6 Two-factor modeling: Convertible bonds with stochastic interest rate
Tải bản đầy đủ - 0trang
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
566
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
4γ
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.
567
568
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
we return to V ≥ nS.
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
569
570
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.
FURTHER READING
• 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).
• See Fabozzi (1996) and Nyborg (1996) for more information about the market practice of
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.
572
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.
573
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?
575