Topic 16 – Trees / Lattice Methods

1. What a tree / lattice really is

Idea:

“Approximate the continuous price process with a recombining grid of up/down moves and price the option by backward induction.”

Simplest version: binomial tree (Cox–Ross–Rubinstein style).

• Time axis: split $[0,T]$ into $N$ steps of size $\Delta t$.
• At each step:
  • Price either goes up by factor $u$,
  • Or down by factor $d$.
• Tree recombines:
  • After $n$ steps, you have $n+1$ possible price levels, not $2^n$.

Under risk-neutral measure:

• Choose risk-neutral probability $p$ so that: $$\mathbb{E}[S_{t+\Delta t} \mid S_t] = S_t e^{r\Delta t}$$
• Then price is:
  • “Expected discounted payoff” computed backwards on the tree.

2. Backward induction and early exercise

Algorithm sketch (European):

1. Build terminal node values:
   • $V_N(j) = \text{payoff}(S_N(j))$.
2. Step backwards: $$V_n(j) = e^{-r\Delta t}\left[p V_{n+1}(j+1) + (1-p) V_{n+1}(j)\right]$$
3. At $n=0$ and $j=0$ you get today’s price.

For American options:

• At each node, take max of:
  • Continuation value (as above),
  • Immediate exercise payoff.

$$V_n(j) = \max\{\text{payoff}(S_n(j)), \text{continuation}\}$$

This is where trees shine:

“They handle early exercise in a natural way by just comparing hold vs exercise at each node.”

3. Why trees are still useful

Pros:

• Intuitive; good for explaining:
  • Risk-neutral probability,
  • Hedging interpretation (replicating portfolio at each node).
• Handle American and some path-dependent features (barriers).
• Flexible:
  • Can be extended to trinomial, non-uniform time steps, local vol approximations.
• Calibration-friendly:
  • You can tweak local vol / up–down factors to match market prices across strikes/tenors (implied tree / local volatility).

Cons:

• Accuracy depends on:
  • Number of steps $N$,
  • Choice of parameters $u, d, p$.
• Higher dimension (many factors) → tree explodes.
• PDE / MC often more natural for complex problems, but trees are still used for:
  • Quick approximations,
  • Some calibration / smile fit routines,
  • Legacy risk/reporting engines.

4. Tree vs PDE vs MC – mental model

• Tree:
  • Discrete time and discrete state (price grid from moves).
  • Good for early exercise in 1D, intuitive hedging picture.
• PDE:
  • Continuous state grid, partial derivatives.
  • Similar complexity, better suited to smooth models and more precise control of numerics.
• MC:
  • Simulation-based, easy for high dimension and path-dependence.
  • Early exercise needs LSMC or dynamic programming.

You can think:

“A binomial tree is basically a crude finite-difference scheme in disguise, wrapped in a probabilistic and hedging interpretation.”

5. Interview soundbite

“Lattice methods approximate the underlying process with a recombining tree and price by backward induction. They’re very natural for American options because at each node you simply compare continuation value to immediate exercise. In low dimension they give quick, intuitive approximations and a nice hedging interpretation, and they can be adapted for calibration via local vol or implied trees. For higher dimension or complex path-dependence we usually switch to PDEs or Monte Carlo, but trees remain a useful workhorse and benchmark.”