Download Pricing Bermudan Style Swaptions Using the Calibrated Hull White

Pricing Bermudan Style Swaptions Using the Calibrated Hull
White Model.
Lisa Larsson∗
Department of Statistics,
The Royal Institute of Technology ,
Lindstedts väg 13
SE-100 44 Stockholm, Sweden
and
Swedbank Markets
Regerings 13
Stockholm SE-105 34, Sweden
April 28, 2004
∗ e-mail
[email protected]
1
Abstract
The Hull and White model for the short rate is reviewed and a trinomial tree for the shortrate is built and adjusted to current term structure. To be able to use the tree for pricing of
a Bermudan swaption, the tree is calibrated to market prices connected with the derivative
that is to be priced. The underlying derivatives for the Bermudan swaption are the European
swaptions which have exercise times that coincide with the exercise times in the Bermudan
swaption. Finally, the risks are calculated so that a hedge for the Bermudan can be made.
The risks considered here are the delta, vega and gamma risks. How the model is implemented
by use of object oriented design is also shown. A numerical example is presented in the end
of the paper.
2
Acknowledgements
This master thesis has been done at Swedbank Markets. I would like to thank Claes Cramer,
Senior Advisor Trading & Strategy, for all invaluable help and support during the process, and
for giving me the opportunity to do my master thesis at Swedbank Markets.
I would also like to thank Harald Lang, Senior Lecturer, at the Department of Mathematics at
the Royal Institute of Technology, for guidance and help.
Finally, I thank my brother, Petter Larsson, for all his support and useful opinions.
Stockholm, April 2004
Lisa Larsson
3
Contents
1 Introduction
5
2 Bermudan Swaptions
2.1 Interest Rate Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 European Swaption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Bermudan Swaption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
6
7
3 Models for Valuing Bermudan Swaptions
3.1 One-Factor Short Rate Model . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The generalized Hull-White Model . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Transformation of the Short-Rate Process . . . . . . . . . . . . . . .
3.3 Trinominal Tree for the Transformed Short-Rate Process . . . . . . . . . . .
3.3.1 Placing the Time Nodes . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Placing the Nodes Representing the Transformed Short-Rate Process
3.4 Adjusting the Tree to the Current Term Structure . . . . . . . . . . . . . .
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4 Valuation and Risk Management
4.1 European Swaption . . . . . . . . . . . . . . .
4.2 Bermudan Swaption . . . . . . . . . . . . . .
4.3 Calibration . . . . . . . . . . . . . . . . . . .
4.3.1 The Calibration Process . . . . . . . .
4.3.2 The Levenberg-Marquardt Algorithm
4.3.3 Calibration of a Bermudan Swaption .
4.4 Hedging . . . . . . . . . . . . . . . . . . . . .
4.4.1 Delta hedging . . . . . . . . . . . . . .
4.4.2 DV01 . . . . . . . . . . . . . . . . . .
4.4.3 Option delta . . . . . . . . . . . . . .
4.4.4 Gamma hedging . . . . . . . . . . . .
4.4.5 Vega hedging . . . . . . . . . . . . . .
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5 Implementing the Model using Object Orientated Model Design
5.1 Sequence Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Class Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
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6 Numerical results
21
7 Discussion
23
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1
Introduction
An interest rate derivative is a contingent claim which has a payoff that is dependent on the levels
of interest rates. The trading in interest derivatives has increased over the past two decades, and
many different new products have been developed. The products have become more specialized
to meet the needs of end users which has lead to a need for new, and often more complicated,
procedures for pricing and hedging these instruments.
One of these more complicated interest rate derivatives that has developed lately and gained
popularity is the Bermudan swaption. The Bermudan swaption is an option to enter into an
interest rate swap at a specified set of times, provided this option has not already been used.
There are several different models that ca be used to price a Bermudan swaption. One of these
models, the Hull-White model, is reviewed and implemented in this paper. When this model is
implemented, a trinomial tree is created for the short-rate, which is the interest rate that applies
over the next short interval of time. This tree is then adjusted to current term structure.
Before pricing, the volatility parameters must be determined. This is done by calibrating the
model to market prices for derivatives connected with the Bermudan swaption.
A Bermudan swaption can be used by a company that wants to insure itself against movements
in the interest rate. This could be useful for example when a company has a loan to pay at a
number of future times where the payments depend on the interest rate at these times.
2
Bermudan Swaptions
This section describes the interest rate derivatives that are connected with a Bermudan swaption.
First the interest rate swap, IRS, is described. This is followed by a description of the European
swap option and the Bermudan swap option, which are two different kinds of options to enter an
IRS.
2.1
Interest Rate Swap
An interest rate swap is an agreement between two parties to exchange a set of fixed interest rate
payments for a set of floating interest rate payments in the future. The floating side of the swap
is called the floating leg and the fixed side is called the fixed leg. The size of each interest rate
payment is calculated on a principal. Since the principal itself is not exchanged, it is called a
notional principal, which here is normalized to one.
The name convention for swaps is based on the fixed side. In a payer swap the holder pays the
fixed side, in a receiver swap the holder receive the fixed side. Fig.(1) shows a receiver swap that
starts at t = 0 and has four fix payment times, of the amount K, with the first payment in T1 . The
fixed payments are made ones a year. For this swap the number of fixed payments and floating
payments are the same. In general, there are more floating payments than fixed payments.
Receive fix
Pay float
t
t=0
T1
T4
Figure 1: A receiver swap.
The floating side of the swap can be rewritten as in Fig.(2), by adding an amount of one in
both the positive and negative direction at each payment time. This procedure starts at the first
payment time and the amount one is added in the positive and negative direction. The floating
rate at this time is the rate for an investment between t = 0 and T1 , so the present value at t = 0
5
for the floating rate and the fixed amount one at T1 is one. This eliminates the floating rate at
T1 . This is carried out for all payment times until all that remains is an positive amount of one at
t = 0 and a negative amount of one at T4 .
t
t=0
t
t=0
T4
T1
T4
T1
t
t
t=0
t=0
T1
T1
T4
T4
Figure 2: Transformation of the floating side of the swap.
=
t
t
t=0
t=0
T1
T4
T1
T4
Figure 3: The fixed and transformed floating side of the swap are of equal present value.
To calculate the fixed swap rate, K, the fact that the two sides of the swap must have the same
present value is used, Fig.(3) This gives the fixed swap rate
1 − P (0, T1 )
K = Pβ
i=N τi P (0, Ti )
(1)
where τi is the year fraction and P (0, Ti ), i = 1, .., N , are the discount factors for each payment
time, Ti . For the swap in Fig(1), N = 4 and the τ value is one since the fixed payment are made
once a year. If the payments were made, for example, twice a year, the value of τ would be 2. More
precise calculations would take the chosen day count convention into account when evaluating the
τ value. The discount factor P (0, T ) is the price at time zero of one money unit at time T . This
discount factor is used when discounting a future cash flow, at time T , into the present time. To
calculate the forward swap rate, that is, the swap rate for a swap starting at a future time Tα with
first payment in Tα+1 , the relation between forward rates and discount factors can be used. This
gives the forward swap rate
P (0, Tα ) − P (0, Tβ )
K = Pβ
i=α+1 τi P (0, Ti )
where Tβ is the time for the last payment.
2.2
European Swaption
A European payer swaption is an option that gives the holder the right, but not the obligation, to
enter into a swap , at a certain time, and pay a fixed rate and receive floating rate. For a European
6
receiver swaption the opposite applies, the holder has the right to enter into a swap at a certain
time and receive fixed rate and pay floating rate.
Consider a swaption that gives the holder the right to pay fixed rate and receive floating rate on a
swap that will start at a given time in the future and last for some number of years. The option will
be exercised if the swap rate for exactly the same swap as the one that can be entered into by the
swaption at the exercise time is higher than the fixed rate guaranteed by the swaption agreement.
An example of how a European swaption can be used is when a company wants to guarantee
that the interest rate they will pay on a loan at some future time does not exceed a certain level.
This can be done if the company buys a European swaption that gives the right to enter into an
interest rate swap to pay a fixed rate of interest and receive floating rate of interest. The company
can now benefit from favorable interest rate movements while having a protection from unfavorable
ones.
2.3
Bermudan Swaption
Following Andersen [8], a definition of the Bermudan swaption is given as an option which at each
date in a schedule of exercise dates gives the holder the right to enter into an interest rate swap,
provided this right has not been exercised at any previous time in the schedule.
The Bermudan swaption can be regarded as a special case of an American swaption which gives
the holder the right to enter into an interest rate swap at all times before maturity. As the American
swaption has more flexibility than the equivalent Bermudan swaption, it is more expensive. The
same is true for a Bermudan swaption compared to a European swaption, since the Bermudans
swaption has more flexibility, it is more expensive than the Bermudan equivalent.
A Bermudan swaption can be used by a company that wants to protect itself from unfavorable
interest rate movements at a number of future times and still benefit from favorable movements of
the interest rate.
3
Models for Valuing Bermudan Swaptions
When choosing models for valuing interest rate derivatives, the derivatives can be divided into two
major groups. The first group consists the European style derivatives. These can be valued using
Black’s model, which is a generalization of the Black-Scholes model. A full description of how to
use this model can be found in [4].
The second group consists of the more complicated derivatives, for which the value of the derivative depends on the holder’s actions during the lifetime of the derivative. For this group, noarbitrage term structure models can be used. The first group of derivatives can also be valued by
these models, even though this is often more complicated than using Black’s model.
The no-arbitrage property ensures that the value of an interest rate derivative generated by the
term structure model is consistent with the bond prices implied by the zero-coupon yield curve.
These models are designed to describe the stochastic nature of the interest rates.
There are two major approaches to model the interest rate in this way. One approach is to
describe the evolution of the forward rate or discount bond prices. The main difficulty with this
kind of model is that Monte-Carlo simulations are needed for the implementation. This makes it
both complicated and time consuming to use for valuing American or Bermudan style swaptions.
The second approach to model the stochastic nature of the interest rate is to use the short rate
which is the rate that applies over the next short interval of time. The short rate is also sometimes
called the instantaneous short rate.
This section starts with a description of the one-factor short-rate model. This is followed by a
study of the generalized Hull-White model, which is the model that will be used for valuation in
this study.
7
3.1
One-Factor Short Rate Model
In a one-factor short rate model, the process of how the short rate, r, evolves is described with one
source of uncertainty. The short rate is assumed to follow the process
dr = m(r, t)dt + s(r, t)dW
(2)
where m(r, t) and s(r, t), the drift and the standard deviation, are assumed to be functions of r
and the time t. The only source of uncertainty for this process is the Wiener process, dW , which
is defined as a continuous-time random-walk process.
An advantage of short rate models is that they often can be modeled in the form of a recombining
tree, which makes the valuation of interest rate derivatives fast and stable.
There are numbers of different short rate models, depending on the choices of m(r, t) and s(r, t).
The model that is focused on in this study is the Hull-White model, which is described below.
Since this one-factor short-rate model is assumed to be a no-arbitrage model, there exists a
risk neutral martingale measure. All prices can be calculated under this measure, independent of
maturity dates.
3.2
The generalized Hull-White Model
In the generalized Hull-White model f (r), which is some function of the short rate, is assumed to
follow the Gaussian diffusion process
df (r) = [θ(t) − a(t)f (r)]dt + σ(t)dW
(3)
where dW is a Wiener process. The function θ(t) is chosen to exactly fit the zero-coupon yield
curve of today. Finally, the functions a and σ are the volatility functions where a describes the
mean reversion and σ the volatility of the short rate.
This model can easily be transformed into other term structure models. When f (r) = r and
a(t) = 0 you get the Ho-Lee model and if f (r) = r and a(t) 6= 0 it is the original Hull-White
model, as will be used here. A shortcoming of the Ho-Lee model compared to the other two is that
it does not contain a mean-reversion term since a(t) = 0.
A mean-reversion term will make the rate drift toward an average level in the long run. This
describes the reality rather well since when rates are high, the economy tends to slow down with
decreasing investments. This would make the rates start to go down. The opposite, when rates
are low, would cause investments to increase which would make the rates go up.
The short rate in the Hull-White model is assumed to be normally distributed, which implies
that the interest rate can become negative. This is a weakness of the model, but the probability
for the rates to become negative is much smaller for this model compared to the Ho-Lee model,
due to the mean reversion. The Hull-White model has become very popular due to is analytical
tractability.
The model also assumes that there are no market frictions, taxes or transaction costs. It is
also assumed that trading takes place at a discrete number of times and that assets are perfectly
divisible.
3.2.1
Transformation of the Short-Rate Process
To make the process easier to analyse the expression (3) can be transformed. Set the current time
to zero and define a deterministic function g(t), which satisfies
dg(t) = [θ(t) − a(t)g(t)]dt
Define a new variable
x(f (r), t) = f (r) − g(t)
with the diffusion process
dx(f (r), t) = df (r) − dg(t) = [θ(t) − a(t)f (r)]dt + σ(t)dW − [θ(t) − a(t)g(t)]dt
8
which gives
dx(f (r), t) = −a(t)[f (r) − g(t)]dt + σ(t)dW = −a(t)x(f (r), t)dt + σ(t)dW
(4)
To calculate the mean and variance for the process of x, set a and σ to constants. This is done to
make the behavior of the process x easier to follow. The calculations can be extended to include
non-constant expressions of a and σ. Let f (r) = r from now since this is the expression for the
interest rate in the Hull White model.
Start by multiplying x by eat differentiate w.r.t t
d(x(r, t)eat ) = eat dx(r, t) + ax(r, t)eat dt + 0 = eat σdW (t)
Integrate this expression
at
x(r, t)e
= x(r, 0) +
Z
t
eas σdW (s)
0
which, together with the assumtion that x(r, 0) is zero, gives the expression for x
x(r, t) =
Z
t
e−a(t−s) σdW (s)
0
This show that x has the expectation value zero.
The variance can now be calculated
V ar[x(r, t)] = E[x(r, t)2 ] =
Z
t
e−2a(t−s) σ 2 ds =
0
σ2
(1 − e−2at )
2a
The expression for the variance shows that, for large values of t, the variance will tend to σ 2 /2a,
which shows that the short-rate will be bounded, an not differ too much from the mean value. This
is an great advantage compared to the Ho-Lee model, that has an infinite variance when t → ∞
will make the probability of a negative interest rate large.
Later on in this paper, when a trinomial tree is built, the expectation and variance over some
time T − t is needed, where t is the start time and T is the end time. The expression for the
expectation is
E[∆x(f (r), t)] = (e−a(T −t) − 1)x(f (r), t)
(5)
and for the variance
V ar[∆x] =
3.3
σ2
(1 − e−2a(T −t) )
2a
(6)
Trinominal Tree for the Transformed Short-Rate Process
The stochastic process for the short rate can be discretely represented in a trinomial interest rate
tree.
In this section the construction of such a trinomial tree is described. This is made with the
same approach as in [7]. First the tree is built for x(r, t) and then adjusted to the current term
structure so that the short rate follows the original diffusion process in (3). The functions a and σ
are assumed to already have been chosen. The actual functions are determined in the calibration
section, after the tree has been constructed.
The tree is here constructed for the short rate function f (r) = r, which is the version of the
Hull-White model where the short rate is assumed to follow a normal distribution.
The construction of the trinominal tree for the transformed short-rate process is divided into
two parts. The first part contains a description of how to choose the times at which nodes are
placed and the second part is to choose values and to place the x nodes.
9
3.3.1
Placing the Time Nodes
The term structure model for the short-rate is intended to be used later on to price an interest
rate derivative, for example an option on a swap. This means that the nodes must be placed so
that they coincide with the payment days for the underlying security. More nodes can then be
added to increase the resolution of the tree. The time values that are represented with nodes are
expressed as ti , i = 0, 1, ..N , where the nodes can be unevenly spaced.
In the case of an option on a swap this means that the time nodes must be placed at the payment
days for the swaption and the exercise days for the option.
3.3.2
Placing the Nodes Representing the Transformed Short-Rate Process
When the time nodes have been calculated, the x(r, t) nodes can be placed. At each time step
ti the first node is placed at x(r, t)i = 0. After that, the rest of√the x(r, t) nodes are placed at
±∆x(r, t)i , ±2∆x(r, t)i , ..., ±j∆x(r, t)i , where ∆x(r, t)i = σ(t) 3∆ti . The value of the node
spacing ∆x is determined by replicating the first five moments of the normal distribution for the
variable x(f (r), ti − 1) − x(f (r), ti ).
Since each time has a unique σ value that will determine the ∆x for all of the x nodes at this
time level, the xi nodes will be evenly spaced, with ∆x spacing.
The tree starts at node (i, j) = (0, 0). Since a trinomial tree is created there will be three
probabilities, one for branching up, pu , one for branching middle, pm , and one for branching down,
pd . The branching from a node i at time ti to time ti+1 is shown in Fig.4
x i+1,k+1
pu
pm
x i,j
x i+1,k
pd
x i+1,k−1
∆t
ti
t i+1
Figure 4: Branching from a node.
10
A Taylor expansion of the expectation value for the change in x(r, t) in (5) gives
M = −a(r, t)(T − t)x(r, t)
The relation between some node j∆x(r, t)i and the three nodes that branch off from this node is
j∆x + M = pu (k + 1)∆xi+1 + pm (k)∆xi+1 + pd (k + −1)∆xi+1
Together with the condition
pd + p m + p u = 1
this gives a simplified expression
j∆x + M = k∆xi+1 + (pu − pd )∆xi+1
(7)
This expression can now be used together with the variance calculated from the diffusion process
in (6) to get the second moment E(dx2 ) = V + M 2 of x(r, t) for the time interval ti+1 , where V is
the Taylor expansion of the variance in (6)
V = −σ 2 (T − t)
V + (j∆xi + M )2 = k 2 ∆x2i+1 + 2k(pu − pd )∆x2i+1 + (pu + pd )∆x2i+1
(8)
The probabilities can now be determined using the equations (7) and (8). This gives
pu
=
pm
=
pd
=
V
α2 + α
+
2
2∆xi+1
2
V
α2 − α
+
2∆x2i+1
2
V
1−
− α2
∆x2i+1
where
j∆xi + M − k∆xi+1
∆xi+1
p
p
from
The probabilities are all positive if − 2/3 < α < 2/3. This means that, when branching
p
a point j∆xi , the central node of the three following nodes should be placed within 2/3∆x of
the expected outcome for this node. The value of k, the index for the central node when branching
from xi , is chosen by
µ
¶
M
k = round
∆xi+1
α=
where round(x) is the closest integer to the real number x.
An example of the shape of a trinomial tree created as described above is shown in Fig.5, where
the time steps are unevenly spaced.
3.4
Adjusting the Tree to the Current Term Structure
In this section the tree is adjusted to fit the initial term structure f (r). Since the expression for
the initial short-rate is f (r) = x(r, t) + g(t), this is done by adding the function g(t) to the value
of x(f (r), t) at each node. Since g(t) is a function of θ(t) and θ(t) is selected so that the model
fits the initial zero-coupon yield curve what is done is that the tree is adjusted to correctly price
discount bonds of all maturities.
A discount bond is a bond that has a payoff of one money unit at the end time in all states of
the world. This means that if a holder buys a discount bound today, with end time T , for some
price, he will get one money unit at time T , whatever has happened in the world.
The adjustment process is recursive and starts at the root node. To describe the process the
following definitions are needed:
11
Figure 5: Branching Structure for a Trinomial Tree.
• Arrow-Debreu (AD) price Q(i, j|h, j) which is the present value of a security in node (h, k)
that has a payoff of $1 at node (i, j) and nothing at any other node.
• Root AD price Qi,j = Q(i, j|0, 0) which is the value in node (0, 0) of a security that pays $1
in node (i, j) and nothing at any other node.
• Probability p(i, j|h, k) as the probability of moving from node (h, k) to node (i, j).
• Present value Pi+1 as the price in node (0, 0) of a discount bond that pays $1 in ti+1 .
The process of determining the values of gi will be done in two steps.
1. Determine the values of Qi,j for each node j in step i.
The root AD price for node (i, j) can be determined once the root AD prices for all nodes
(i − 1, k) has been determined. To see this, start with the expression for the AD price at
node (i, j)
Q(i, j|i − 1, k) = p(i, j|i − 1, k)e−ri−1,k (ti −ti−1 )
Use this to determine the root AD price in node (i, j)
X
X
Qi,j =
Q(i, j|i − 1, k)Qi−1,k =
p(i, j|i − 1, k)e−ri−1,k (ti −ti−1 ) Qi−1,k
k
k
where the summation is made over all the x nodes at time ti−1
2. Use these Qi,j values to create an expression Pi+1 . This is done by discounting a bond that
pays one money unit at all the nodes in level ti+1 back to ti . Multiply each discount factor
with the corresponding Qi,j value and summarize:
X
X
−1
Pi+1 =
Qi,j e−r(i,j)(ti −ti−1 ) =
Qi,j e−f (x(i,j)+gi )(ti −ti−1 )
j
j
This will give the value for a zero-coupon bond maturing in ti+1 . To describe the present
value in this way gives an expression that is separated into two parts, one part which is
known and a second part which contains the gi value to be calculated. This expression is set
equal to the value of a zero-coupon bond at ti+1 on the market. The gi value is then used to
calculate the values for Qi+1,j and so on.
12
When the Hull-White model is studied, the function for the short rate is simply f (r) = r. This
means that gi can be expressed analytically
gi =
P
ln( k Qi,j )e−x(i,j)∆t − ln(Pi+1 )
∆t
Note that the short rate function first comes into the calculations at this stage, where the term
structure is being fit to the current term structure.
Figure 6: Branching Structure for a Adjusted Trinomial Tree
An example of a tree in which the short-rate has been adjusted is shown Fig (6).
4
Valuation and Risk Management
This section describes how the trinomial tree for the short rate is used to price interest rate
derivatives. All parameters, such as the volatility functions, are assumed to already have been
determined. First the method to value a European swaption is described and after that the value
of a Bermudan swaption.
4.1
European Swaption
Consider a European payer swaption to enter into a swap starting at time tm , which is the exercise
time for the swaption, and first paying at tm+1 with strike rate K and final time tn . To calculate
the value for this swaption the IRS values are needed in each node at exercise time tm . The IRS
value for a specific node is calculated in the tree by starting from the end and calculating the
discount factors needed. If the discount factors for the payment times for a specific node at time
tm are P (tm , tj ), j = m + 1, ..., n, and n − (m + 1) is the number of payment times, the value for
the IRS in this node is
n
X
P (tm , tj )
IRS = 1 − P (tm , tn ) − K
j=m+1
The values will in general not be zero since each node represents one way that the world will
evolve in terms of the short rate and it is all these values that the IRS might take that is discounted
back to today that is zero. To decide if the holder will exercise the right to enter the swap, the
13
strike rate K is compared to the swap rate, Kswap , that an IRS would offer at this node. The swap
rate is the rate that gives the value zero for an IRS at this time
1 − P (tm , tn )
Kswap = Pn
j=m+1 P (tm , tj )
where the discount factors are the same as above. This is the same expression as in (1). The holder
will enter the swap if K < Kswap . This comparison is then made for all nodes at time level t. For
the nodes where the holder will enter the swap, the value of the node is set to the value of the IRS.
The values will finally be discounted back to today, and this gives the value of the swaption.
4.2
Bermudan Swaption
Consider a payer swaption starting at a start time with fixed rate K and a number of exercise
times. If the holder exercises the option at a exercise time, he will receive a swap paying the fixed
rate K for the underlying swap starting at this exercise time.
The Bermudan swaption can be defined as in [5]:
Definition (BermudanSwaption) A (payer) Bermudan swaption is a swaption characterized by three dates Tk < Th < Tn , giving its holder the right to enter at any time Tl in-between
Tk and Th (included) into an interest-rate swap with first reset in Tl , last payment in Tn and
fixed-rate K. Thus, the swap start and length depend on the instant Tl when the option is exercised.
The valuation of a Bermudan swap option in a trinomial tree can be divided in three parts to
make the procedure easier to follow.
• Start at the end time of the tree, with the last payment time for the underlying swap. Set
the value of the discount factor for this time to one and discount back through the tree to
previous swap payment day. Set the value for a new discount factor to one and continue
to discount the discount factors back through the tree adding new discount factors for each
swap payment time. When the last exercise time is reached, calculate the IRS value for each
node and determine if the option should be exercised in this particular node and set the node
value to the IRS value if the option is exercised. This step is analogous to the valuation of
an European swaption.
• The node values are discounted back through the tree to the previous exercise time together
with the discount factors. During this process new discount factors are added whenever a
swap payment time is reached. At the previous exercise time the IRS is calculated for each
node using the discount factors and the exercise is checked in the same way as before. If
the option is exercised, the IRS value is compared to the backwardly cumulated value. The
largest value is then set as the value of the node. This step is repeated until the first exercise
time is reached.
• When the first exercise time is reached, the node values are calculated as above. These values
are finally discounted back through the tree until the start time is reached. The node value
of this node is the value of the Bermudan swaption.
4.3
Calibration
In the Hull-White model the volatility functions a and σ have to be determined before a derivative
is valued. This is done by adjusting the tree to a number of market prices. These market prices
are the prices of the derivatives which are connected to the derivative that is to be valued and are
called the underlying derivatives.
4.3.1
The Calibration Process
The volatility functions are assumed to be sufficiently well described by first, second or third
order functions where the constants are to be determined during the minimization process. The
14
algorithm used in the minimization process is the Levenberg-Marquardt process which is described
in this section.
4.3.2
The Levenberg-Marquardt Algorithm
The Levenberg-Marquardt algorithm is a stable and fast minimization algorithm often used when
solving nonlinear least squares problems.
To determine the volatility functions parameters, the root mean square error
v
u n
uX
RM Se = t (PM odel,i − PM arket,i )2 /n
i=1
is minimized. In the equation above, PM odel,i and PM arket,i is, respectively, the price for derivative
i calculated by the model and the price observed in the market and n is the number of market
prices the model is calibrated to.
The Levenberg-Marquardt algorithm is a combination of the Gauss-Newton and the steepest descent method. When the root mean square error is large, the steepest descent method is used,
which is a method that finds the approximate location of the minimum. When minimum is approached and the root mean square error is small, the Levenberg-Marquardt method continuously
switches to the Gauss-Newton method, which is a method for a more exact determination of the
location of the minimum.
4.3.3
Calibration of a Bermudan Swaption
A Bermudan swaption is connected to a collection European swap options where each exercise time
in the Bermudan swap option corresponds to an European swaption with the same exercise time.
When valuing a Bermudan swaption the Hull-White model is calibrated to these underlying European swaptions. This means that a different set of European swap options has to be used to
determine the volatility functions to value different Bermudan swap options.
When calibrating the model to price, for example, a 3-year Bermudan swap option, the European
swaptions 1x2, and 2x1 are used. The notation used here for the European swaption is on the form
(number of years to exercise)x(swap length).
4.4
Hedging
When a financial institution writes a Bermudan swaption it is exposed to risk. The risk lies in
the swap price, that is, changes in the yield curve or in the prices for the underlying European
swaptions. Prices of the European swaptions are quoted in implied Black volatilities. The same
European swaptions as the ones used for calibration, that is the Eurpean swaptions corresponding
to each exercise time in the schedule, can be used when hedging a Bermudan swaption.
The portfolio can be hedged against different sorts of risks. These risks are all part of what is
called the ”Greek letters”.
This section describes three different risk measures, delta, gamma and vega, and how these can
be used to hedge a Bermudan swaption. The problem is to manage the Greeks so that all risks are
at acceptable levels.
4.4.1
Delta hedging
Delta hedging of an Bermudan swaption is hedhing agains movements in the prices of the underlying
swap and Eurpean swaptions. These two kinds of delta risks are explained below.
4.4.2
DV01
The DV01, the ”dollar value of one basis point”, of an Bermudan swaption is the rate of change
of the price if the yield curve is shifted up by one basis point (0.01%). The value is calculated
by shifting the zero curve up by one basis point and calculating the price for the same Bermudan
15
swaption. The difference between the price before and after the shifting of the zero curve is the
DV01 value. To hedge against this risk, the trader can buy the opposite position of the underlying
swap, that is, the swap starting at the first exercise day and ending at the Bermudan swaptions
maturity.
4.4.3
Option delta
The option delta of an Bermudan swaption consists of the option deltas of the underlying European
swaptions. The option delta for a European swaption consists of two parts. The first part is the
gamma risk that depends on the rate of change of European swaption price with respect to the
underlying swap price and the second part is the gamma risk that depends on the option delta.
The second part can be calculated analytically using Black’s model, se J. Hull [4]. To calculate
the part of the option delta that depends on the underlying swap, the yield curve is divided into
”buckets”, each bucket containing the part of the yield curve that corresponds to each of the
European swaptions life times. This part of the gamma value is then obtained by shifting the
bucket for the European swaption up one basis point, bp, and recalculate the Bermudan price.
The difference between this price and the original price for the Bermudan with an unchanged zero
curve, is the part of the option delta dependent on the underlying swap.
One option delta is calculated for each European swaption, and the hedge consists in short selling
the underlying swaptions that has a to high option delta value.
4.4.4
Gamma hedging
The gamma, Γ, of an Bermudan swaption is the rate of change of the portfolio’s delta with respect
to the price of the underlying asset. This means that the gamma risk is the second partial derivative
of the Bermudan swaption price with respect to the price of the underlying. The gamma value
for the Bermudan with respect to the underlying swap is called the convexity. Each underlying
European swaption produces a gamma value called the option gamma.
A small option gamma or convexity value means that the delta changes slowly and that adjustments to keep the portfolios delta at an acceptable level does not need to be done often. If the
gamma value is high, there is a need to change the hedge more often. This risk measure is therefore
used to help manage the delta risks.
4.4.5
Vega hedging
The vega, V, of the Bermudan swaption is the volatility risks in the underlying European swaptions.
As with the vega and gamma risks, each European swaption produces one vega value. This value
is calculated by changing the implied Black volatilities with a small amount and repricing the
Bermudan swaption. The vega risks are then hedged by buying or selling enough of each European
swaption so that the total vega risk (all of the vega risks together) is at an acceptable level.
16
5
Implementing the Model using Object Orientated Model
Design
This section will describe how the model is implemented using object oriented model design. First
a sequence diagram is shown to describe how the objects collaborate. After that, a class diagram
is presented.
5.1
Sequence Diagram
A sequence diagram describes how the objects collaborate. A sequence diagram for the problem
studied here is shown in Fig. (7). This diagram shows how operations are carried out according
to time.
Trader
TreeDirector
TreeBuilder
Calibrator
HullWhiteFunction
Create
Construct
Build
GetTree
Create
CalibrateTree
Create
Ecal
Value
Figure 7: The sequence diagram.
5.2
Class Diagram
A class diagram gives an overview of a system by showing its classes and the relationship among
them. Class diagrams are static which means that they display what interacts but not what
happens when they actually do interact. A class diagram for the implementation of the Hull
White model is shown in Fig.(8).
17
Value
LevenbergMarquardt−
Optimizer
Swaption
ParameterFunction
TradeDate
IndexParameterList
IndexStateList
ExerciseDate
HullWhiteFunction
Eval()
BuildTree()
Underlying
K
Price
Minimize()
0
1
IFunction
BermudanSwaption
HullWhiteFunction
VolatilityFunctionBuilder
SwaptionSchedule
Strikes
parameterList
function
type
Eval()
BuildTree()
TrinomialTree
BuildVolatilityFunction()
Build()
GetFunction()
treeMatrix
1
TrinomialTree()
GetNodeEnumerator()
GetTimesEnumerator()
TreeDates()
Accept()
<<Singleton>>
HullWhiteSwaption−
ValuationModel
ArrowDebreuVisitor
daycountType
zeroCurve
0
Instance()
Value()
0,...,*
VisitTrinomialTree()
Visit()
TrinomialNode
TrinomialTreeBuilder
1
IVisitor
BuildTree()
Build()
GetTree()
ancestors
prices
TrinomialNode()
AddAncestor()
AddPrices()
GetPrices()
Figure 8: The class diagram.
The data dictionary that now follows explains more about each class in the class diagram.
TrinomialTree A trinomial tree.
Attributes
treeMatrix A matrix with the nodes.
Operations
TrinomialTree() Creates an empty tree.
GetNodeEnumerator() Gets an enumerator for the nodes at a certain time in the tree.
GetTimesEnumerator() Gets an enumerator for the time nodes in the tree.
TreeDates() Gets a vector of all the dates in the tree.
Accept() Accepts a visitor of the type IVisitor.
18
TrinomialNode A trinomial node.
Attributes
ancestors A list with all the ancestors.
prices A list with all the discount prices.
Operations
TrinomialNode() Creates a node.
AddAncestor() Adds an ancestor to the nodes ancestor list.
AddPrices() Adds a price to the price list.
GetPrices() Returns the price list.
TrinomialTreeBuilder Builds a trinomial tree.
Operations
BuildTree() Creates an empty tree.
Build() Builds for one time level.
GetTree() Returns the tree.
ArrowDebreuPriceVisitor .
Attributes
daycountType Specification of daycount type.
zeroCurve A zero curve.
Operations
VisitTrinomialTree() Visits an calculates and sets the Arrow Debreu prices.
Visit() Determines if to visit a node or the tree.
inherits from IVisitor
HullWhiteFunction
Operations
Eval() Builds a tree and calculates the price for a swaption.
BuildTree() Builds a tree for some volatility functions.
inherits from IFunction
HullWhiteSwaptionValuationModel
Operations
Instance() Creates an instance of the class if it does not already exists one.
Value() Values a swaption given the tree.
LevenbergMarquardtOptiomizer
Operations
Minimize() Minimizes the objective function to the target function.
VolatilityFunctionBuilder
Attributes
19
parameterList List of the volatility parameters.
function A piecewise volatility function.
type Specifies if the volatility function is variable or constant.
Operations
BuildVolatilityFunction() Creates a new empty piecewise function.
Build() Adds a function to the piecewise function.
GetFunction() Returns the function.
IVisitor
Operations
Visit() Defines that a method that takes an object must be implemented.
IFunction
Operations
Eval() Defines that a method Eval must be implemented.
Swaption
Attributes
TradeDate Defines the trade date.
ExerciseDate Defines the exercise date.
Underlying Defines the underlying swap.
K Defines the strike rate.
Price Defines the swaption price.
BermudanSwaption
Attributes
SwaptionSchedule List of underlying European swaptions.
Strikes List of strikes.
ParameterFunction
Attributes
IndexParameterList List of parameters.
IndexStateList State of parameters.
20
6
Numerical results
This section gives an numerical example of the pricing of a Bermudan payer swaption. The valuation date is 13 April, 2004, the day convention is 30/360, and the currency is Euro. The yield
curved used in the pricing process can be seen in Fig.(9). The interest rates and instruments used
to create this curve is shown in Table (1).
5.5
5
4.5
Yield
4
3.5
3
2.5
2
1.5
2005
2010
2015
2020
Year
2025
2030
2035
Figure 9: The Current Term Structure
Instrument
d
d
d
d
d
d
d
d
s
s
s
s
s
Year
1m
2m
3m
4m
5m
6m
9m
12m
2y
3y
4y
5y
7y
Interest Rate
2.052
2.044
2.038
2.036
2.039
2.044
2.091
2.1728
2.5590
2.91
3.217
3.475
3.883
Table 1: Instruments and interest rates that builds the curve for the current term structure.
The Bermudan swaption to be priced is an option to enter a swap at five different times with
the first exercise time in one year and a new exercise time for every year that follows until the
final exercise time in six years from the start time. The strike rate is 3.99726% at every exercise
time. The undelying European swaptions will be the swaptions with exercise times at each of the
times in the exercise schedule for the Bermudan. Table (2) gives the Black volatilities and the
prices that these volatilities gives for the underlying swaptions which are used in the calibration
process. The volatility functions a and σ are assumed to be of second order and the resulting
functions after the calibration process is finisched are shown in Table (3). The volatility functions
used for building the final tree to price the Bermudan swaption is shown in Fig.(10) and Fig.(11).
This table also shows the prices for the underlying swaptions when calculated using the trinomial
21
Mat. x Length
1x5
2x4
3x3
4x2
5x1
Vol
19.9
18.58
17.09
16.35
15.95
Price
140.9
157.3
137.3
103.2
57.5
Table 2: Prices and Black volatilities for the underlying European payer swaptions.
Mat. x Length
1x5
2x4
3x3
4x2
5x1
a
0.02992
0.03000
0.03000
0.03004
0.03000
σ
0.008933
0.008703
0.008398
0.008321
0.008352
Price
140.9
157.3
137.3
103.2
57.5
Table 3: The values for a, σ after calibration and the Hull White price for the underlying European
swaptions.
tree. These prices are the same as the market prices which shows that the minimization process
has succeeded.
0.035
0.034
0.033
0.032
a
0.031
0.03
0.029
0.028
0.027
0.026
0.025
2004
2005
2006
2007
2008
2009
t
Figure 10: The volatility function a.
Finally, the price for this Bermudan swaption is 284.3 bp. As expected, the price for the
Bermudan is higher than any individual price of the European swaptions. The price is less than
the sum of all of the European swaption prices. This is correct, since it otherwise would be no
need to buy this contract instead of all the underlying European swaptions .
22
−3
9.4
x 10
9.2
σ
9
8.8
8.6
8.4
8.2
2004
2005
2006
2007
2008
2009
t
Figure 11: The volatility function σ
7
Discussion
This study has found that practical pricing of an Bermudan swaption using an implementation of
the Hull-White model for the short-rate is feasible. The process is not very time consuming and if
the resultant prices are compared to, for example, Bloomberg, they are similar. There are no market
prices for Bermudan swaptions since the market for these derivatives is not fully developed. This
makes it rather difficult to compare results to evaluate the implementation made in this master
thesis. Different calculation programs for interest rate derivatives often gives different prices since
there is no market standard for which model to use when pricing Bermudan swaptions.
The largest challenge in pricing a Bermudan lies in calibrating the volatility functions. The
underlying derivatives of an Bermudan swaption is the European swaptions and the prices for
these are used to calibrate the model. The volatility functions are assumed to follow a linear,
quadratic or cubic functions. In the section with a numerical example, the volatility functions are
assumed to follow a second order polynomial. The other options will give a very similar result in
the final price.
To be able to hedge against movements in the short-rate, different risk measures can be used.
Here the delta, gamma and vega risks are studied. Which risk measures to choose and how to
interpret them is rather complicated, different traders have different methods.This is an area that
could be studied in much more detail than what is done here.
23
References
[1] R. Rebonato 1996, Interest-Rate Option Models, John Wiley & Sons
[2] A. Pelsser 1998, Efficient Methods for Valuing Interest Rate Derivatives, Springer-Verlag
Berlin Heidelberg New York
[3] J. Hull and A. White 1996, Hull-White on Derivatives, Risk Publications
[4] John C. Hull 2000, Options, Futures and Other Derivatives, Prentice-Hall International
[5] D. Brigo and F. Mercurio 2001, Interest Rate Models, Theory and Practice, Springer-Verlag
Berlin Heidelberg New York
[6] Lawrence C. Galitz 1995, Financial Engineering, Pitman Publishing
[7] J. Hull and A. White 2000, The General Hull-White Model and Super Calibration, Benjamin/Cummings
[8] L. Andersen 1999, A Simple Approach to the Pricing of Bermudan Swaptions in the MultiFactor Libor Market Model,
[9] M. T. Heath 1997, Scientific Computing, McGraw-Hill International Editions
[10] W. Press, S. Teukolsky, W Vetterling, B. Flannery 1992, Numerica Recipies in C, Cambridge
University Press
24