(19d)
Proof. See Appendix A.
The accuracy and performance of these drift approximation methods are
discussed in section IV.
- THE LATTICE PROCEDURE IN THE LMM
The “lattice” is the generic term for any graph we build for the
pricing of financial products. Each lattice is a layered graph that
attempts to transform a continuous-time and continuous-space underlying
process into a discrete-time and discrete-space process, where the nodes
at each level represent the possible values of the underlying process in
that period.
There are two primary types of lattices for pricing financial products:
tree lattices and grid lattices (or rectangular lattices or Markov chain
lattices). The tree lattices, e.g., traditional binomial tree, assume
that the underlying process has two possible outcomes at each stage. In
contrast with the binomial tree lattice, the grid lattices (see Amin
[1993], Gandhi-Hunt [1997], Martzoukos-Trigeorgis [2002],
Hagan [2005], and Das [2011]) shown in Exhibit 1, which permit
the underlying process to change by multiple states, are built in a
rectangular finite difference grid (not to be confused with finite
difference numerical methods for solving partial differential
equations). The grid lattices are more realistic and convenient for the
implementation of a Markov chain solution.
This article presents a grid lattice model for the LMM. To illustrate
the lattice algorithm, we use a callable exotic as an example. Callable
exotics are a class of interest rate derivatives that have Bermudan
style provisions that allow for early exercise into various underlying
interest rate products. In general, a callable exotic can be decomposed
into an underlying instrument and an embedded Bermudan option.
We will simplify some of the definitions of the universe of instruments
we will be dealing with for brevity. Assume the payoff of a generic
underlying instrument is a stream of payments for i=1,…,N ,
where is the structured coupon. The callable exotic is a Bermudan style
option to enter the underlying instrument on any of a sequence of
notification dates . For any notification date , we define a
right-continuous mapping function by . If the option is exercised att , the reduced price of the underlying instrument, from the
structured coupon payer’s perspective, is given by
(20)
where the ratio is usually called the reduced value of the underlying
instrument or the reduced exercise value or the reduced intrinsic value.
Lattice approaches are ideal for pricing early exercise products, given
their “backward-in-time” nature. Bermudan pricing is usually done by
building a lattice to carry out a dynamic programming calculation via
backward induction and is standard. The lattice model described below
also uses backward induction but exploits the Gaussian structure to gain
extra efficiencies.
First we need to create the lattice. The random process we are going to
model in the lattice is the LMM (12). Unlike traditional trees, we only
position nodes at the determination dates (the payment and exercise
dates). At each determination date, the continuous-time stochastic
equation (12) shall be discretized into a discrete-time scheme. Such
discretized schemes basically convert the Brownian motion into discrete
variables. There is no restriction on discretization schemes. At any
determination date t , for instance, we discretize the Brownian
motion to be equally spaced as a grid of nodes , for i =
1,…, . The number of nodes and the space between nodes at each
determination date can vary depending on the length of time and the
accuracy requirement. The nodes should cover a certain number of
standard deviations of the Gaussian distribution to guarantee a certain
level of accuracy. We have the discrete form of the forward rate as
(21)
The zero-coupon bond (2) can be expressed in discrete form as
(22)
We now have expressions for the forward rate (21) and discount bond
(22), conditional on being in the state at time t , and from these
we can perform valuation for the underlying instrument.
At the maturity date, the value of the underlying instrument is equal to
the payoff, i.e.,
(23)
The underlying state process in the LMM (11) is a Brownian motion. The
transition probability density from state (, ) to state (, ) is given by
(24)
Applying the variable substitution (8), equation (24) can be expressed
as
(25)
Equation (20) can be further expressed as a conditional value on any
state (, ) as:
(26)
This is a convolution integral. Some fast integration algorithms, e.g.,
Cubic Spline Integration, Fast Fourier Transform (FFT), etc., can be
used for optimization. We use the Trapezoidal Rule Integration in this
paper for ease of illustration.
Incomplete information handling . Convolution is widely used in
Electrical Engineering, particularly in signal processing. The important
part is that the far left and far right parts of the output are based on
incomplete information. Any models that try to compute the transition
values using integration will be inaccurate if this problem is not
solved, especially for longer maturities and multiple exercise dates.
Our solution is to extend the input nodes by padding the far end values
on each side and only take the original range of the output nodes.
Next, we determine the option values in each final notification node. On
the last exercise date, if we have not already exercised, the reduced
option value in any state is given by
(27)
Then, we conduct the backward induction process that is performed by
iteratively rolling back a series of long jumps from the final exercise
date across notification dates and exercise opportunities until we reach
the valuation date. Assume that in the previous rollback step , we
calculated the reduced option value: . Now, we go to . The reduced
option value at is
(28a)
where the reduced continuation value is given by
(28b)
We repeat the rollback procedure and eventually work our way through the
first exercise date. Then the present value of the Bermudan option is
found by a final integration given by
(29)
The present value or the price of the callable exotic from the coupon
payer’s perspective is:
(30)
This framework can be used to price any interest rate products in the
LMM setting and can be easily extended to the Swap Market Model (SMM).
- Calibration
First, if we choose the LMM as the central model, we need to price
interest rate derivatives that depend on either or both of cap and
swaption markets. Second, we will undoubtedly use various swaptions to
hedge a callable exotic. It is a reasonable expectation that the
calibrated model we intend to use to price our exotic, will at least
correctly price the market instruments that we intend to hedge with.
Therefore, in an exotic derivative pricing situation, recovery of both
cap and swaption markets might be desired.
The calibration of the LMM to caplet prices is quite straightforward.
However, it is very difficult, if not impossible, to perfectly recover
both cap and swaption markets. Fortunately for the LMM, there also exist
extremely accurate approximate formulas for swaptions implied
volatility, e.g., Rebonato’s formula.
We introduced a parameter and set where denotes the market Black caplet
volatility. One can choose different for different . For simplicity we
describe one situation here. By choosing , we have perfectly calibrated
the LMM to the caplet prices in the market. However, our goal is to
select a to minimize the sum of the squared differences of the
volatilities derived from the market and the volatilities implied by our
model for both caps and swaptions combined.
In the optimization, we use Rebonato’s formula for an efficient
expression of the model swaption volatilities, given by
(31a)
where =1 under one-factor LMM. The swap rate is given by
(31b)
(31c)
Assume the calibration containing caplets and G swaptions. The error
minimization is given by
(32)
where denotes the market Black swaption volatility. The optimization can
be found at a stationary point where the first derivative is zero; that
is,
(33)
In terms of forward volatilities, we use the time-homogeneity assumption
of the volatility structure, where a forward volatility for an option is
the same or close to the spot volatility of the option with the same
time to expiry. The time-homogeneous volatility structure can avoid
non-stationary behavior.
In the LMM, forward swap rates are generally not lognormal. Such
deviation from the lognormal paradigm however turns out to be extremely
small. Rebonato [1999] shows that the pricing errors of swaptions
caused by the lognormal approximation are well within the market bid/ask
spread. For most short maturity interest rate products, we can use the
lattice model without calibration (33). However, for longer maturity or
deeply in the money (ITM) or out of the money (OTM) exotics we may need
to use the calibration and even some specific skew/smile adjustment
techniques to achieve high accuracy.
- NUMERICAL IMPLEMENTATION
In this section, we will elaborate on more details of the
implementation. We will start with a simple callable bond for the
purpose of an easy illustration and then move on to some typical
callable exotics, e.g., callable capped floater swap and callable range
accrual swap. The reader should be able to implement and replicate the
model after reading this section.