Multi-period lattice valuation

Repeat the one-period argument backward

The one-period binomial lesson valued two terminal payoffs. A multi-period lattice applies the same pricing step at every node, beginning with the last row and moving backward to valuation-time.

At the node with lattice-time-index $\explain{lattice-time-index}{i}$ and lattice-state-index $\explain{lattice-state-index}{j}$, let the lattice-node-up-weight be $\explain{lattice-node-up-weight}{q_{\explain{lattice-time-index}{i},\explain{lattice-state-index}{j}}}$ and the lattice-node-discount-factor be $\explain{lattice-node-discount-factor}{d_{\explain{lattice-time-index}{i},\explain{lattice-state-index}{j}}}$. Let the local next-date cash flow $\explain{lattice-next-cash-flow}{CF_{i+1,j}}$ be paid at the successor node $(\explain{lattice-time-index}{i}+1,\explain{lattice-state-index}{j})$; the node value $\explain{lattice-node-value}{V}_{\explain{lattice-time-index}{i}+1,\explain{lattice-state-index}{j}}$ excludes it. The lattice-node-value at node $(\explain{lattice-time-index}{i},\explain{lattice-state-index}{j})$ is the discounted risk-neutral expectation of the successor cash flows plus the successor node values:

$$\explain{lattice-node-value}{V_{i,j}} =\explain{lattice-node-discount-factor}{d_{i,j}} \left[ \left(1-\explain{lattice-node-up-weight}{q_{i,j}}\right) \left(\explain{lattice-next-cash-flow}{CF_{i+1,j}} +\explain{lattice-node-value}{V_{\explain{lattice-time-index}{i}+1,\explain{lattice-state-index}{j}}}\right) +\explain{lattice-node-up-weight}{q_{i,j}} \left(\explain{lattice-next-cash-flow}{CF_{\explain{lattice-time-index}{i}+1,\explain{lattice-state-index}{j}+1}} +\explain{lattice-node-value}{V_{\explain{lattice-time-index}{i}+1,\explain{lattice-state-index}{j}+1}}\right) \right]$$

(4.5.1)

The two successors of node $(\explain{lattice-time-index}{i},\explain{lattice-state-index}{j})$ are the nodes $(\explain{lattice-time-index}{i}+1,\explain{lattice-state-index}{j})$ and $(\explain{lattice-time-index}{i}+1,\explain{lattice-state-index}{j}+1)$. The labels “up” and “down” are conventions. The input up weights and values must use the same order.

Timing is part of the model

In (4.5.1), each next-date cash flow is added to the value of its successor node before discounting. So the value of a coupon bond at a node is ex-coupon: it excludes the coupon paid at that node. A change of the order of exercise and coupon payment changes the contract, and it can change the option payoff.

Diagram 4.5.1Backward-induction order. Start from terminal node values. At each earlier row, add any next-date cash flow to each successor value, form the local risk-neutral weighted average, multiply by the local discount factor, and repeat.

Worked example

The terminal values are USD 0, USD 20, and USD 40. At time index 1, both nodes have discount factor 0.90, and their up weights are 0.25 and 0.75. At time index 0, the discount factor is 0.95 and the up weight is 0.50. Applying (4.5.1) row by row gives the node values:

$$\begin{aligned} \explain{lattice-node-value}{V}_{1,0}&=0.90\left[\left(1-0.25\right)\left(0\right)+0.25\left(20\right)\right]=4.50\\ \explain{lattice-node-value}{V}_{1,1}&=0.90\left[\left(1-0.75\right)\left(20\right)+0.75\left(40\right)\right]=31.50\\ \explain{lattice-node-value}{V}_{0,0}&=0.95\left[\left(1-0.50\right)\left(4.50\right)+0.50\left(31.50\right)\right]\\ &=17.10\ \text{USD} \end{aligned}$$

(4.5.2)

backwardInductionValue reproduces the three node rows [[17.1], [4.5, 31.5], [0, 20, 40]] in its independent reference test.

Knowledge check 4.5.1 Multi-period lattice

1. DirectQuestion 1Not answered

2. TransferQuestion 2Not answered

Sources

The replication, risk-neutral weighting, and backward-induction construction follow Hull, Chapter 12 §§12.1–12.3 [1]. The node discount factors are model inputs. Calibration and short-rate dynamics are outside this lesson.

References

1. Hull, Options, Futures, and Other Derivatives (8th ed., 2012). draft ↩