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6 Two-factor modeling: Convertible bonds with stochastic interest rate

# 6 Two-factor modeling: Convertible bonds with stochastic interest rate

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convertible bonds Chapter 33

The analysis is much as before; the choice

2

=

∂V

∂r

∂Z

∂r

and

∂V

∂S

eliminates risk from the portfolio. Terms involving T1 and T2 may be grouped together separately

to ﬁnd that

1

=

∂ 2V

∂V

∂ 2V

∂ 2V

+ 12 σ 2 S 2 2 + ρσ Sw

+ 12 w2 2

∂t

∂S

∂S∂r

∂r

∂V

∂V

+ rS

+ (u − λw)

− rV = 0.

∂S

∂r

(33.4)

where again λ(r, S, t) is the market price of interest rate risk. This is exactly the same market

price of risk as for ordinary bonds with no asset dependence and so we would expect it not to

be a function of S, only of r and t.

This is the convertible bond pricing equation. There are two special cases of this equation

that we have seen before:

• When u = 0 = w we have constant interest rate r; Equation (33.4) collapses to the Black–

Scholes equation.

1.5

1.35

1.2

1.05

0.9

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

0.75 Asset

0.6

0.45

0.3

0.15

0

2

4

6

0

8

10 12 14

16

Spot interest rate

Figure 33.8 The value of a CB with stochastic asset and interest rates.

565

0

2

4

6

8 10

12

14

Spot interest rate

16

1.2

1.35

1.5

1.05

0.9

0.75

0.6

Asset

0.15

0.3

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.45

Part Three ﬁxed-income modeling and derivatives

0

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Figure 33.9 ∂V/∂S for a CB with stochastic asset and interest rates.

0

−0.002

−0.004

−0.006

−0.008

1.5

1.25

1

−0.01

−0.012

−0.014

−0.016

−0.018

−0.02

0.75

Asset

0.5

0.25

0

2

4

6

0

8

10

12

14

16

Spot interest rate

Figure 33.10

∂V/∂r for a CB with stochastic asset and interest rates.

• When there is no dependence on an asset ∂/∂S = 0 we return to the basic bond pricing

equation.

Dividends and coupons are incorporated in the manner discussed in Chapter 8 and earlier in

this chapter. For discrete dividends and discrete coupons we have the usual jump conditions.

The condition at maturity and constraints are exactly as before; there is one constraint for

each of the convertibility feature, the call feature and the put feature.

convertible bonds Chapter 33

In Figure 33.8 is shown the value of a CB when the underlying asset is lognormal and interest

rates evolve according to the Vasicek model ﬁtted to a ﬂat 7% yield curve. In Figures 33.9 and

33.10 are shown the two hedge ratios ∂V /∂S and ∂V /∂r.

33.7

A SPECIAL MODEL

In some circumstances, and for a very narrow choice of interest rate model, we can ﬁnd a

similarity reduction. For example, if we use the Vasicek model for the risk-neutral interest rate

dr = (η − γ r) dt + β 1/2 dX2

then we can look for convertible bond prices of the form

V (S, r, t) = g(r, t)H

S

,t .

g(r, t)

Skipping some of the details, the function g(r, t) must be the value of a zero-coupon bond with

the same maturity as the CB:

g(r, t) = Z(r, t; T ) = eA(t;T )−rB(t;T )

where

B(t; T ) =

1

(1 − e−γ (T −t) )

γ

and

A(t; T ) =

βB(t; T )2

1

(B(t; T ) − T + t)(ηγ − 12 β) −

.

2

γ

The function H (ξ , t) then satisﬁes

∂ 2H

∂H

+ 12 ξ 2 σ 2 + 2B(t; T )ρβ 1/2 σ + B(t; T )2 β

= 0.

∂t

∂ξ 2

This problem must be solved subject to

H (ξ , T ) = max(nξ , 1)

and

H (ξ , t) ≥ nξ .

Unfortunately call and put features do not ﬁt into this similarity formulation. This is because

the constraint

V (S, t) ≤ MC

becomes

H (ξ , t) ≤

MC

Z(r, t; T )

the right-hand side of which cannot be written in terms of just ξ and t.

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Part Three ﬁxed-income modeling and derivatives

33.8 PATH DEPENDENCE IN CONVERTIBLE BONDS

The above-mentioned convertible bond is the simplest of its kind; they can be far more complex.

One source of complexity is, as ever, path dependency. A typical path-dependent bond would

be the following.

The bond pays \$1 at maturity, time t = T . Before maturity, it may be converted, at any time,

for n of the underlying. Initially, n is set to some constant nb . At time T0 the conversion ratio

n is set to some function of the underlying at that time, na (S(T0 )). Restricting our attention

to a deterministic interest rate, this three-dimensional problem (for which there is no similarity

solution) satisﬁes

∂V

∂V

∂ 2V

+ 12 σ 2 S 2 2 + rS

− rV ≤ 0.

∂t

∂S

∂S

Here V (S, S, t) is the bond value with S being the value of S at time T0 . Dividends and coupons

are to be added to this equation as necessary. The CB value satisﬁes the ﬁnal condition

V (S, S, T ) = 1,

the constraint

V (S, S, t) ≥ n(S, S),

where

n(S, S) =

for t ≤ T0

nb

na (S) for t > T0 .

and the jump condition

V (S, S, T0− ) = V (S, S, T0+ ).

This problem is three-dimensional, it has independent variables S, S and t. We could introduce

stochastic interest rates with little extra theoretical work (other than choosing the interest rate

model) but the computing time required for the resulting four-dimensional problem might make

this infeasible.

33.9 DILUTION

In reality, the conversion of the bond into the underlying stock requires the company to issue n

new shares in the company. This contrasts with options for which exercise leaves the number

of shares unchanged.

I’m going to subtly redeﬁne S as follows. If N is the number of shares before conversion

then the total worth of the company is N S − V before conversion. The −V in this is due to

the company’s obligations with respect to the CB. This means that the share price is actually

NS − V

N

and not S. The constraint that the CB value must be greater than the share price is

V ≥n

NS − V

N

convertible bonds Chapter 33

which can be rewritten as

V ≥

N

nS.

n+N

(33.5)

We must also have

V ≤S

(33.6)

and

V (S, T ) = 1.

Constraint (33.5) bounds the bond price below by its value on conversion, which is lower

than it was previously. Constraint (33.6) allows the company to declare bankruptcy if the bond

becomes too valuable. The factor N/(n + N ) is known as the dilution. In the limit n/N → 0

33.10 CREDIT RISK ISSUES

The risk of default in CBs is very important. If the issuing company goes bankrupt, say, you

will not receive any coupons and nor will the ability to convert into the stock have much

value. We have said that in the absence of default issues, the CB trades like a bond when the

1.6

1.4

1.2

V

1

0.8

0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

S

Figure 33.11 The value of a CB allowing for risk of default.

1

1.2

1.4

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Part Three ﬁxed-income modeling and derivatives

stock price is low. This is true, except that the bond behaves like a risky bond, one that may

default. Furthermore, if the stock price is very low it is usually indicative of a none-too-healthy

company. We can expect, and this is seen in the markets, that the price of a CB versus the

stock looks rather like the plot in Figure 33.11. Here the CB price goes to zero as S → 0. We

will see the model that led to this picture in Chapter 40.

33.11 SUMMARY

Convertible bonds are a very important type of contract, playing a major role in the ﬁnancing

of companies. From a pricing and hedging perspective they are highly complex instruments.

They have the early exercise feature of American options but in three guises—the option to

convert, call and put—sometimes behaving like a bond and sometimes like a stock. They have

long lifespans, ﬁve years, say, meaning that the assumption of constant interest rates is not

valid. In more complicated cases they can be path-dependent. Finally, they are not without the

risk of default. Put all these together and you have quite a sophisticated contract.

• For details of the effect of the issue of new shares on the value of convertible bonds see

Brennan & Schwartz (1977), Cox & Rubinstein (1985) and Gemmill (1992).

valuing CBs.

• For state-of-the-art modeling of converts look up research by Ayache et al. (2002, for

example).

• Cross-currency convertibles are considered by Ouachani & Zhang (2004).

CHAPTER 34

mortgage-backed

securities

In this Chapter. . .

types of mortgages

what makes mortgage-backed securities special

modeling prepayment

34.1

INTRODUCTION

Most homeowners know about mortgages. And mortgage-backed securities are simply many

mortgages lumped together for onward sale. The most interesting aspect of mortgage-backed

securities is that you must model the behavior of homeowners, whether or not they pay off

their mortgages early, reﬁnance etc. Because there are so many individual mortgages in each

security it is only necessary to model the average behavior of the homeowners.

34.2

INDIVIDUAL MORTGAGES

A mortgage is a loan, usually to private individuals, to help them with the purchase of a home.

They come in a variety of forms. The rate of interest could be ﬁxed, or ﬂoating, or ﬁxed for a

while and then left to ﬂoat. Sometimes the monthly payment to the lender covers both interest

and gradual repayment of the principal. Sometimes only the interest is covered and the principal

must be repaid at the end of the loan via some other form of

funding. The lifetime of most mortgages is long, typically 20,

25 or 30 years.

The size of the loan that an individual can borrow will depend

on their income and estimated credit-worthiness. The loan itself

is backed by the property. There is risk to the lender above normal interest risk because of the possibility of default combined

with the possibility of falling property values.

• Fixed rate mortgages have a rate of interest that is ﬁxed for the life of the mortgage.

Monthly payments remain at the same level and include payment of interest and a gradual

repayment of principal. These are the most common form of mortgage in the US.

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Part Three ﬁxed-income modeling and derivatives

• Floating rate mortgages have interest payments linked to a variable interest rate. This variable rate is set by the mortgagee, the lender, and varies discretely and is loosely associated

with the interest rates of the currency. These are the most common mortgages in the UK.

34.2.1

Monthly Payments in the Fixed Rate Mortgage

Since the monthly payments are constant, they represent both payment of interest and gradual

repayment of principal. To start with the interest forms the bulk of the payment later in the life

of the mortgage it is the repayment of principal that makes up the major part of each payment.

If the interest rate is rM and the amount of the loan is scaled to one and each monthly payment

is x then

12 N

1=

i=1

x

rM

1+

12

i

,

where N is the number of years over which the loan is paid off, the mortgage maturity. The

monthly payment can be calculated from

x=

rM /12

1 − (1 + rM /12)−12 N

.

The remaining balance, when there are M payments left, is then given by

M

P =

i=1

x

rM

1+

12

i

.

These calculations use a discrete-time interest rate. If we were to use a continuous-time

version then the sums would become integrals.

For more details of how mortgages work play a game of Monopoly.

34.2.2

Prepayment

Although mortgages are very much like other ﬁxed-income securities they do have one novel

feature; the borrower may decide to pay back the outstanding balance before the maturity of the

loan. This prepayment is rather like the callable feature in a more usual bond. The difference

between the two situations is important. We’ve seen how to model the callability of a bond

in Chapter 32. There the bond callability was treated very much like the early exercise of an

American option, with the call being made at an optimal time. With mortgages the time of

prepayment depends very much on the individual’s circumstances and his personal deﬁnition

of ‘optimal.’ (There is more on what this might mean for the lender in Chapter 63.)

The reasons for prepayment could be any of the following:

• The owner of the property comes into some money, he is risk averse and pays off the

mortgage early (perhaps mortgage rates are higher than bank interest rates).

• The owner decides to move house and pays off the mortgage with the proceeds from

the sale.

mortgage-backed securities Chapter 34

• The house falls down in an earthquake and the insurance goes to the lender.

• The householder defaults, and the insurance pays off the loan.

• Interest rates fall and the owner ﬁnds a better deal from another lender; this is known as

reﬁnancing.

Some of these are more important than the others, but most of them are outside the normal

pricing assumption that the owner pays off when interest rates change suitably. Only the last

one in the list approaches this ‘rational’ explanation, and even then there is a great deal of

inertia stopping people from bothering with reﬁnancing, or even knowing that the possibility

exists.

We’ll look at the stats of prepayment and how to model it later on.

34.3

MORTGAGE-BACKED SECURITIES

Mortgage-backed securities (MBS) are created by pooling together many individual mortgages. Investors then buy a piece of this pool and in return get the sum of all the interest

and principal payments. These MBSs can often then be bought

and sold through a secondary market.

By buying into this pool of mortgages the investor gets a

stake in the housing-loan market, but with less of a prepayment

risk. Most prepayers do not act ‘rationally’ on an individual

basis, but when there are a lot of them the ‘average’ prepayment can be considered. This is rather like diversiﬁcation. We’ll

see how the pricing is done via a prepayment function that

represents the average behavior of the borrower.

Collateralized mortgage obligations (CMO) are securities based on MBSs but in which

there has been further pooling and/or splitting so as to create securities with different maturities

for example. A typical CMO might receive interest and principal only over a certain future

time frame.

MBSs can be stripped into principal and interest components. Principal only (PO) MBSs

receive only the principal payments and become worth more as prepayment increases. Interest

only (IO) MBSs receive only the interest payments. The latter can be very risky since high

levels of prepayment mean much fewer interest payments.

34.3.1

The Issuers

In the US most MBSs are issued by the following organizations:

• Government National Mortgage Association (GNMA or, colloquially, Ginnie Mae)

• Federal National Mortgage Association (FNMA or Fannie Mae)

• Federal Home Loan Mortgage Corporation (FHLMC or Freddie Mac)

These are all US agencies or government sponsored.

Others, such as investment banks and house builders, also issue private-label MBSs.

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Part Three ﬁxed-income modeling and derivatives

34.4 MODELING PREPAYMENT

If prepayment is what distinguishes MBSs from other ﬁxedincome securities and derivatives then we need a model for

this prepayment.

We could go to the extreme of calculating the optimum time

for prepayment as we did for American options. This would

give the theoretical maximum value for the MBS. This is not

the usual practice since the mortgagors (the homeowners) do

not behave in this ‘rational’ fashion, and the resulting MBS value would be unrealistically high.

Instead we model what the mortgagors actually do in practice. I’ve put the word rational in

inverted commas because rationality is in the eyes of the beholder, as it were (see Chapter 63).1

In the next few sections we’ll look at prepayment models. We’ll assume that the underlying

mortgages are all ﬁxed rate.

34.4.1

The Statistics of Repayment

13

12.5

12

11.5

11

10.5

10

9.5

9

8.5

8

7.5

7

6.5

6

5.5

1995

1993

1991

1989

1987

1985

1983

1981

1979

45–50

40–45

35–40

30–35

25–30

20–25

15–20

10–15

5–10

0–5

Coupon

In Figure 34.1 is shown a contour map of the value of new GNMA 30-year loans in billions of

US dollars by year of origination and by coupon. Periods of falling interest rates are accompanied by periods of falling coupon rates. Figure 34.2 shows US 30-year rates for the same

period.

1977

574

Origination year

Figure 34.1 New Ginny Mae 30-year loans by year and coupon rate. Source: Davidson & Herskovitz

(1996).

1

Is there an arbitrage here? If so, it would involve the cooperation of millions of homeowners.

mortgage-backed securities Chapter 34

16

14

12

Rate

10

8

6

4

2

0

28-Aug-76 25-May-79 18-Feb-82 14-Nov-84 11-Aug-87 07-May-90 31-Jan-93 28-Oct-95

Figure 34.2 US 30-year interest rates.

Clearly, there is a wide spread of coupon rates. So, in any model of MBSs we are going

to have to work in terms of averages of coupons. Thus we tend to think in terms of the

Weight-averaged Coupon (WAC), where the weighting is by value of the loan.

Here are a couple of deﬁnitions used in prepayment models.

Single Monthly Mortality (SMM) is the amount prepaid in any month as a percentage of

the expected mortgage balance:

SMM =

Scheduled balance − Actual balance

.

Scheduled balance

Conditional Prepayment rate (CPR) is an annualized version of the SMM:

CPR = 1 − (1 − SMM)12 .

34.4.2

The PSA Model

The simplest model is the Public Securities Association Model (PSA) which takes no prepayment effects into account other than the age of the mortgage. The model assumes that

prepayment starts at zero, for a mortgage just initiated, rising at 0.2% per month for the ﬁrst

30 months and then stays constant at 6%. These numbers are in annualized, CPR, terms. See

Figures 34.3 and 34.4 for the CPR and the percentage of remaining mortgages for the PSA

model. How did I work out the data for Figure 34.4?

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