Chapter 16. Project: Interest Rate Trees: Calibration and Pricing
Tải bản đầy đủ - 0trang
380
C H A P T E R 16
. Project: Interest rate trees: Calibration and pricing
conditional probability density that the short rate will attain a value ri at time Ti , given ri−1
at time Ti−1 , by p ri ri−1 t , then (for a generally stochastic rt diffusion process) the price
can be accurately approximated by an M-dimensional integral,
M
P0 r0 T =
0
···
0
t e−ri−1
p ri ri−1
t
rM T drM
dr1
(16.2)
i=1
where TM = T is the terminal time. In the limit t → 0 (or M → since T = M t) this
gives an exact path integral representation of the price. Lattice methods arise by choosing a
finite number M of time slices and evaluating equation (16.2) by using efficiently recombining
lattice point integral approximations. For zero-coupon bonds we have a pay-off of one dollar
with certainty ( rT T = 1); hence
Z0 T = Z0 r0 T = E0 exp −
T
0
rs ds
(16.3)
Of interest are the Arrow–Debreu prices, denoted by G r0 0 r T and given by
G r0 0 r T = E0 exp −
T
0
rs ds
rT − r rt=0 = r0
(16.4)
which is the expectation of an infinitely narrow butterfly spread pay-off (i.e., the Dirac delta
function) conditional on the short rate’s starting at r0 at time t = 0. These correspond to the
worth at time t = 0, given (i.e., conditional on) current state r0 , of a riskless security that
pays one dollar if state rT = r is attained at any later time T > 0. The zero-coupon bonds are
expressed in terms of the Arrow–Debreu values as follows:
Z0 T =
0
G r0 0 r T dr
(16.5)
An important consistency requirement is the continuity relation
G r0 0 ri Ti =
0
G r0 0 ri−1 Ti−1 G ri−1 Ti−1 ri Ti dri−1
(16.6)
This formula is the basis for a discrete version that is used in the sections that follow
to generate a forward induction procedure for propagating the Arrow–Debreu prices. The
function G ri−1 Ti−1 ri Ti is the Arrow–Debreu value conditional on the short rate’s having
value ri−1 at time Ti−1 and attaining a value of ri at a later time Ti > Ti−1 . We conclude this
section by noting that the quantity Z r t t + t = Zt r t + t defined by the conditional
expectation
Z r t t + t = Et exp −
=
0
t+ t
t
rs ds
rt = r
drT G r t rT T = t + t
(16.7)
gives the price of a discount bond at time t ≥ 0 (any time later than current time), with
time to maturity of t, conditional on the short rate’s having value r at time t. Note that
here we have explicitly denoted the conditional nature of the expectation. This formula, in
conjunction with concatenating equation (16.6) for every time step Ti − Ti−1 , forms the basis
for producing lattice pricing formulas of derivatives, such as caplets, floorlets, and swaptions
dealt with later.
16.2 Binomial Lattice Calibration for Discount Bonds
381
16.2 Binomial Lattice Calibration for Discount Bonds
In developing binomial interest rate trees we subdivide calendar time t ∈ 0 T into M subintervals T0 = 0 T1 T2
TM with time spacing t = Ti − Ti−1 . At each time t = Ti there
are i + 1 nodes corresponding to the attainable values of the short rate r j i ≡ r j Ti ,
j=0 1
i. Note that throughout we use a notation for the short rate whereby rt denotes
the continuous short rate variable at calendar time t, whereas r j i ≡ r j Ti corresponds
to the discretized short rate value at the node with state j and time Ti . Also, note that the
indexing of the nodes in binomial models is such that the index has nonnegative value: j ≥ 0.
Figure 16.1 gives a schematic of the binomial interest rate tree. The two binomial models
considered in this project are the Ho–Lee (HL) and Black–Derman–Toy (BDT) models. The
HL model is the simplest, with no mean reversion. The HL model follows a normal stochastic
process
drt =
t dt +
t dWt
(16.8)
where t and t are deterministic drift and volatility functions, respectively. One obvious
shortcoming of this model is the admittance of negative interest rates. The BDT model
removes these deficiencies by considering the logarithm of the short rate, which is assumed
to follow a stochastic process of the form
d log rt =
t +
d
log
dt
t log rt dt +
t dWt
(16.9)
where t is the lognormal volatility and the drift function allows for a drift component
as well as a mean-reversion component for the variable log rt . Note that the speed of the
mean reversion is zero for the case of constant volatility. Note that throughout this study we
shall assume a constant volatility. Hence, mean reversion shall remain zero in the current
implementation of the BDT model. In contrast, mean reversion is introduced in later sections
where we implement the Hull–White and Black–Karasinski models using trinomial lattices.
The HL lattice model can be defined by a set of nodes placed according to
r j i = r j −1 i +2
√
t
(16.10)
r (M, M)
r (1, 1)
r (2, 2)
r ( j, i – 1)
r ( j, i )
r (1, 2)
r (0, 0)
r (0, 1)
r (0, 2)
r ( j, M)
r ( j – 1, i – 1)
r (0, M)
T0
T1
T2
Ti – 1
Ti
TM
FIGURE 16.1 A binomial lattice originating at the current short-rate node r 0 0 .
382
C H A P T E R 16
. Project: Interest rate trees: Calibration and pricing
whereas the BDT lattice can be taken as
r j i = r j − 1 i exp 2
√
t
(16.11)
M. Here
is a (lattice) volatility parameter
for given time slice Ti = i t, i = 0 1
for the short rate. Throughout the calibration to market zero-coupon bonds, as discussed in
this section, the volatility shall be preset to a fixed value, independent of time. It should be
noted that there exist a whole range of fixed values that can produce identical matches
to the same set of market zero-coupon bond prices. Different values for this parameter have
the effect of shifting the overall drift of the tree so as to still keep it risk neutral. Note that
the assumption of a fixed volatility eliminates the reversion component in the BDT model.
By allowing the volatility to be time dependent, one can further calibrate to a larger set of
market instruments besides zero-coupon bonds. The first step to consider in the calibration of
discount bond prices is the interpolation of yields from given treasury yield data. Consider the
M, and set the current time t = T0 = 0. The discount
set of maturities T = Ti , i = 1 2
Z0 TM derived
curve for the calibration consists of the set of prices Z0 T1 Z0 T2
y0 TM . This set of yields
from the set of corresponding yields yt Ti = y0 T1 y0 T2
does not, in practice, match the actual input market set of N maturity yields given at a fixed
y0 T¯ N . The latter are the actual treasury
set of times denoted by the set y0 T¯ 1 y0 T¯ 2
yields at times T¯ 1 = 3 months, T¯ 2 = 6 months, etc. The foregoing discount prices are therefore
obtained after having interpolated for the yields yt Ti at each ith time step. This must be
done either by employing a simple linear interpolation or by using a spline-fitting algorithm
of higher order, such as a cubic spline. The MFFit numerical library class is useful for this
purpose.
Lattice methods correspond to fixing the number of integrations in all the equations of
the previous section into some fixed integer, such as the number of time steps in the case of
pricing, and, in turn, evaluating each integral using only two (for the case of a binomial lattice)
or three (for trinomial lattices) points of integration. An important approximation underlying
the binomial lattice methodology is to set the conditional transition density for every time
step t simply as a constant, p ri ri−1 t = 21 . Moreover, the short rate is taken as locally
constant ri−1 within time intervals Ti−1 Ti , hence giving the conditional Arrow–Debreu
values for t maturity as the simple form G rk Ti−1 rj Ti ≡ G k Ti−1 j Ti = 21 e−ri−1 t .
By adopting the binomial short-rate lattices defined by equation (16.10) (for the HL model)
or (16.11) (for the BDT model), we now are in a position to obtain the discrete-time versions
of the equations in the previous section.
To begin with, equation (16.6) takes the discrete form
G 0 0 j Ti =
G 0 0 k Ti−1 G k Ti−1 j Ti
(16.12)
1
exp −r k i − 1
2
(16.13)
k=j−1 j 0 ≤ k ≤ i−1
where
G k Ti−1 j Ti =
t
Note that equation (16.12) describes a procedure that takes into account the Arrow–Debreu
prices at intermediate nodes r k i − 1 for the previous time Ti−1 , which are subsequently
used for time stepping by an amount t until a terminal node r j i is reached at the time
slice Ti = Ti−1 + t. The sum involves only two possible values for k: k = j − 1 and k = j,
with the restriction that 0 ≤ k ≤ i − 1. For the extreme (highest or lowest) node there is
only one term in the sum. This is the forward induction equation that is used in practice to
generate all Arrow–Debreu prices G 0 0 j Ti for each jth node at terminal time Ti . It is
16.2 Binomial Lattice Calibration for Discount Bonds
r( j, i – 1 )
G(
k=
;k
(0, 0
)
T –1
= j, i
r(0, 0 )
j, T
i–
G (0, 0 ; k = j – 1, Ti –1)
1;
j, T
i )r
)
G (0, 0 ; j, Ti
G
383
r( j, i )
)
; j, T i
1, T i –1
=j–
G (k
r( j – 1, i – 1 )
Ti – 1
T0
Ti
FIGURE 16.2 A pictorial representation of the forward propagation of Arrow–Debreu prices on a
binomial lattice originating at the current short-rate node r 0 0 . Their values, i.e., G 0 0 j Ti , at
nodes corresponding to a later time step ti are obtained as a sum of contributions from (at most) two
intermediate time Ti−1 two-legged paths.
important to note that this forward induction equation can generally be used for any type of
short-rate model (see Figure 16.2). The discrete-time version of the security price given by
equation (16.2) takes the form
i
P 0 r0 T i =
G 0 0 j Ti
r j Ti Ti
(16.14)
j=0
where G 0 0 j Ti are computed using the forward induction relation in equation (16.12).
Specializing this formula to the case of zero-coupon bonds, which have a riskless pay-off of
one dollar, we have the discrete-time lattice version of equation (16.5):
i
Z0 Ti =
G 0 0 j Ti
(16.15)
j=0
Hence, the Arrow–Debreu prices at all the nodes of a given maturity T are sufficient for
determining the price of a discount bond of that maturity (see Figure 16.3). In the calibration
procedure the market zero-coupon prices at times T = Ti are used as input to the left-hand
side of equation (16.15). By solving for the nodes at the i − 1 th time step, for every time
slice Ti , we imply the whole lattice and hence obtain the market prices of all discount
bonds correctly. In practice, the right-hand side, for each T = Ti , is computed by using the
vector of Arrow–Debreu values G 0 0 k Ti−1 , k = 0 1
i − 1, which are assumed to
be known from the previous time step, as well as a trial vector of nodes r k i − 1 . These are
plugged into forward induction equation (16.12) while using equation (16.13) and summing
up all node contributions via equation (16.15). At the same time, one also makes use of the
constraint among the r k i − 1 , namely, equation (16.10) or (16.11), depending on whether
one is calibrating the Ho–Lee or BDT model, respectively. Hence, this reduces the discount
bond calibration problem to a succession of M root-finding problems that make the leftand right-hand sides of equation (16.15) equal for each Ti . Note also the single-variable
nature of the problem, since the expressions are reduced to finding just one node, i.e., the
lowest one r 0 i − 1 , and the rest follow for time slice Ti−1 . Observe that at each maturity
the nodes being computed are lagged by one time step. One can use the MFZero library
class for the purpose of finding roots. To start the procedure off, one uses G 0 0 0 0 = 1
384
C H A P T E R 16
. Project: Interest rate trees: Calibration and pricing
r ( i, i )
, 0;
)
•••
G(0
Ti
j = i,
r ( j + 1, i )
T)
; j + 1, i
G(0, 0
G(0, 0 ; j,Ti )
r ( j, i )
•••
r (0, 0)
G(0,
0; j =
0, T )
i
T0
r (0, i )
Ti
FIGURE 16.3 Lattice calibration of zero-coupon bonds of maturity Ti uses a sum of Arrow–Debreu
i, at time
prices beginning from the current time T0 and node r 0 0 up to all nodes r j i , j = 0
Ti . Note, however, that calibration up to maturity time Ti determines the short-rate lattice points up to
the previous time step with time Ti−1 .
and solves for r 0 0 . Note that since there are only two branches in this case, giving
G 0 0 k = 0 1 t = 21 exp −r 0 0 t , one has
r 0 0 =−
log Z0 T1 = t
t
(16.16)
where the first node, r 0 0 , is given by the smallest-maturity zero-coupon bond price (i.e., the
initial term structure). If one assumes continuous compounding, then one can also avoid the
numerical root-finding procedure in the case of the Ho–Lee model, which admits a simple
analytical solution for the node positions r j i − 1 in terms of Z0 Ti and the Arrow–Debreu
prices for terminal time Ti−1 .
16.3 Binomial Pricing of Forward Rate Agreements, Swaps,
Caplets, Floorlets, Swaptions, and Other Derivatives
Recall from Chapter 2 the price of a plain-vanilla FRA of a given tenor = Ti+1 − Ti .
Assuming continuous compounding, equation (2.8) can be used to give the net present value
of an FRA (to the party receiving an initial nominal amount) with one dollar nominal:
PV FRA
0
= −Z0 T1 + e
f0 T1 T2
Z0 T2
(16.17)
where the forward is given by
f0 T1 T2 =
1
log
Z0 T1
Z0 T2
(16.18)
Since all expressions are completely determined by the prices of the zero-coupon bonds, it
necessarily follows that all FRAs are also exactly priced by the binomial lattices obtained
from the calibration procedure in the previous section. Moreover, as recalled from Chapter 2,
a swap is just a collection of FRAs. Plain-vanilla swaps are, therefore, also priced exactly
within the foregoing calibration framework.
16.3 Binomial Pricing of Forward Rate Agreements
385
The pricing of options such as caps (or caplets) is not as straightforward as that for
FRAs. In particular, recall from Chapter 2, the pay-off of a caplet struck at fixed interest rate
RK , maturing at time T, on a floating reference rate R T of tenor applied to the period
T T + in the future. The floating rate is typically the three- or six-month LIBOR. The
pay-off of this caplet option is given by
T
Cpl R
T = R
T − RK
+
Z T rT T +
(16.19)
where ZT rT T + is the discount function over that period, since the cash flow occurs at
time T + . Here we define x + ≡ max x 0 as usual. In order to obtain the price at current
time t = 0 of this caplet one must take an expectation, or integral, of the pay-off with a
risk-neutral distribution in the reference rate Rt , where t = T , i.e., the expiry or maturity
time of the option on the (call-type) pay-off. The latter is, however, expressed in terms of a
rate applied to the period of the tenor (i.e., the reference forward rate) and not the short rate
used in the rate lattice calibration of the previous section. In particular, the short rate lattice
gives the conditional distribution of the short rate.
To price the caplet, one must relate the short rate to this reference forward rate. In
particular, the values of the short rate at the nodes j i , r j i must be related to the values of
the reference forward rates, denoted by R j i , at these nodes. This is achieved by using the
continuous-time relation for the forward rate, and this is where the conditional zero-coupon
prices Z r t T are useful. In particular, for continuous compounding,
R
t =
1
R
t =
1
log
Zr t t
Z r t t+
(16.20)
log
1
Z r t t+
(16.21)
and since Z r t t = 1,
Choosing t = Ti and T = Ti + n t, where it is assumed that the tenor is exactly n periods of
the lattice time step, for some integer n, = n t, we arrive at the discrete time value at the
jth node:
R
j i =
1
log
1
Z j Ti Ti + n
t
(16.22)
Here Z j Ti Ti + n t ≡ Z j Ti Ti+n is the zero-coupon value maturing at time Ti+n (n
time steps in the future), conditional on the short rate’s having value r j i at time Ti . Based
on equations (16.19) and (16.22), we can write all components of the payoff vector of the
caplet at each node j i , denoted by C j i , as
C
j i = R
j i − RK
+
Z j Ti Ti+n
(16.23)
where equation (16.22) is plugged in for R j i . Note that the preceding equations assume
continuous compounding, while a similar set of equations obtain for the case of discrete
compounding, where the log x function is simply replaced by x. The foregoing payoff vector
introduces an extra procedural step, requiring one to compute the quantities Z j Ti Ti+n ,
which involve a separate forward induction starting from the nodes r j i . In practice, these
are computed using the discrete-time version of equation (16.7):
j+n
Z j Ti Ti+n =
G j Ti k Ti+n
k=j
(16.24)