Pricing Bermudan Style Swaptions Using the Calibrated Hull

White Model.

Lisa Larsson∗

Department of Statistics,

The Royal Institute of Technology ,

Lindstedts vag 13

SE-100 44 Stockholm, Sweden

and

Swedbank Markets

Regerings 13

Stockholm SE-105 34, Sweden

April 28, 2004

∗e-mail lisal@kth.se

1

Abstract

The Hull and White model for the short rate is reviewed and a trinomial tree for the short-

rate 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, andfor 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 atthe 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 52.1 Interest Rate Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 European Swaption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Bermudan Swaption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Models for Valuing Bermudan Swaptions 73.1 One-Factor Short Rate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 The generalized Hull-White Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2.1 Transformation of the Short-Rate Process . . . . . . . . . . . . . . . . . . . 83.3 Trinominal Tree for the Transformed Short-Rate Process . . . . . . . . . . . . . . . 9

3.3.1 Placing the Time Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3.2 Placing the Nodes Representing the Transformed Short-Rate Process . . . . 10

3.4 Adjusting the Tree to the Current Term Structure . . . . . . . . . . . . . . . . . . 11

4 Valuation and Risk Management 134.1 European Swaption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Bermudan Swaption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.3.1 The Calibration Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3.2 The Levenberg-Marquardt Algorithm . . . . . . . . . . . . . . . . . . . . . 154.3.3 Calibration of a Bermudan Swaption . . . . . . . . . . . . . . . . . . . . . . 15

4.4 Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.4.1 Delta hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.4.2 DV01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.4.3 Option delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4.4 Gamma hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4.5 Vega hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5 Implementing the Model using Object Orientated Model Design 175.1 Sequence Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.2 Class Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6 Numerical results 21

7 Discussion 23

4

1 Introduction

An interest rate derivative is a contingent claim which has a payoff that is dependent on the levelsof interest rates. The trading in interest derivatives has increased over the past two decades, andmany different new products have been developed. The products have become more specializedto 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 gainedpopularity is the Bermudan swaption. The Bermudan swaption is an option to enter into aninterest 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 thesemodels, the Hull-White model, is reviewed and implemented in this paper. When this model isimplemented, a trinomial tree is created for the short-rate, which is the interest rate that appliesover 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 themodel 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 movementsin the interest rate. This could be useful for example when a company has a loan to pay at anumber 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 Europeanswap option and the Bermudan swap option, which are two different kinds of options to enter anIRS.

2.1 Interest Rate Swap

An interest rate swap is an agreement between two parties to exchange a set of fixed interest ratepayments for a set of floating interest rate payments in the future. The floating side of the swapis called the floating leg and the fixed side is called the fixed leg. The size of each interest ratepayment is calculated on a principal. Since the principal itself is not exchanged, it is called anotional 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 thefixed side, in a receiver swap the holder receive the fixed side. Fig.(1) shows a receiver swap thatstarts at t = 0 and has four fix payment times, of the amount K, with the first payment in T1. Thefixed payments are made ones a year. For this swap the number of fixed payments and floatingpayments are the same. In general, there are more floating payments than fixed payments.

41

t = 0Pay float

Receive fix

t

TT

Figure 1: A receiver swap.

The floating side of the swap can be rewritten as in Fig.(2), by adding an amount of one inboth the positive and negative direction at each payment time. This procedure starts at the firstpayment time and the amount one is added in the positive and negative direction. The floatingrate 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 atT1. This is carried out for all payment times until all that remains is an positive amount of one att = 0 and a negative amount of one at T4.

T

t

T T

t

T

T T

t

t

TT

t = 0

t = 0

1

4 1 4

1 4 1 4

t = 0

t = 0

Figure 2: Transformation of the floating side of the swap.

4141

t = 0t = 0TT

t

=

TT

t

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 samepresent value is used, Fig.(3) This gives the fixed swap rate

K =1 − P (0, T1)

∑β

i=N τiP (0, Ti)(1)

where τi is the year fraction and P (0, Ti), i = 1, .., N , are the discount factors for each paymenttime, Ti. For the swap in Fig(1), N = 4 and the τ value is one since the fixed payment are madeonce a year. If the payments were made, for example, twice a year, the value of τ would be 2. Moreprecise calculations would take the chosen day count convention into account when evaluating theτ value. The discount factor P (0, T ) is the price at time zero of one money unit at time T . Thisdiscount factor is used when discounting a future cash flow, at time T , into the present time. Tocalculate the forward swap rate, that is, the swap rate for a swap starting at a future time Tα withfirst payment in Tα+1, the relation between forward rates and discount factors can be used. Thisgives the forward swap rate

K =P (0, Tα) − P (0, Tβ)∑β

i=α+1 τiP (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, toenter 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 certaintime 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 aswap that will start at a given time in the future and last for some number of years. The option willbe exercised if the swap rate for exactly the same swap as the one that can be entered into by theswaption 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 guaranteethat 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 aninterest rate swap to pay a fixed rate of interest and receive floating rate of interest. The companycan now benefit from favorable interest rate movements while having a protection from unfavorableones.

2.3 Bermudan Swaption

Following Andersen [8], a definition of the Bermudan swaption is given as an option which at eachdate 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 givesthe holder the right to enter into an interest rate swap at all times before maturity. As the Americanswaption has more flexibility than the equivalent Bermudan swaption, it is more expensive. Thesame is true for a Bermudan swaption compared to a European swaption, since the Bermudansswaption 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 unfavorableinterest rate movements at a number of future times and still benefit from favorable movements ofthe interest rate.

3 Models for Valuing Bermudan Swaptions

When choosing models for valuing interest rate derivatives, the derivatives can be divided into twomajor groups. The first group consists the European style derivatives. These can be valued usingBlack’s model, which is a generalization of the Black-Scholes model. A full description of how touse this model can be found in [4].

The second group consists of the more complicated derivatives, for which the value of the deriva-tive depends on the holder’s actions during the lifetime of the derivative. For this group, no-arbitrage term structure models can be used. The first group of derivatives can also be valued bythese 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 theterm 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 todescribe the evolution of the forward rate or discount bond prices. The main difficulty with thiskind of model is that Monte-Carlo simulations are needed for the implementation. This makes itboth 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 ratewhich is the rate that applies over the next short interval of time. The short rate is also sometimescalled the instantaneous short rate.

This section starts with a description of the one-factor short-rate model. This is followed by astudy of the generalized Hull-White model, which is the model that will be used for valuation inthis 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 onesource 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 rand the time t. The only source of uncertainty for this process is the Wiener process, dW , whichis 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 recombiningtree, 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 arisk neutral martingale measure. All prices can be calculated under this measure, independent ofmaturity 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 tofollow 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 yieldcurve of today. Finally, the functions a and σ are the volatility functions where a describes themean reversion and σ the volatility of the short rate.This model can easily be transformed into other term structure models. When f(r) = r anda(t) = 0 you get the Ho-Lee model and if f(r) = r and a(t) 6= 0 it is the original Hull-Whitemodel, as will be used here. A shortcoming of the Ho-Lee model compared to the other two is thatit 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. Thisdescribes the reality rather well since when rates are high, the economy tends to slow down withdecreasing investments. This would make the rates start to go down. The opposite, when ratesare 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 impliesthat the interest rate can become negative. This is a weakness of the model, but the probabilityfor 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 analyticaltractability.

The model also assumes that there are no market frictions, taxes or transaction costs. It isalso assumed that trading takes place at a discrete number of times and that assets are perfectlydivisible.

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 timeto zero and define a deterministic function g(t), which satisfies

dg(t) = [θ(t) − a(t)g(t)]dt

Define a new variablex(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 tomake the behavior of the process x easier to follow. The calculations can be extended to includenon-constant expressions of a and σ. Let f(r) = r from now since this is the expression for theinterest rate in the Hull White model.

Start by multiplying x by eat differentiate w.r.t t

d(x(r, t)eat) = eatdx(r, t) + ax(r, t)eatdt + 0 = eatσdW (t)

Integrate this expression

x(r, t)eat = x(r, 0) +

∫ t

0

easσdW (s)

which, together with the assumtion that x(r, 0) is zero, gives the expression for x

x(r, t) =

∫ t

0

e−a(t−s)σdW (s)

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] =

∫ t

0

e−2a(t−s)σ2ds =σ2

2a(1 − e−2at)

The expression for the variance shows that, for large values of t, the variance will tend to σ2/2a,which shows that the short-rate will be bounded, an not differ too much from the mean value. Thisis 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 sometime T − t is needed, where t is the start time and T is the end time. The expression for theexpectation is

E[∆x(f(r), t)] = (e−a(T−t) − 1)x(f(r), t) (5)

and for the variance

V ar[∆x] =σ2

2a(1 − e−2a(T−t)) (6)

3.3 Trinominal Tree for the Transformed Short-Rate Process

The stochastic process for the short rate can be discretely represented in a trinomial interest ratetree.

In this section the construction of such a trinomial tree is described. This is made with thesame approach as in [7]. First the tree is built for x(r, t) and then adjusted to the current termstructure so that the short rate follows the original diffusion process in (3). The functions a and σare assumed to already have been chosen. The actual functions are determined in the calibrationsection, after the tree has been constructed.The tree is here constructed for the short rate function f(r) = r, which is the version of theHull-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 intotwo parts. The first part contains a description of how to choose the times at which nodes areplaced 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 interestrate derivative, for example an option on a swap. This means that the nodes must be placed sothat they coincide with the payment days for the underlying security. More nodes can then beadded to increase the resolution of the tree. The time values that are represented with nodes areexpressed 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 paymentdays 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 stepti 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 thevariable 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 thistime 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 threeprobabilities, 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

pu

m

d

i,j x

x

x

x

i+1,k+1

i+1,k

i+1,k−1

t ti i+1

p

∆ t

p

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 conditionpd + pm + pu = 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 processin (6) to get the second moment E(dx2) = V + M2 of x(r, t) for the time interval ti+1, where V isthe Taylor expansion of the variance in (6)

V = −σ2(T − t)

V + (j∆xi + M)2 = k2∆x2i+1 + 2k(pu − pd)∆x2

i+1 + (pu + pd)∆x2i+1 (8)

The probabilities can now be determined using the equations (7) and (8). This gives

pu =V

2∆x2i+1

+α2 + α

2

pm =V

2∆x2i+1

+α2 − α

2

pd = 1 − V

∆x2i+1

− α2

where

α =j∆xi + M − k∆xi+1

∆xi+1

The probabilities are all positive if −√

2/3 < α <√

2/3. This means that, when branching from

a point j∆xi, the central node of the three following nodes should be placed within√

2/3∆x ofthe expected outcome for this node. The value of k, the index for the central node when branchingfrom xi, is chosen by

k = round

(

M

∆xi+1

)

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 forthe initial short-rate is f(r) = x(r, t) + g(t), this is done by adding the function g(t) to the valueof x(f(r), t) at each node. Since g(t) is a function of θ(t) and θ(t) is selected so that the modelfits the initial zero-coupon yield curve what is done is that the tree is adjusted to correctly pricediscount 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 ofthe world. This means that if a holder buys a discount bound today, with end time T , for someprice, 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 thefollowing 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 $1in 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 atnode (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)

Qi,j =∑

k

Q(i, j|i − 1, k)Qi−1,k =∑

k

p(i, j|i − 1, k)e−ri−1,k(ti−ti−1)Qi−1,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 thatpays one money unit at all the nodes in level ti+1 back to ti. Multiply each discount factorwith the corresponding Qi,j value and summarize:

Pi+1 =∑

j

Qi,je−r(i,j)(ti−ti−1) =

∑

j

Qi,je−f−1(x(i,j)+gi)(ti−ti−1)

This will give the value for a zero-coupon bond maturing in ti+1. To describe the presentvalue in this way gives an expression that is separated into two parts, one part which isknown and a second part which contains the gi value to be calculated. This expression is setequal to the value of a zero-coupon bond at ti+1 on the market. The gi value is then used tocalculate 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. Thismeans that gi can be expressed analytically

gi =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 termstructure 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 ratederivatives. All parameters, such as the volatility functions, are assumed to already have beendetermined. First the method to value a European swaption is described and after that the valueof a Bermudan swaption.

4.1 European Swaption

Consider a European payer swaption to enter into a swap starting at time tm, which is the exercisetime for the swaption, and first paying at tm+1 with strike rate K and final time tn. To calculatethe value for this swaption the IRS values are needed in each node at exercise time tm. The IRSvalue for a specific node is calculated in the tree by starting from the end and calculating thediscount factors needed. If the discount factors for the payment times for a specific node at timetm are P (tm, tj), j = m + 1, ..., n, and n − (m + 1) is the number of payment times, the value forthe IRS in this node is

IRS = 1 − P (tm, tn) − K

n∑

j=m+1

P (tm, tj)

The values will in general not be zero since each node represents one way that the world willevolve in terms of the short rate and it is all these values that the IRS might take that is discountedback 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 swaprate is the rate that gives the value zero for an IRS at this time

Kswap =1 − P (tm, tn)

∑n

j=m+1 P (tm, tj)

where the discount factors are the same as above. This is the same expression as in (1). The holderwill enter the swap if K < Kswap. This comparison is then made for all nodes at time level t. Forthe 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 exercisetimes. If the holder exercises the option at a exercise time, he will receive a swap paying the fixedrate 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 character-ized by three dates Tk < Th < Tn, giving its holder the right to enter at any time Tl in-betweenTk and Th (included) into an interest-rate swap with first reset in Tl, last payment in Tn andfixed-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 tomake the procedure easier to follow.

• Start at the end time of the tree, with the last payment time for the underlying swap. Setthe value of the discount factor for this time to one and discount back through the tree toprevious swap payment day. Set the value for a new discount factor to one and continueto discount the discount factors back through the tree adding new discount factors for eachswap payment time. When the last exercise time is reached, calculate the IRS value for eachnode and determine if the option should be exercised in this particular node and set the nodevalue to the IRS value if the option is exercised. This step is analogous to the valuation ofan European swaption.

• The node values are discounted back through the tree to the previous exercise time togetherwith the discount factors. During this process new discount factors are added whenever aswap payment time is reached. At the previous exercise time the IRS is calculated for eachnode using the discount factors and the exercise is checked in the same way as before. Ifthe option is exercised, the IRS value is compared to the backwardly cumulated value. Thelargest value is then set as the value of the node. This step is repeated until the first exercisetime is reached.

• When the first exercise time is reached, the node values are calculated as above. These valuesare finally discounted back through the tree until the start time is reached. The node valueof 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 derivativeis valued. This is done by adjusting the tree to a number of market prices. These market pricesare the prices of the derivatives which are connected to the derivative that is to be valued and arecalled the underlying derivatives.

4.3.1 The Calibration Process

The volatility functions are assumed to be sufficiently well described by first, second or thirdorder functions where the constants are to be determined during the minimization process. The

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algorithm used in the minimization process is the Levenberg-Marquardt process which is describedin this section.

4.3.2 The Levenberg-Marquardt Algorithm

The Levenberg-Marquardt algorithm is a stable and fast minimization algorithm often used whensolving nonlinear least squares problems.

To determine the volatility functions parameters, the root mean square error

RMSe =

√

√

√

√

n∑

i=1

(PModel,i − PMarket,i)2/n

is minimized. In the equation above, PModel,i and PMarket,i is, respectively, the price for derivativei calculated by the model and the price observed in the market and n is the number of marketprices the model is calibrated to.The Levenberg-Marquardt algorithm is a combination of the Gauss-Newton and the steepest de-scent 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 ap-proached and the root mean square error is small, the Levenberg-Marquardt method continuouslyswitches to the Gauss-Newton method, which is a method for a more exact determination of thelocation 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 timein 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 Eu-ropean swaptions. This means that a different set of European swap options has to be used todetermine the volatility functions to value different Bermudan swap options.When calibrating the model to price, for example, a 3-year Bermudan swap option, the Europeanswaptions 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 inthe swap price, that is, changes in the yield curve or in the prices for the underlying Europeanswaptions. Prices of the European swaptions are quoted in implied Black volatilities. The sameEuropean swaptions as the ones used for calibration, that is the Eurpean swaptions correspondingto 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 iscalled the ”Greek letters”.

This section describes three different risk measures, delta, gamma and vega, and how these canbe used to hedge a Bermudan swaption. The problem is to manage the Greeks so that all risks areat acceptable levels.

4.4.1 Delta hedging

Delta hedging of an Bermudan swaption is hedhing agains movements in the prices of the underlyingswap 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 changeof the price if the yield curve is shifted up by one basis point (0.01%). The value is calculatedby shifting the zero curve up by one basis point and calculating the price for the same Bermudan

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swaption. The difference between the price before and after the shifting of the zero curve is theDV01 value. To hedge against this risk, the trader can buy the opposite position of the underlyingswap, that is, the swap starting at the first exercise day and ending at the Bermudan swaptionsmaturity.

4.4.3 Option delta

The option delta of an Bermudan swaption consists of the option deltas of the underlying Europeanswaptions. The option delta for a European swaption consists of two parts. The first part is thegamma risk that depends on the rate of change of European swaption price with respect to theunderlying 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 calculatethe 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 theEuropean swaptions life times. This part of the gamma value is then obtained by shifting thebucket 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 zerocurve, 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 sellingthe 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 respectto the price of the underlying asset. This means that the gamma risk is the second partial derivativeof the Bermudan swaption price with respect to the price of the underlying. The gamma valuefor the Bermudan with respect to the underlying swap is called the convexity. Each underlyingEuropean swaption produces a gamma value called the option gamma.

A small option gamma or convexity value means that the delta changes slowly and that adjust-ments to keep the portfolios delta at an acceptable level does not need to be done often. If thegamma value is high, there is a need to change the hedge more often. This risk measure is thereforeused 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 valueis calculated by changing the implied Black volatilities with a small amount and repricing theBermudan swaption. The vega risks are then hedged by buying or selling enough of each Europeanswaption so that the total vega risk (all of the vega risks together) is at an acceptable level.

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5 Implementing the Model using Object Orientated Model

Design

This section will describe how the model is implemented using object oriented model design. Firsta sequence diagram is shown to describe how the objects collaborate. After that, a class diagramis presented.

5.1 Sequence Diagram

A sequence diagram describes how the objects collaborate. A sequence diagram for the problemstudied here is shown in Fig. (7). This diagram shows how operations are carried out accordingto time.

Trader TreeDirector TreeBuilder Calibrator HullWhiteFunction Value

Create

Construct

Create

CalibrateTree

Build

GetTree

Value

Create

Ecal

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 amongthem. Class diagrams are static which means that they display what interacts but not whathappens when they actually do interact. A class diagram for the implementation of the HullWhite model is shown in Fig.(8).

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IFunction

1

GetTree()Build()BuildTree()

TrinomialTreeBuilder

0,...,*VisitTrinomialTree()Visit()

daycountTypezeroCurve

ArrowDebreuVisitor

BuildTree()Eval()

HullWhiteFunction

<<Singleton>>

ValuationModelHullWhiteSwaption−

IndexStateListIndexParameterList

Strikes

PriceK

TradeDate

ExerciseDate

Underlying

Swaption

BermudanSwaption

SwaptionSchedule

1

0BuildTree()

Instance()

Value()

0

1

parameterListfunctiontype

BuildVolatilityFunction()Build()GetFunction()

VolatilityFunctionBuilder

IVisitor

HullWhiteFunction

Eval()

AddPrices()AddAncestor()TrinomialNode()

prices ancestors

TrinomialNode

ParameterFunction

TrinomialTree

Accept()

LevenbergMarquardt−Optimizer

Minimize()

treeMatrix

TrinomialTree()GetNodeEnumerator()GetTimesEnumerator()TreeDates()

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.

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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.

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6 Numerical results

This section gives an numerical example of the pricing of a Bermudan payer swaption. The val-uation date is 13 April, 2004, the day convention is 30/360, and the currency is Euro. The yieldcurved used in the pricing process can be seen in Fig.(9). The interest rates and instruments usedto create this curve is shown in Table (1).

2005 2010 2015 2020 2025 2030 20351.5

2

2.5

3

3.5

4

4.5

5

5.5

Year

Yie

ld

Figure 9: The Current Term Structure

Instrument Year Interest Rated 1m 2.052d 2m 2.044d 3m 2.038d 4m 2.036d 5m 2.039d 6m 2.044d 9m 2.091d 12m 2.1728s 2y 2.5590s 3y 2.91s 4y 3.217s 5y 3.475s 7y 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 withthe first exercise time in one year and a new exercise time for every year that follows until thefinal exercise time in six years from the start time. The strike rate is 3.99726% at every exercisetime. The undelying European swaptions will be the swaptions with exercise times at each of thetimes in the exercise schedule for the Bermudan. Table (2) gives the Black volatilities and theprices that these volatilities gives for the underlying swaptions which are used in the calibrationprocess. The volatility functions a and σ are assumed to be of second order and the resultingfunctions after the calibration process is finisched are shown in Table (3). The volatility functionsused 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

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Mat. x Length Vol Price1x5 19.9 140.92x4 18.58 157.33x3 17.09 137.34x2 16.35 103.25x1 15.95 57.5

Table 2: Prices and Black volatilities for the underlying European payer swaptions.

Mat. x Length a σ Price1x5 0.02992 0.008933 140.92x4 0.03000 0.008703 157.33x3 0.03000 0.008398 137.34x2 0.03004 0.008321 103.25x1 0.03000 0.008352 57.5

Table 3: The values for a, σ after calibration and the Hull White price for the underlying Europeanswaptions.

tree. These prices are the same as the market prices which shows that the minimization processhas succeeded.

2004 2005 2006 2007 2008 20090.025

0.026

0.027

0.028

0.029

0.03

0.031

0.032

0.033

0.034

0.035

t

a

Figure 10: The volatility function a.

Finally, the price for this Bermudan swaption is 284.3 bp. As expected, the price for theBermudan is higher than any individual price of the European swaptions. The price is less thanthe sum of all of the European swaption prices. This is correct, since it otherwise would be noneed to buy this contract instead of all the underlying European swaptions .

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2004 2005 2006 2007 2008 20098.2

8.4

8.6

8.8

9

9.2

9.4x 10

−3

t

σ

Figure 11: The volatility function σ

7 Discussion

This study has found that practical pricing of an Bermudan swaption using an implementation ofthe Hull-White model for the short-rate is feasible. The process is not very time consuming and ifthe resultant prices are compared to, for example, Bloomberg, they are similar. There are no marketprices for Bermudan swaptions since the market for these derivatives is not fully developed. Thismakes it rather difficult to compare results to evaluate the implementation made in this masterthesis. Different calculation programs for interest rate derivatives often gives different prices sincethere 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. Theunderlying derivatives of an Bermudan swaption is the European swaptions and the prices forthese 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 areassumed to follow a second order polynomial. The other options will give a very similar result inthe 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 tointerpret them is rather complicated, different traders have different methods.This is an area thatcould be studied in much more detail than what is done here.

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References

[1] R. Rebonato 1996, Interest-Rate Option Models, John Wiley & Sons

[2] A. Pelsser 1998, Efficient Methods for Valuing Interest Rate Derivatives, Springer-VerlagBerlin 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-VerlagBerlin 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, Ben-jamin/Cummings

[8] L. Andersen 1999, A Simple Approach to the Pricing of Bermudan Swaptions in the Multi-Factor 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, CambridgeUniversity Press

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