(19d)
Proof. See Appendix A.
The accuracy and performance of these drift approximation methods are discussed in section IV.
  1. 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).
  1. 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.
  1. 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.