Trinomial Tree Option Pricing: Three-State Lattice and Early Exercise

For educational purposes only; not investment advice.

Direct answer

A trinomial tree prices an option on a discrete time-and-price lattice where the underlying moves up, stays at a middle level, or moves down at each step. After calibrating risk-neutral transition probabilities, it calculates expiration payoffs and works backward to today by discounting the probability-weighted value at every node.

The extra middle branch gives more freedom than a binomial tree to match moments, align nodes with strikes or barriers, and improve numerical behavior. It does not automatically make the price more accurate. The parameterization, number of steps, dividends, exercise rule, boundary treatment, and market inputs determine the result.

Trees are especially useful for American-style options because every node can compare continuation with immediate exercise. They are numerical models, not claims about the three prices the stock will actually take.

Lattice and backward induction

For step length Δt, define stock multipliers u > m > d and probabilities pᵤ, pₘ, p_d. A recombining tree is parameterized so paths such as up-then-down return to an existing node, keeping the node count manageable.

Risk-neutral probabilities must satisfy at least:

pᵤ + pₘ + p_d = 1

and, with continuous dividend yield q,

pᵤu + pₘm + p_dd = e^((r−q)Δt).

A specific trinomial scheme adds variance or moment-matching conditions to determine the three probabilities and multipliers. All probabilities must be between zero and one. Negative “probabilities” indicate an invalid grid or incompatible parameters, not a tradable prediction.

At expiration, set V = max(S−K,0) for a Call or max(K−S,0) for a Put. At each earlier node, the European continuation value is

V_continue = e^(−rΔt)(pᵤVᵤ + pₘVₘ + p_dV_d).

For an American option, use

V = max(V_continue, intrinsic value).

Repeat until the root. Discrete cash dividends require the stock tree or node transitions to reflect the ex-dividend drop. Greeks can be estimated from neighboring nodes, but grid alignment and finite differences introduce numerical noise.

One-step worked example

Let S₀ = $100, Put strike K = $105, and one-step risk-free growth factor R = 1.02. Use u = 1.10, m = 1.00, d = 0.90 and probabilities

pᵤ = 0.35, pₘ = 0.50, p_d = 0.15.

They sum to one, and expected stock growth is

0.35(1.10) + 0.50(1.00) + 0.15(0.90) = 1.02 = R.

Terminal stock	Put payoff `max(105−S,0)`	Probability-weighted payoff
`$110`	`$0`	`$0`
`$100`	`$5`	`$2.50`
`$90`	`$15`	`$2.25`

The European Put value is

V₀ = ($0 + $2.50 + $2.25) / 1.02 = $4.6569.

For an American Put, immediate exercise at the root is $105 − $100 = $5. Because $5 > $4.6569, this one-step model chooses exercise and values the American Put at $5.

As a cross-check, the corresponding European Call payoff is $5 only in the up state, so C = 0.35($5)/1.02 = $1.7157. Then C − P = −$2.9412, matching S₀ − K/R = $100 − $105/1.02 = −$2.9412.

Model checks and risks

State the exact tree parameterization; “trinomial” alone does not define u, d, or probabilities.
Reject negative probabilities and verify the risk-neutral growth and variance conditions.
Increase steps and record convergence rather than trusting one grid.
Compare European results with Black-Scholes under matching assumptions and check intrinsic/no-arbitrage bounds.
Test American values against European values and inspect the exercise boundary.
Model discrete dividends, rates, borrow, exercise style, settlement, and contract adjustments correctly.
Align important strikes, barriers, and dividend dates with the grid when possible.
Stress volatility, step count, time discretization, and boundary choices.
Separate model value from executable Bid/Ask, fees, slippage, and available size.
Use independent implementation or benchmark cases to detect indexing and discounting errors.

Common misconceptions

“Three branches are forecasts for up, flat, and down.” They are numerical states calibrated for pricing.
“The probabilities are historical odds.” They are risk-neutral weights determined by the model.
“More branches guarantee accuracy.” A poorly calibrated trinomial tree remains wrong.
“One-step early exercise proves real holders should exercise now.” The result depends on the coarse example and omitted inputs.
“A model price is a fair executable quote.” Market supply, spreads, inventory, and costs determine fills.
“Convergence means the model is true.” It shows numerical stability within assumptions, not that the assumptions fit the market.