 𝑇2 )𝑋)
𝑁 ′ = 𝑁�1 + 𝛼(𝑇1 , 𝑇2 )�
It is now possible to get the price of the corresponding floorlet:
More precisely:
𝑭𝒍𝒍(𝑡, 𝑇1 , 𝑇2 , 𝑁, 𝑋) = 𝑁 ′ 𝒁𝑩𝑪(𝑡, 𝑇1 , 𝑇2 , 𝑋 ′ ) = 𝑍𝐵𝐶(𝑡, 𝑇1 , 𝑇2 , 𝑁 ′ , 𝑁)
𝑁 ′ 𝑃(𝑡, 𝑇2 )
𝑁 ′ 𝑃(𝑡, 𝑇2 )
� 𝛴(𝑡, 𝑇 , 𝑇 )
� 𝛴(𝑡, 𝑇 , 𝑇 )
𝑙𝑛 �
𝑙𝑛 �
)
𝑁
×
𝑃(𝑡,
𝑇
𝑁
× 𝑃(𝑡, 𝑇1 )
1
2
1 2
1
𝑭𝒍𝒍(𝑡, 𝑇1 , 𝑇2 , 𝑁, 𝑋) = 𝑁 ′ 𝑃(𝑡, 𝑇2 )∅ �
+
� − 𝑃(𝑡, 𝑇1 )𝑁∅ �
+
�
𝛴(𝑡, 𝑇1 , 𝑇2 )
𝛴(𝑡, 𝑇1 , 𝑇2 )
2
2
The pricing of caps and Floors
Consider the cap that reset at time (𝑇𝑖 ) with a cap rate of X, and a notional of N. The price at time t of the cap is:
𝑪𝒂𝒑(𝑡, 𝑇𝑖 , 𝜏, 𝑁, 𝑋)
𝑃(𝑡, 𝑇𝑖−1 )
𝑙𝑛 �
� 𝛴(𝑡, 𝑇 , 𝑇 )
(1 + 𝑋𝜏𝑖 ) × 𝑃(𝑡, 𝑇𝑖 )
𝑖−1 𝑖
= � −𝑁(1 + 𝑋𝜏𝑖 )𝑃(𝑡, 𝑇𝑖 )∅ �
−
�
𝛴(𝑡, 𝑇𝑖−1 , 𝑇𝑖 )
2
𝑛
𝑖=1
𝑃(𝑡, 𝑇𝑖−1 )
𝑙𝑛 �
� 𝛴(𝑡, 𝑇 , 𝑇 )
(1 + 𝑋𝜏𝑖 ) × 𝑃(𝑡, 𝑇𝑖 )
𝑖−1 𝑖
+ 𝑁𝑃(𝑡, 𝑇𝑖−1 )∅ �
+
�
𝛴(𝑡, 𝑇𝑖−1 , 𝑇𝑖 )
2
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Likewise, the value of the corresponding floor is:
𝑭𝒍𝒐𝒐𝒓(𝑡, 𝑇𝑖 , 𝜏, 𝑁, 𝑋)
(1 + 𝑋𝜏𝑖 )𝑃(𝑡, 𝑇𝑖 )
𝑙𝑛 �
� 𝛴(𝑡, 𝑇 , 𝑇 )
𝑃(𝑡, 𝑇𝑖−1 )
𝑖−1 𝑖
= � 𝑁(1 + 𝑋𝜏𝑖 )𝑃(𝑡, 𝑇𝑖 )∅ �
+
�
𝛴(𝑡, 𝑇𝑖−1 , 𝑇𝑖 )
2
𝑛
𝑖=1
The European Swaptions
(1 + 𝑋𝜏𝑖 )𝑃(𝑡, 𝑇𝑖 )
𝑙𝑛 �
� 𝛴(𝑡, 𝑇 , 𝑇 )
𝑃(𝑡, 𝑇𝑖−1 )
𝑖−1 𝑖
− 𝑁𝑃(𝑡, 𝑇𝑖−1 )∅ �
−
�
𝛴(𝑡, 𝑇𝑖−1 , 𝑇𝑖 )
2
Consider a European swaption with strike rate X, maturity T and nominal value N, which gives the holder the
option to enter at time T an IRS with payments time (𝑇𝑖 . 𝑖 = 1, … , 𝑛), where he pays (receives) at fixed rate X and
receives (pays) LIBOR set in arrears.
The value of this swaption, under the G2++ model, can only be solved numerically as:
“See the official syllabus on page 158”
Recommendation for potential exam question: I strongly recommend that under the G2++ model, given the formula for the call
option of a zero coupon bond of a unit face value, you are able to derive the following:
o
o
o The call option on a zero coupon bond of face value N,
The put option on a zero coupon bond of face value 1, and that of face value N,
The price of a caplet, the price of a floorlet, the price of a cap and that of a floor.
Good luck!!
Let us march ahead with G2++ model.
The analogy with the Hull-White Two-Factor Model
The Hull-White two factor model assumes the dynamics:
𝑑𝑟(𝑡) = [𝜃(𝑡) + 𝑢(𝑡) − 𝑎�𝑟(𝑡)]𝑑𝑡 + 𝜎1 𝑑𝑍1 (𝑡)
Where the stochastic mean reversion level satisfies:
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𝑑𝑢(𝑡) = −𝑏�𝑢(𝑡)𝑑𝑡 + 𝜎2 𝑑𝑍2 (𝑡)
Where:
IRM-Theory-and-Practice-4
𝑑𝑍1 (𝑡). 𝑑𝑍2 (𝑡) = 𝜌̅ 𝑑𝑡
And the fixed parameters:
𝑢(0) = 0
𝑟0 ; 𝑎�; 𝜎1 ; 𝑏�; 𝜎2 𝑎𝑟𝑒 𝑎𝑙𝑙 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒
Simple integration and great algebra leads to:
Under the Hull-White two factor model, the risk neutral dynamics of the short rate is captured by the equation:
𝒕
𝒕
𝒕
�
𝒓(𝒕) = 𝒓𝟎 𝒆−𝒂�𝒕 + � 𝜽(𝒗)𝒆−𝒂�(𝒕−𝒗) 𝒅𝒗 + 𝝈𝟑 � 𝒆−𝒂�(𝒕−𝒗) 𝒅𝒁𝟑 (𝒕) + 𝝈𝟒 � 𝒆−𝒃(𝒕−𝒗) 𝒅𝒁𝟐 (𝒕)
𝟎
𝟎
But earlier on, we established that:
𝟎
Conditional on the information up to time s (𝐹𝑠 ), the short rate under the G2++ model is normally distributed with:
And
In particular:
𝑉𝑎𝑟�𝑟(𝑡)� =
𝐸�𝑟(𝑡)� = 𝑥(𝑠)𝑒 −𝑎(𝑡−𝑠) + 𝑦(𝑠)𝑒 −𝑏(𝑡−𝑠) + 𝜑(𝑡)
𝜎2
𝜂2
𝜎𝜂
�1 − 𝑒 −2𝑎×(𝑡−𝑠) � + �1 − 𝑒 −2𝑏×(𝑡−𝑠) � + 2𝜌
�1 − 𝑒 −(𝑎+𝑏)×(𝑡−𝑠) �
2𝑎
2𝑏
𝑎+𝑏
𝒕
𝒓(𝒕) = 𝝈 � 𝒆
𝟎
−𝒂(𝒕−𝒖)
𝒕
𝒅𝑾𝟏 (𝒖) + 𝜼 � 𝒆−𝒃(𝒕−𝒖) 𝒅𝑾𝟐 (𝒖) + 𝝋(𝒕)
𝟎
With the appropriate change of variable, we recover the same model either way. Thus:
Given the Hull-White two factors model, we recover the G2++ model by setting:
𝑎� = 𝑎;
𝑏� = 𝑏; 𝜎 = 𝜎3 ; 𝜂 = 𝜎4 ;
𝜎1 𝜌̅ − 𝜎4
𝜌=
;
𝜎3
𝜑(𝑡) = 𝒓𝟎
𝒆−𝒂� 𝒕
Likewise:
𝒕
+ � 𝜽(𝒗)𝒆−𝒂� (𝒕−𝒗) 𝒅𝒗
𝟎
Given the G2++ model, we recover the Hull-White two factors model by setting:
𝑎� = 𝑎;
𝑏� = 𝑏; 𝜎1 = �𝜎 2 + 𝜂 2 + 2𝜌𝜎𝜂;
𝜎2 = 𝜂(𝑎 − 𝑏); 𝜌̅ =
�𝜎 2
𝜎𝜌 + 𝜂
+
𝜂2
+ 2𝜌𝜎𝜂
;
𝜃(𝑡) =
𝑑𝜑(𝑡)
+ 𝑎𝜑(𝑡)
𝑑𝑡
“The going from G2++ to the HW, back and forth are both documented on the formula sheet”
Construction of an Approximating Binomial Tree
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We build a binomial tree in both dimensions. The dynamics of the model is:
Where
And
𝑟(𝑡) = 𝑥(𝑡) + 𝑦(𝑡) + 𝜑(𝑡);
𝑑𝑥(𝑡) = −𝑎𝑥(𝑡)𝑑𝑡 + 𝜎𝑑𝑊1 (𝑡),
𝑑𝑦(𝑡) = −𝑏𝑦(𝑡)𝑑𝑡 + 𝜂𝑑𝑊2 (𝑡),
𝑟(0) = 𝑟0
𝑥(0) = 0
𝑦(0) = 0
𝑊1 𝑎𝑛𝑑 𝑊2 𝑎𝑟𝑒 𝑡𝑤𝑜 𝐵𝑟𝑜𝑤𝑛𝑖𝑎𝑛 𝑚𝑜𝑡𝑖𝑜𝑛 𝑤𝑖𝑡ℎ 𝑖𝑛𝑠𝑡𝑎𝑛𝑡𝑎𝑛𝑒𝑜𝑢𝑠 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝜌
𝑟0 , 𝑎, 𝜎, 𝑏, 𝜂 𝑎𝑟𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠
First Step: We build two binomial trees approximating the dynamics of x and y.
Remember these basic calculations here:
Also:
𝐸{𝑥(𝑡 + ∆𝑡)} = 𝑥(𝑡)𝑒 −𝑎∆𝑡
�
𝜎2
(1 − 𝑒 −2𝑎∆𝑡 )
𝑉𝑎𝑟{𝑥(𝑡 + ∆𝑡)} =
2𝑎
𝐶𝑜𝑣�𝑥(𝑡 + ∆𝑡), 𝑦(𝑡 + ∆𝑡)� =
𝜌𝜎𝜂
�1 − 𝑒 −(𝑎+𝑏)∆𝑡 � ≈ 𝜌𝜎𝜂∆𝑡
𝑎+𝑏
If at time time t if we have the values x(t) and y(t), the process x (resp. y) can either go up to x(t)+Δx (y(t)+Δy) or
down to x(t)-Δx (y(t)-Δy). Graphically:
We chose the quantities: p, q, Δx and Δy in order to match the conditional mean and variance of the continuoustime processes x and y.
Neglecting all terms with higher order than sqrt(Δt), we get:
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For the G2++ model, the appropriate parameters to build the binomial tree for the processes x(t) and y(t) are:
And
∆𝑥 = 𝜎√∆𝑡
𝑝=
1 𝑥(𝑡)𝑎∆𝑡 1 𝑥(𝑡)𝑎
−
= −
√∆𝑡
2
2∆𝑥
2
2𝜎
𝑝=
1 𝑦(𝑡)𝑏∆𝑡 1 𝑦(𝑡)𝑏
−
= −
√∆𝑡
2
2∆𝑦
2
2𝜂
∆𝑦 = 𝜂√∆𝑡
Realize that p and q depend on the values of x(t) and y(t) respectively and not explicitly on t.
Second Step: We approximate the tree for r(t)
The process for r(t) is approximated through a quadrinomial tree as represented below:
We solve the system:
Subject to:
𝜋1 + 𝜋2 = 𝑝
𝜋3 + 𝜋4 = 1 − 𝑝
�
𝜋1 + 𝜋3 = 𝑞 𝑎𝑛𝑑 𝜋2 + 𝜋4 = 1 − 𝑞
𝜋1 + 𝜋2 + 𝜋3 + 𝜋4 = 1
The solution is given out on page 166 and right below for your convenience:
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For the G2++ model, the appropriate parameters to build the binomial tree for the process r(t) are:
𝜋1 =
𝜋2 =
𝜋3 =
𝜋4 =
1 + 𝜌 𝑏𝜎𝑦(𝑡) + 𝑎𝜂𝑥(𝑡)
−
√∆𝑡
4
4𝜎𝜂
1 − 𝜌 𝑏𝜎𝑦(𝑡) − 𝑎𝜂𝑥(𝑡)
+
√∆𝑡
4
4𝜎𝜂
1 − 𝜌 𝑏𝜎𝑦(𝑡) − 𝑎𝜂𝑥(𝑡)
−
√∆𝑡
4
4𝜎𝜂
1 + 𝜌 𝑏𝜎𝑦(𝑡) + 𝑎𝜂𝑥(𝑡)
+
√∆𝑡
4
4𝜎𝜂
Example of Calibration to Real Market Data
We consider two calibrations:
1.
2.
A calibration to cap volatilities and
A calibration to swaption volatilities.
Cap volatilities calibration





The goal is to minimize the sum of the squares of the percentage differences between the market prices
and the model prices.
Giving the high number of parameters of the two factors Gaussian model, you shall expect the quality of
the calibration to be good.
The model actually reproduces the market cap-volatilities very accurately.
Too many time varying parameters can lead to over-fitting, thus the future volatility structures implied by
a G2++ model with more time varying parameters are likely to be unrealistic.
The rho coefficient (𝝆) is closed to minus 1. The model tends to degenerate into a one factor model. In
fact, caps prices do not depend on the correlation of forward rates (IRM-Theory-and-Practice-1).
Swaption volatilities calibration


The rho coefficient (𝝆) is far from minus 1. This is expected since swaption prices contain information
between the correlations of forward rates.
The calibration result is rather satisfactory.
Exam Question
None
Practice Questions
What is the motivation for using two factors model in interest rate pricing/modeling?
With one factor model, all the points of the yield curve are perfectly correlated. In reality, yield curve exhibit
decorrelation, and some instruments have payoffs depending on more than one interest rates. For these
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QFI Exam 2013-2014
PAK Study Manual
IRM-Theory-and-Practice-4
instruments, the structure of the correlations is important in calculating the payoff. With the one factor model, the
price is misleading at best. The two factor model is a better process for capturing this correlation.
Explain the relationship between the HW 2 factor model and the G2++ model?
It is possible to move from one to the other, given a proper change of variable.
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