A lattice is a discrete time representation of the evolution of the underlying asset’s price. It divides a certain time interval, like the interval between the option initial date (year 0) and the maturity date (year T ), into n equal time steps. The length of each time step ∆t is equal to T /n. It approximates the distribution of the underlying asset’s price at each time step. A lattice consists of nodes and branches connect them.
Each node at time i can be viewed as a possible asset’s price at time i. Each branch that connects two nodes located at adjoint time steps denotes a possible evolution of the underlying asset’s price. The well-known CRR binomial lattice will be introduced next as an example to show what a lattice looks like, how it is constructed, and how an option is priced with the lattice model [15].
2.5.1 The Structure of CRR Binomial Lattice
A 2-time-step CRR binomial lattice is illustrated in Fig. 2.4(a). Recall that Si denotes the value of the underlying asset at time i. Si+1 equals Siu with probability Pu and Sid with probability Pd(≡ 1 − Pu), where d < u. u is equal to eσ√∆t, where σ denotes the annual volatility of the underlying asset’s price (see Eq. (2.1)). The identity
ud = 1 (2.11)
holds in this lattice model. The probability Pufor an up move is set to (er∆t−d)/(u−
d). Both d ≤ er∆t ≤ u and 0 < Pu < 1 must hold to avoid arbitrage. The asset’s price at time i that results from j down moves and i − j up moves therefore equals S0ui−jdj with probability i
Figure 2.4: The CRR Binomial Lattice.
A 2-time-step CRR binomial lattice model are illustrated in (a) and (b). The price of the underlying asset for each node is illustrated in (a) and the alias (N (∗, ∗)) for each node is illustrated in (b). N (i, j) stands for the node at time i with j cumulative down moves. The probability of reaching each node is listed under the node.
We now map the asset’s prices to nodes on the CRR binomial lattice used for pricing. Node N(i, j) stands for the node at time i with j cumulative down moves.
Its associated asset’s price is hence S0ui−jdj. The asset’s price can move from N(i, j) to N(i + 1, j) with probability Pu and to N(i + 1, j + 1) with probability Pd. As a consequence, node N(i, j) can be reached from the root with probability i
j
Pui−jPdj. See Fig. 2.4(b) for illustration.
2.5.2 How to Construct a Lattice
We use the well-known Cox-Ross-Rubinstein (CRR) binomial lattice to illustrate how a lattice is constructed in principle. The logarithmic asset’s price mean (µ) and variance (Var) one time step from now are derived from Eq. (2.1) as
µ ≡ (r − 0.5σ2)∆t, (2.12)
Var ≡ σ2∆t. (2.13)
To make sure that the lattice converges to the continuous-time asset’s price process, the mean and the variance of the logarithmic price process should be calibrated by matching those of the lattice and those of the continuous-time model:
Puln u + Pdln d = µ, (2.14) Pu(ln u − µ)2+ Pd(ln d − µ)2 = Var. (2.15) Note that
Pu+ Pd= 1. (2.16)
The 4 parameters (Pu, Pd, u, and d) are uniquely obtained by solving Eqs. (2.11), (2.14)–(2.16). The branching probabilities Pu and Pd should be between 0 and 1 to meet the no-arbitrage requirements. In the CRR lattice, this demand can always be met by suitably increasing n [35].
Construction of a Multinomial Lattice
If each node in a lattice can branch to # nodes at the next time step, we call it an
#-nomial lattice. The above idea can be applied to construct an #-nomial lattice. Note that 2# degrees of freedoms are provided by an #-nomial lattice. They include # price multiplicative factors (like u and d in the CRR binomial lattice) and # branching probabilities (like Pu and Pd in the CRR binomial lattice). These branching proba-bilities must be between 0 and 1 to meet the no-arbitrage requirements. We need 2#
independent equations to determine these 2# variables uniquely. The calibration of mean and variance gives 2 equations. The branching probabilities sum to 1, giving another one. Additional 2# − 3 equations must be added. For example, Eq. (2.11) is the additional equation used in the CRR binomial model.
Generally speaking, most of these extra equations are enforced to meet specific requirements. The number of these extra equations determines the lattice model that will be constructed. In Chapter 3, a trinomial lattice (# = 3) will be constructed to price an Asian option.
2.5.3 Pricing an Ordinary Option with the Lattice Approach
Options can be priced by the backward induction method on a lattice. Let us focus on pricing a European-style call on the CRR binomial lattice first. Recall that option
value can be obtained by taking expectation of the future discounted payoffs as in Eq.
(2.9), which can be decomposed into a combination of numerous formulae as follows:
C = e−r∆t(PuCu+ PdCd), (2.17) where C denotes the option value of an arbitrary node N, Cu denotes the option value of the node that can be reached from N by a upward movement, and Cd denotes the option value of the node that can be reached from N by a downward movement. The option value for each node at the maturity date (at time n) is defined by Eq. (2.2).
The option value for each node at time i (0 ≤ i ≤ n − 1) is evaluated by substituting the option values of the nodes at time i + 1 into the right hand side of Eq. (2.17).
The pricing result is the option value at the root node.
100 112.5
200 200
50 25
400 350
100 50
25 0
Figure 2.5: Pricing an Ordinary Option on a CRR Lattice.
The upper cell of each node (colored of gray) denotes the price of the underlying asset, and the lower cell of each node denotes the option value at that node.
Take a simple 2-time-step CRR binomial lattice model illustrated in Fig. 2.5 as an example. The upward factor u and the downward factor d are 2 and 0.5, respectively.
The upward branching probability Pu and the downward branching probability Pd
are both 0.5. The risk-free rate is set as 0, and the strike price is 50. The upper cell of each node (colored of gray) denotes the price of the underlying asset, and the lower cell of each node denotes the option value at that node. The option value of a node at time 2 (the maturity date) is computed by Eq. (2.2). For example, the option value for the uppermost node at time 2 is
max(400− 50, 0) = 350.
For each node at time 0 or time 1 can be evaluated by applying Eq. (2.17). For example, the option value for the upper node at time 1 is
e0(0.5 × 350 + 0.5 × 50) = 200.
The final pricing result is 112.5.
An American-style option can be exercised before the maturity date. An option will be exercised early if the option holder finds that it is more beneficial to exercise the option than to hold the option. The lattice approach can handle this by modifying Eq. (2.17) as follows:
C = max(Exercise value, e−r∆t(PuCu + PdCd)), (2.18) where the “Exercise value” denotes the payoff to exercise the option immediately (see Eq. (2.3)).
In the above method, the option value of each node is evaluated once by applying Eq. (2.2) (for the nodes at time n), Eq. (2.17) (for pricing European-style options), or Eq. (2.18) (for pricing American-style option). Since the above mentioned equations can be evaluated in constant time and there are O(n2) nodes in the lattice, the computational time complexity is O(n2).
2.5.4 The PDE Approach
The value of the option may be represented as the solution of a PDE. When the PDE cannot be solved analytically, the solution can be approximated by the finite difference method numerically. A finite difference method places a grid of points on the space over which the desired function takes value and then approximates the function value at each of these points. The approximation method is similar to the backward induction used in the lattice approach.