Pricing of convertible bonds with credit risk and stochastic interest rate

I shall present a general framework for valuing convertible bonds, with a Black-Scholes stock price, and the Hull White model for the interest rate.. I hope that this report will impart

NATIONAL UNIVERSITY OF SINGAPORE

DEPARTMENT OF MATHEMATICS

AN ACADEMIC EXERCISE PRESENTED IN PARTIAL FULFILLMENT

FOR THE DEGREE OF MASTER OF SCIENCE IN FINANCIAL MATHEMATICS

OU GUOQING HT091241U August, 2010

PRICING OF CONVERTIBLE BONDS WITH CREDIT

RISK AND STOCHASTIC INTEREST RATE

SUPERVISOR: PROF TAN HWEE HUAT, PROF DAI

MIN

Acknowledgement

I sincerely thank all those who have helped me in one way or another in this project

I would like to take this opportunity to thank Professor Dai Min and Professor Tan

Hwee Huat for their guidance and assistance throughout the realization of the thesis,

despite their tight schedule in teaching, research and supervision of students

Furthermore, I appreciate all professors in National University of Singapore who

imparted the essential foundations in stochastic calculus, computational mathematics

for tackling this project, such as Dr Xia Jianming, Prof Bao Weizhu and Prof Liu Jie

My thanks also go to Zhang Bixuan who provided me the up-to-date market data,

enlightening advices in programming, and several other friends whose technical

advices help me overcome the major or minor problems encountered

I am deeply grateful to my parents who have physically come to support me in

Singapore Their encouragement has certainly motivated me at many difficult times

in the process of realizing this project

Abstract

The convertible bond is an interesting security with its hybrid nature from both debt

and equity Complications in pricing convertible bonds arise due to additional

contractual features such as callability and puttability, soft call provision

Since 1991, most practitioners have used the binomial tree models to evaluate

convertibles bonds In this thesis, a partial differential equation is formulated from

the Two-Factor model, attempting a consistent treatment of equity, interest rate and

credit risk as well as the incorporation of the call and put provisions I shall present a

general framework for valuing convertible bonds, with a Black-Scholes stock price,

and the Hull White model for the interest rate

By no-arbitrage, the Hull-White model is calibrated to fit the initial term structure of

interest rates as well as the volatility surface of European swaptions, which are

readily quoted from the financial market The closed form formula of the European

swaption under the Hull-White model is deduced With the Levenberg-Marquardt

algorithm, I seek to find model parameters that lead to a least-square fit to its market

prices

The approach for solving the PDE is based on the numerical solution of linear

complementarity problems brought up in E Ayache, P Forsyth, K Vertzal (2003) and

the penalty method A convergence study is conducted in the report

I hope that this report will impart on the subject of convertible bond pricing,

calibration of Hull-White model and the structure of convertible bonds to students

and others readers interested in financial products pricing with PDE approach

Keywords: convertible bonds, credit risk, stochastic interest rate, Hull-White model,

Two-Factor model, calibration, Levenberg-Marquardt algorithm, penalty method

Content page

Acknowledgement 3

Abstract 4

Content page 6

List of figures 9

List of tables 10

1 INTRODUCTION 11

1.1 Convertible Bonds 11

1.1.1 Hybrid nature of convertible bonds 12

1.1.2 Callable and puttable features of convertible bonds 13

1.2 Literature Review 14

1.3 Outline of the Report 15

2 HULL AND WHITE MODEL IN BOND PRICING 18

2.1 HJM Model 18

2.1.1 Definition and value of a zero coupon bond 18

2.1.2 Value of the short rate 19

2.1.3 Link with the Hull and White model 20

2.2 Rate Processes 21

2.2.1 Short rate and forward rate 21

2.2.2 Beta 22

2.2.3 Zero coupon bonds 22

3 CALIBRATION OF THE HULL-WHITE MODEL 25

3.1 Pricing European Swaptions 25

3.1.1 Numeraire change 25

3.1.2 The case of the Hull and White model 26

3.1.3 Value of a call on a zero coupon bond 27

3.1.4 Value of a swaption 28

3.2 General Mechanism of Calibration 30

3.3 Levenberg Marquardt Minimization Algorithm 33

3.3.1 The Gauss Newton Algorithm 34

3.3.2 The Gradient Descent 35

3.3.3 The Levenberg Marquardt Algorithm 35

4 PRICING MODEL OF CONVERTIBLE BONDS 37

4.1 Convertible bonds with credit risk and stochastic interest rate: Two-Factor model 37

4.1.1 Model structure 37

4.1.2 Modeling credit risk with a Poisson process 38

4.1.3 Setting upon default in the Two-Factor model 39

4.2 PDE Formulation 40

4.2.1 Delta Hedging 40

4.2.2 Terminal and boundary conditions 43

4.2.3 Coupon payments and interest accrual 44

4.2.4 Formulation as a linear complementarity problem 45

4.2.5 Recovery under the Two-Factor model 47

5 IMPLEMENTATION 49

5.1 Treating the Swap Curve 49

5.1.1 Interpolation of the rate curves 49

5.1.2 Cubic spline interpolation 52

5.2 Discretization 54

5.2.1 Discretization of the PDE 54

5.2.2 Discretization on the boundary 56

5.3 Two Methods for Solving the Linear Complementarity Problem 59

5.3.1 Penalty Method 59

5.3.2 Direct method for reinforcing the constraints of the bond price 61

6 NUMERICAL RESULTS 63

6.1 Results from Calibraion 63

6.1.1 a, σ , MSE 63

6.1.2 Implied volatility surface from a and σ 64

6.1.3 Implied volatility smile from a and σ 66

6.2 Convertible Bond Price 68

6.2.1 Parameters and explanation for parameters chosen 68

6.2.2 Comparing the penalty method and the direct method 70

6.2.3 Convergence of the finite-difference scheme 71

6.2.4 CB price and the initial stock price, spot rate 73

6.2.5 Relationship with correlation coefficient, hazard rate and maturity 76

6.2.6 With and without coupon payment 80

7 CONCLUSION 81

7.1 Result Evaluation 81

7.2 Further Studies 83

8 REFERENCES 84

9 APPENDICES 87

9.1 Codes of the Numerical Implementation 87

9.1.1 C++ code on calibration 87

9.1.2 Matlab code on convertible bonds pricing 87

9.1.3 <convbond_fi_sir.m> 88

9.1.4 <convbond_fi_sir_penalty.m> 91

9.1.5 <convbond_fi_sir_cpc.m> 94

9.2 C++ Programme Flow 97

9.2.1 Main code 97

9.2.2 Inside the class G1analytics, interpolation and LMnumerics: 98

9.3 Class Variables of C++ code 99

9.4 Class Functions of C++ code 100

List of figures

Figure 1 Payoff of a convertible bond with κ=1 as the conversion ratio 13

Figure 2 Input yield curve on annual intervals for deducing the discount factors

51

Figure 3 Discount factors on monthly intervals 52

Figure 4 Initial short forward rates at t=0 on monthly intervals 52

Figure 5 C++ calibration output window showing the a and σ and the

mean-square errors in swapiton price 63

Figure 6 Graph of extrapolated volatility surface 65

Figure 7 Input volatility surface for comparison 66

Figure 8 Volatility skew for 2y2y swaption based our input data and calibration

results 68

Figure 9 CB price V(S,r,0), given initial stock price S and spot rate r 73

Figure 10 Two dimensional graph of V(S,r,0) against S, with speciific r values 74

Figure 11 Two dimensional graph of V(S,r,0) against r, with specific S values 75

Figure 12 Graph of V(80,0.05,0, )ρ against ρ 79

Figure 13 Graph of V(80,0.05,0, )p against p 77

Figure 14 Graph of V(80,0.05,0, )T against T 78

List of tables

Table 1 The input volatility surface from real market quotations 31

Table 2 Weight table indicating the swaptions used for calibration 32

Table 3 Implied swaption price from Black’s model with input volatilities in Table 1 33

Table 4 Implied volatility surface with a and σ obtained from calibration 64

Table 5 Error between the extrapolated volatility surface and the market data.64 Table 6 Implied volatility for various strikes 67

Table 7 Data for numerical implementation 69

Table 8 Comparison of convertible bond prices from two methods 71

Table 9 CB price at various mesh sizes and time step sizes, Ns=Nr 72

Table 10 CB price at various mesh sizes and time step sizes, Nt=Ns=Nr 72

Table 11 Convertible bond price without coupon and with semi-annual coupon payment of $4, S0=80, Ns=Nr=20 80

1 INTRODUCTION

1.1 Convertible Bonds

The market for convertible bonds has expanded tremendously in the past 10 years At

the beginning of the decade, just over $60 bn were outstanding in the US which is

considered highly liquid as compared to other domestic markets In early 2002, there

were approximately $270bn convertibles outstanding in the global market, $500bn in

2003 (E Ayache, P A Forsyth, K.R Vertzal 2004), $600bn in 2004 (Sungard report,

2004) and, by some estimations (V Gushchin, E Curien, 2007), reached $700bn in

2006 and exceeded $800bn in 2007 As convertible bonds become an increasingly

popular source of finance for firms, new contractual features of convertibles were

continually developed including different types of call clauses with or without a

hurdle, trigger prices and “soft call” feature, clauses which restrict the conversion

right of holders to contingent events (CoCo clause, conversion based on stock price,

CoCoCB clause, conversion based on trading price condition), mandatory clauses

(Arzac, 1997), “death spiral” convertible bonds (Hillion and Vermaelen, 2001),

option to change the conversion ratio (Hoogland, Neumann, Bloch 2001), perpetual

feature (Sirbu, Pikovsky, Shreve, 2002) The development and sophistication in the

contractual features resulted in increasing technical challenges of the bond valuation,

which have certainly aroused the research interests of academics and practitioners

alike

1.1.1 Hybrid nature of convertible bonds

Convertible bonds are financial products, typically having the feature that the holder

can convert into shares of common stock in the issuing company or cash of equal

value at an agreed-upon price It carries additional value to the holder through the

conversion right provided for the upside potential, the issuer on the other hand

benefits from the reduced interest rate

If the bond holder chooses to convert during the lifetime of the bond, the bond is

redeemed the holder receives some common shares from the issuer As long as the

bond holder does not convert the bond, he receives a coupon periodically and is still

repaid his principal at maturity If the convertible bond remains live till maturity, the

payoff at maturity is

It becomes clear that convertible bonds are hybrid financial products with bond-like

and equity-like features (Shown in Figure 1) The underlying risks come from both

the stock price and interest rate variation The hybrid nature has inspired some

models to consider the convertible bond value to be composed of a bond component

and an option on the stock

S

Figure 1 Payoff of a convertible bond with κ=1 as the conversion ratio

1.1.2 Callable and puttable features of convertible bonds

Among the wide variety of contractual features, this thesis focuses on the call and put

provision A put provision allows the holder to return the convertible to the issuer in

exchange for a predetermined amount of cash at certain points in time, and hence

provides a downside protection in case of rising interest rates This adds a further

layer of protection to the conversion right that bond hoders already dispose of

When convertibles are callable, the issuer has the option to purchase back the bond at

a predetermined strike price which often changes during the lifetime of the bond

However, the holder still has the priority to convert the bond when the call

announcement is made; hence the call provision is often used to force early

conversion of the bond Early conversion of a convertible bond is not optimal for the

holder under certain conditions; hence this call provision reduces the value of the

convertible It limits the investor's return if interest rates fall or the stock price rises

Often, convertible bonds are call-protected for some years and become callable only

after that

1.2 Literature Review

Interest rate models can be divided into two categories: equilibrium models, starting

with assumptions about economic variables and no-arbitrage model, taking today’s

term structure as an input and hence avoiding arbitrage opportunities The second

category is more popular for its empirical realism, i.e being able to fit initial term

structure

In the second category, we capture the term structure of interest rates in two

approaches One approach is to model the evolution of either forward rates or

discount bond prices This approach was initialized by Heath, Jarrow and Morton

(HJM, 1992) In the paper, they specify the behavior of instantaneous forward rates

The method is both easily comprehensible and powerful, as it contains many other

term structure models as special cases It exactly fits the initial term structure of

interest rates and is compatible with complex volatility structures On top of that, it

can readily be extended to as many sources of risk as desired

More recently the HJM model has been modified by Brace, Gatarek and Musiella

(1997), Jamshidian (1997), and Miltersen, Sandmann, and Sondermann (1997) to

apply to non-instantaneous forward rates This modification is known as the Libor

Market Model (LMM) In one version, 3-month forward rates are modeled This

allows the model to exactly replicate observed cap prices that depend on 3-month

forward rates In another version forward swap rates are modeled This allows the

model to exactly replicate observed European swap option prices The main

disadvantage of the HJM –LMM models is that they are difficult to implement by

any means other than Monte Carlo simulation Consequently, such models are

computationally slow and difficult to use for American or Bermudan style options

The other major approach of the second category is to describe the evolution of the

instantaneous rate of interest, the rate that applies over the next short interval of time

Short rate models are often more difficult to understand than models of the forward

rate, but they are computationally fast and useful for valuing all types of interest-rate

derivatives They are often implemented in the form of a recombining tree similar to

the stock price tree first developed by Cox, Ross, and Rubinstein (1979)

The Hull-White model has mean-reverting feature and extends on the models of

Vasicek and Cox-Ingersoll-Ross to be arbitrage free It contains many popular term

structure models as special cases, such as the Ho-Lee model.By introducing a

time-dependent drift, the resulting term structure of the Hull and White model is

consistent with current market prices of bonds The Hull-White model is also chosen

for scope of this thesis for its convenience in model calibration as compared to the

Cox, Ingersoll and Ross model

1.3 Outline of the Report

The main aim of this project is to calibrate the Hull-White model with real market

data and to study the pricing of the convertible bond, with the occurrence of default

considered in the pricing model

Section 2, Hull-White model: This section first details the link between the Heath,

Jarrow and Morton model (HJM) and the Hull-White Model, then gives the

stochastic formulas for all the variables (short rate, forward rate, zero coupon)

Section 3, Calibration: Firstly a closed formula for swaptions is established, which is

very useful for the calibration procedure The section then explains how to choose

aand σ so that the model swaption volatilities best fit the market volatilities, and

gives a short overview of the Levenberg Marquardt minimization algorithm which is

used to minimize the error function

Section 4, Pricing model of convertible bonds: This section introduces the

Two-Factor model which captures the credit risk with a Poisson process and makes some

reasonable assumptions upon default, while incorporating an additional stochastic

process of short term interest rate in response to the long life-span feature of

convertible bonds The complete PDE is formulated by delta hedging arguments for

subsequent numerical implementation

In Section 5, Implementation: This section explains how the swap curve is

transformed into a discounting curve, using cubic spline interpolation and

bootstrapping, the discretization of the PDE with two state variables inside the

solution domain and on the boundary, and how the penalty method and the direct

method are applied to solve the linear complementarity problem

In Section 6, Numerical results: This section present the numerical results obtained

from calibration and from solving the PDE in the Two-Factor model, under the

assumption of zero recovery rate and total default I will also demonstrate the

convergence in the numerical results as the mesh size and time step are reduced and

study the correlation between CB price and its various parameters

In Section 7, Conclusion: This section evaluates the numerical results obtained and

discusses possible future extensions in the subject

2 HULL AND WHITE MODEL IN BOND PRICING

2.1 HJM Model

2.1.1 Definition and value of a zero coupon bond

HJM is a class of models containing all the models diffusing zero coupon bonds and

assuming the following dynamic under the risk neutral measure:

t

t dt t T dW r

T t B

T t dB

),()

,(

),(

Γ+

=

(2) where B ( T t, )is the value of a zero coupon bond, r is the short rate, t Γ( T t, )is a

volatility function, and W is a Brownian motion We can notice that as the price of a t

zero coupon is known when t = T (and is worth 1), Γ(T,T)=0

The solution of the stochastic equation is:

+

t

s ds s T dW s T ds r

T B T

t

B

0

2 0

0

),(2

1)

,(exp

),0(),

(

(3) Taking T =tin the previous equation and using B(t,t)=1 we get:

+

t

s ds s t dW s t ds r

t B

0

2 0

0

),(2

1)

,(exp

),0(1

−Γ

−Γ

t B

T B T

t

B

0

2 2

0

),(),(2

1)

,(),(exp),0(

),0()

,

(

(5)

2.1.2 Value of the short rate

The forward continuous nominal rate between T and T +θ, fixed at t , is defined as

1))

,(exp(

θθ

θ

+

=

T T B T

R

t

t

(6) whereB t(T,T +θ) is the forward value of a zero coupon which satisfies, by absence

of arbitrage:

),(

),(),(

T t B

T t B T

(ln(

1),

θ

θθ

0 0

),(),(2

1)

,(),(),(

θθ

The forward short rate f ( T t, )is defined as

),(lim

),(t T θ 0 R T θ

),(),()

,(),

,(

0 0

),(),()

,(),0

(12)

2.1.3 Link with the Hull and White model

The Hull and White model assumes that, under the risk neutral measure, the short

rate follows the dynamic:

)()())

()(()

)(),(s t =−σ s −a t−s

Then

a

s t a s

2

2 0

1(0, ) exp( ) ( ) exp( ) exp( ) ( ) exp( )

dt dW as s

at a

dW t dt t

t f dr

t

s t

0

)exp(

)()exp(

)()

,0

(17)

t t

2exp( ) 2exp( 2 ) ( ) exp(2 ))

()

Equivalently,

t t

t

t f dt t af dt ar

∂

∂+

=

(18) Therefore the HJM model with the above choice for γ( t s, )is equivalent to the Hull

and White model, provided:

∫

+

∂

∂+

t

t f t af t

0

2

),(),0(),0()

θ

(19) This condition needs to be satisfied to allow the Hull and White model to enter the

HJM framework and so to be arbitrage free

2.2 Rate Processes

From now one we place ourselves in the case of a Hull and White model with a

constant volatility parameter, ( )σ t = The diffusion equation becomes: σ

)())

()(()

2.2.1 Short rate and forward rate

The equations representing the short rate and the forward rate in the Hull and White

Model can be deduced from the equations presented above in the more generic case

of the HJM model We only need to replace γ( T s, )and )Γ( T s, with their values in

the case of the Hull and White model We get the following equations:

0

2 2

2

)exp(

)exp(

)exp(

12),0

2.2.2 Beta

Beta represents the money market account, that is the amount of money that one

would get by placing 1 unit of currency at the risk neutral rate More precisely:

β

(22) Using the value calculated for r tabove, we can deduce the value of β Indeed: (t)

t

a ds s f ds

2 0

0

)exp(

)exp(

)exp(

12)

,0()

(

(23)

Using the relation between the forward rate and the zero coupon bonds to simplify

the first term and Fubini’s theorem to exchange the integrals in the last term, we get:

2.2.3 Zero coupon bonds

The value of a Zero coupon bond at tcan be calculated using the formula:

t

B(, ) exp ( , )

(26)

The forward rate has already been calculated in the previous parts so this formula can

be computed We replace f ( s t, )with its value in the integral to calculate a closed

formula for the value of the zero coupon bonds as shown below:

2 2 2

T

t t

The first term f s ds f s ds f s ds

t T

T

0 0

),0()

,0()

,0

relation between the zero coupon bonds and the forward rate, and the last term can be

simplified using the fact that the integral inside the bracket is independent of s and

hence can be seen as a constant term in the other integral, which means that the two

integrals can be computed separately Finally we get:

2 3 2

4 exp( ) exp( ) exp( )

2 3 2 3

This formula can also be rewritten as

)),(exp(

),(),(t T X t T Y t T r t

(

),0

B

T B

3 CALIBRATION OF THE HULL-WHITE MODEL

3.1 Pricing European Swaptions

Although swaption pricing isn’t directly necessary for evaluating convertible bonds, we

need to be able to price them in order to calibrate correctly the parameters a and σof the

Hull and White Model It is necessary to establish a closed formula for their price

We first establish theoretical formulas for numeraire change in the general case then in

the case of the Hull and White model in the previous section Then we calculate the value

of a call or a put on a zero coupon bond, and finally in the last subsection we calculate the

value of a put on a bond with coupons from which we can deduce a closed formula for

swaptions

3.1.1 Numeraire change

The price P of an asset giving a payoff h(X T)at time T is given by the following

expectation under the risk neutral measure:

[ (0, ) ( T)]

Q D T h X E

⎝∫ ⎠is the risk neutral numeraire

We define a new probability measureQ T, called the forward measure, corresponding

to the numeraire B(t,T), a zero coupon bond It is defined by:

),0(

exp

0

T B

ds r dQ

dQ

T s

T ⎜⎜⎝⎛− ⎟⎟⎠⎞

(34)

Under this new probability measure, the price P can be computed as:

[ (0, ) ( )] (0, ) [ ( )]

)(),0

T Q

T T

dQ

dQ X h T D E

Q t

T T

t s

means that the forward value of a zero coupon

),(

),(),(

T t B

S t B S T

under the forward measureQ T

3.1.2 The case of the Hull and White model

We have shown that in the HJM model,

+

t

s ds s t dW s t ds r

t

B

0

2 0

0

),(2

1)

,(exp

)

,

0

(

under the risk-neutral measure Q

),(2

1)

,(exp

under the forward measure Q T

This means that using the value calculated for r in the Hull and White model, we get: t

∫

−+

=

t

T u

r

0

)exp(

)exp(

),

(38)

where m ( T t, )is a deterministic function

This means that r tconditional on F s(information at time s) is a Gaussian variable

under the forward measureQ T, following a law N(m,v2(s,t))with

2)2exp(

)2exp(

),(

2 2

a du au at

t s

v

t s

3.1.3 Value of a call on a zero coupon bond

We define asZBC(t,T,S,K) the price at time t of a Call option with strike K and

maturity T written on a Zero Coupon maturing at time S

E K S T t

T t

),(

),()

,(

T t B

S t B F S T B

(41) Moreover since r Tconditional onF t follows a lawN(m,v2(t,T)), Y(T,S)r Tfollows a

lawN(m,'Y(T,S)2v2(t,T)).(42)

We are exactly in the framework of the pricing of a call in the Black’s model and

similar calculations lead to:

),()(),(),

,(

),(ln

p B t T K

S t B

(Φ represents the cumulative Gaussian distribution)

Similarly, the price of a put on a zero coupon is given by:

)(),()(

),(),,,

(46)

3.1.4 Value of a swaption

We first study the pricing of a put on a coupon-bearing bond We consider a bond

paying couponsc1, c2,…, c at time steps n T1, T2,…, T n

The price of this coupon-bearing bond at time T is written CB(T0,T,C) In the Hull

and White model, it only depends on the short rate at time T and is worth: 0

i n

i

i

i B T T c X T T Y T T r c

C T T

CB

1

0 0

1 0

0, , ) ( , ) ( , )exp( ( , ) )(

T X c K P

)),(exp(

),

(48) Since the value of the coupon-bearing bond is continuous and decreasing (between

T X c

n i

i i

∑

*)),(exp(

),(

(49)

The unique *r solution of this equation can be found using Newton algorithm (the

function studied is convex so the algorithm always converges)

This means that the payoff of the put can be rewritten as:

i n

i

i i

Since each term in the sum is a decreasing function of r , the difference between two

corresponding terms in the two sums always has the same sign as the difference

between two other corresponding terms, so:

i i

i X T T Y T T r X T T Y T T r c

P

1

0 0

0

0, )exp( ( , ) *) ( , )exp( ( , ) )

(51) This allows us to compute the price of a put on this coupon-bearing bond:

T X T T t ZBP c K

C T T t

CBP

0, , , ) , , , ( , )exp( ( , ) *,

(

(52) Swaptions can now be priced since they can be viewed as an option on a coupon

bearing bond Indeed, consider a payer swaption with strike rate S, maturity T and 0

nominal value N, which gives the holder the right to enter at time T an interest rate 0

swap with payment times T1, T2,…, T , where he pays the fixed rate n Sand receives

the Libor rate This corresponds to a put on a coupon bearing bond of strike

N

K = and with coupon payments at dates T worth i

)( − −1

i NS T T

))(

3.2 General Mechanism of Calibration

The Hull and White model assumes that under the risk neutral probability, the short

rate r follows the equation: t dr(t)=(θ(t)−ar(t))dt+σdW(t) where W(t) is a

Brownian motion, and θ(t), a and σ are parameters to be determined The aim of

calibration is to determine the Hull and White paramaters aand σ which allow us to

best fit the market conditions

A swap curve is given as a first input to calibrate this model By interpolation and

bootstrapping, it is transformed into a discounting curve, where a zero coupon value

is known for every maturity by time steps of 1 month This allows us to compute the

forward rate f(0,t)as the derivative of the discounting curve

)

(t

θ can then be determined (as a function of aand σ ) by absence of arbitrage using

this Discounting curve (and the forward rates deduced from this curve)

To determine the values of a and σ one needs more market information This

information will come from the second input used to calibrate the model, swaption

volatilities a and σ will be chosen in order to allow the swaption model prices (for

which there exists a closed formula in the Hull and White model, calculated in

section above) to fit as well as possible the swaption market prices More precisely a

and σ are chosen in order to minimize the sum of the squares of the difference

between the swaption market prices and the swaption model prices

More precisely, we choose a set of n swaptions with different tenors and maturities

on which we want to calibrate our model, and we associate weights w irepresenting

the importance these swaptions should have in our calibration We minimize the

th i

i Swaption a Swaption w

a F

1

2

),()

,

(55) where the term Swaption th(a,σ)

i is the price of the Swaption given by the model, and Swaption i real is the price of the Swaption given by the market In practice we

always takew i =1 An example of the weight table is shown in Table 1

Mid-market volatilities for at the money swap options

The swap is assumed to start at the expiry of the option, so the total life of

the transaction is the sum of the option life and the swap life

Table 2 Weight table indicating the swaptions used for calibration

We work on at the money payer swaptions In order to compute the market prices

and the model prices of these swaptions, we begin by computing the swap rate S

For a swaption of maturity T , tenor t , and with payment exchanged every f Δt, the

swap rate must satisfy:

),

(),

()

,

t t k

t T t B t k T t B t S T t B

f

++

Δ+Δ

f t k T t B t

t T t B T t B S

p

),

(

),

(),(

(57) The market price Swaption i real is then deduced from the market volatilities using the

Black’s formula (which is by convention used to give the implied volatilities quoted

(t T k t SN d KN d B

t Swaption

f

t t k

t T K

S d

−

−+

The model price Swaption i th(a,σ) is a function of a and σ calculated using the

theorical formulas obtained in the Hull and White model in the previous section

3.3 Levenberg Marquardt Minimization Algorithm

We use the Levenberg Marquardt minimization algorithm in order to minimize the

function F(a,σ) This algorithm is a combination of two well known algorithms

which we will present below: the Gauss-Newton algorithm and the method of

gradient descent

3.3.1 The Gauss Newton Algorithm

The Gauss Newton algorithm is used to minimize a function S which is the sum of

the squares of mfunctions r , ,1 r m of nvariables β1, ,βm (with m≥ ), that is n

The algorithm works recursively, starting from an initial guess β0of the minimum

and calculating better approximations β1,β2, Starting from an approximation of

the minimum βs, we try to find βs+ 1such that 2

2

1)( s+

r β is as small as possible

We use the linear approximation ( + 1)≈ ( )+ ( s)Δ

r s

r β β β where J ris the Jacobian matrix of r and Δ=βs+1−βs Finding the optimal Δ is equivalent to minimizing

2 2

)(

)

r

s J

r β β which is a linear least squares problem Δ is the solution of the

set of linear equations J J J t r

r r

t

r )Δ=−( (61) This set of equations can be solved using the QR factorization of J r

This algorithm tends to work well close to the minimum when the linear

approximation is almost true, but can fail to converge if we start too far away from a

minimum

3.3.2 The Gradient Descent

We now explain how the gradient descent algorithm works, in the particular case of

the function Sdefined above Once again, the algorithm works recursively, starting

from an initial guessβ0 of the minimum and calculating better approximations

β such that S(βs+ 1) is as small as possible

In order to do that, we search βs+ 1 in the direction of the steepest descent, that is in

the direction of ∇S, the gradient ofS This means that we take βs+ 1 such that

)(

β + − =− ∇ that is such that Δ=−λ∇S(βs) for a positive value of λ to

be determined In the case of ∑

=

= m

i i

r S

1

2( ))

(β β , ∇S(β)=2J r t r so Δ=−2λJ r t r.(62)

Gradient descent works better than the Gauss Newton Algorithm far from the

minimum, because it always makes sure to take a step in a direction in which the

slope decreases, but it can be very slow to converge when it is close to the minimum,

as well as for functions which have a narrow curved valley when it can zig-zag

3.3.3 The Levenberg Marquardt Algorithm

We still want to minimize the same function ∑

=

= m

i i

r S

1

2( ))

(β β as before recursevily

This time we choose Δ=βs+1−βssuch that (J r t J r + )λI Δ=−J r t r (63)If λ is very

small this algorithm is very close to the Gauss Newton Algorithm If λ is very big,

this algorithm is very close to the Gradient Descent In practice λ is adjusted at each

time step (by a multiplication or a division by a constant parameter) in order to try to

get )S(βs+ 1 as small as possible This should enable us to benefit from the

advantages of both algorithms while minimizing their drawbacks

Thanks to the minipack C code, I was able to modify and adopt the

Levenberg-Marquardt algorithm for finding the optimization parameters in C++ Substantital

time was spent to study the lmdif framework and the algorithm used A series of

functions such as G1ModelSingle was written to provide the essential error function

for initiating lmdif

4 PRICING MODEL OF CONVERTIBLE BONDS

4.1 Convertible bonds with credit risk and stochastic interest rate:

Two-Factor model

4.1.1 Model structure

The maturity of a convertible bond is typically longer than a traded option and the

effect of interest rate variation over its lifetime can be quite significant to the bond

price I hence incorporate the Hull-White short rate model to the classic Black

Scholes model, the combination of which is named the Two-Factor model

The risk-neutral short-term interest rate is given by:

2

where a is the mean reversion parameter, σ is the volatility and they are assumed to

be constant, while the ( )θ t is assumed to be a (locally bounded) deterministic

function of time, used to calibrate to the observed term structure of interest rates

The risk neutral stock price is given by:

where S is the stock price, μ is the drift rate, σ is the volatility of stock price, { t W1t}

is a standard Brownion Motion and r is the short rate diffused by the Hull and White t

Model

In addition,

1t 2t

where ρ negative as the interest rate and the stock price are negatively correlated

When the interest rate increases, funds are attracted to the bond market from the

stock market, hence the stock price drops due to a lower demand and vice versa

4.1.2 Modeling credit risk with a Poisson process

In the real market, the bonds issued by corporate are generally defaultable, thus the

credit risk should hence be considered in the pricing model of the CB Inspired by

the common use of Poisson distribution for modeling rare events like default, we

extend the Poisson distribution to a continuous time frame, which is a Poisson

process Let { ( ) :N t t≥0}be a Poisson process, where N(t) is the number of defaults

that have occurred up to time t

where λ is the intensity of default

As Δt becomes infinitely small,

[( ( ) ( )) 1]

Clearly, the first default occurrence is what we are interested in practice Define the

hazard rate p S t in such a way that the probability of default in the time period t ( , )

to t dt+ , conditional on no-default in [0, ]t is ( , ) p S t dt

( , )

( default from to | No default in [0, ])

Because of the Independent increment property of a Poisson process, we have

In the framework of this model, we assume λ and hence ( , )p S t to be deterministic

Given that no default event occurs prior to t, the probability of a default event and no

default in the time period t to t dt+ is respectively

( , )

p S t dt and 1- ( , ) p S t dt (67)

4.1.3 Setting upon default in the Two-Factor model

In the event of default, the stock price jumps from S+ to S- instantaneously:

(1 )

Where 0≤ ≤ When η 1 η= , the stock price lose all its value and it is called “total 1

default”; when η = , the stock price remains unaffected, this situation is called 0

“partial default” E Ayache, P.A Forsyth, K.R Vertzal(2003) η is the percentage

loss in share value upon default

The convertible bond holder has two options upon default,

• Receive the amount, RX, where R∈[0,1] is the recovery factor X can be the

face-value or the pre-default value of the bond portion of the convertible

• Convert the bond into shares worth κS(1−η)

Therefore, the value of the bond upon default is

max(κS(1−η),RX) (69)

A further assumption is that the default risk is diversifiable, that is the expected value

gains and loss due to default is zero and is hence not compensated under the risk

neutral measure Another implication of this assumption is that the real world and

risk-neutral world default probabilities are identical

4.2 PDE Formulation

4.2.1 Delta Hedging

When both the interest rate and the stock price are stochastic, the convertible bond

has a value of the form V V S r t= ( , , ) with two state variables Without loss of

generality, we assume the conversion is permitted at maturity of upon default, and

there is no call and put provision The continuous rights will be reinstalled later

through the constraints on V

Since the CB has two sources of randomness, we must hedge our option with two

other contracts, one being the underlying stock and the other being another CB to

hedge the interest rate risk We can use CB with the same contractual feature except

a different maturity T to hedge away the interest risk, and the price of this CB is 1