17.2 Binomial Trees

Binomial Trees

The binomial model is the FRM workhorse for discrete-time option pricing and for American early-exercise decisions that closed-form Black–Scholes cannot handle. You build a recombining tree for the underlying, compute terminal payoffs, and induce backward under a risk-neutral measure.

One-Step European Call and Put

Spot S0. Over time Δt = T, the stock moves to S0·u or S0·d. A European call payoff is max(S_T − K, 0); a put is max(K − S_T, 0). With continuous risk-free rate r and no dividends, the risk-neutral probability of an up move is

p* = (e^(r Δt) − d) / (u − d)

and the option value is

f = e^(−r Δt) [p* f_u + (1 − p*) f_d]

Worked one-step European call

S0 = 20, K = 21, r = 10%, T = 0.25 years, u = 1.10, d = 0.90.

S_u = 22, S_d = 18.

Call payoffs: f_u = max(22 − 21, 0) = 1; f_d = max(18 − 21, 0) = 0.

e^(r Δt) = e^(0.10 × 0.25) = e^0.025 ≈ 1.0253.

p* = (1.0253 − 0.90) / (1.10 − 0.90) = 0.1253 / 0.20 = 0.6265.

f = e^(−0.025) [0.6265 × 1 + 0.3735 × 0] ≈ 0.9753 × 0.6265 ≈ 0.611.

Worked one-step European put (same tree)

Put payoffs: P_u = max(21 − 22, 0) = 0; P_d = max(21 − 18, 0) = 3.

P = e^(−0.025) [0.6265 × 0 + 0.3735 × 3] ≈ 0.9753 × 1.1205 ≈ 1.093.

Check put-call parity: c + Ke^(−rT) = 0.611 + 21 × e^(−0.025) ≈ 0.611 + 20.481 = 21.092; p + S0 = 1.093 + 20 = 21.093. Match within rounding.

Two-Step Trees

Split T into two steps of length Δt = T/2. Nodes: S0; after one step S0u, S0d; after two steps S0u², S0ud, S0d² (recombining).

At each penultimate node, compute the continuation value as the discounted risk-neutral average of the two successor option values. For Europeans, that continuation is the node value. For Americans, node value = max(exercise, continuation).

Worked two-step European call

S0 = 50, K = 50, r = 5%, T = 0.50, σ calibration giving u = 1.20, d = 0.8333 (≈ 1/u), Δt = 0.25.

e^(r Δt) = e^(0.05 × 0.25) = e^0.0125 ≈ 1.0126.

p* = (1.0126 − 0.8333) / (1.20 − 0.8333) = 0.1793 / 0.3667 ≈ 0.489.

Terminal spots: S_uu = 72, S_ud = 50, S_dd = 34.72.

Call payoffs: 22, 0, 0.

At up node (S = 60): f_u = e^(−0.0125) [0.489 × 22 + 0.511 × 0] ≈ 0.9876 × 10.758 ≈ 10.62.

At down node (S = 41.67): f_d = e^(−0.0125) [0.489 × 0 + 0.511 × 0] = 0.

At time 0: f = e^(−0.0125) [0.489 × 10.62 + 0.511 × 0] ≈ 5.13.

Worked two-step American put

Same tree, American put, K = 50.

Terminal puts: max(50 − 72, 0) = 0; max(50 − 50, 0) = 0; max(50 − 34.72, 0) = 15.28.

At up node (S = 60): continuation = e^(−0.0125) [0.489 × 0 + 0.511 × 0] = 0; exercise = max(50 − 60, 0) = 0 → node = 0.

At down node (S = 41.67): continuation = e^(−0.0125) [0.489 × 0 + 0.511 × 15.28] ≈ 0.9876 × 7.808 ≈ 7.71; exercise = 50 − 41.67 = 8.33. American chooses 8.33 (early exercise).

At time 0: continuation = e^(−0.0125) [0.489 × 0 + 0.511 × 8.33] ≈ 0.9876 × 4.257 ≈ 4.20; exercise = 0 → American put ≈ 4.20. A European put at the down node would have kept 7.71, yielding a lower time-0 value ≈ e^(−0.0125)[0.511 × 7.71] ≈ 3.89. Early exercise premium ≈ 0.31.

Volatility in the Binomial Model

Cox–Ross–Rubinstein (CRR) sets

u = e^(σ √Δt), d = e^(−σ √Δt) = 1/u

so that the local variance matches σ² Δt for small Δt. The risk-neutral mean is matched through p* using the asset’s cost of carry (e^(r Δt) for non-dividend stock; e^((r−q)Δt) with yield q).

Worked CRR factors

σ = 30%, Δt = 0.25 → √Δt = 0.50 → σ√Δt = 0.15.

u = e^0.15 ≈ 1.1618; d = e^(−0.15) ≈ 0.8607.

Higher σ widens (u − d), raising both call and put values through convexity—the discrete analog of vega.

Convergence with More Steps

As n → ∞ with Δt = T/n and CRR (or JR) calibration, the European binomial price converges to the Black–Scholes–Merton price. Convergence is typically oscillatory; practitioners use n = 50–200 for vanillas and carefully chosen n for barriers. American prices converge to the true American value (no closed form in general). Exam point: more steps → closer to BSM for Europeans; Americans need the tree (or PDE/LS) because early exercise is path/node dependent.

Delta from the Tree

At any node with successors f_u, f_d and spots S_u, S_d,

Δ = (f_u − f_d) / (S_u − S_d)

This is the stock position in the replicating portfolio over the next step.

Worked delta (one-step call)

From the first example: f_u = 1, f_d = 0, S_u = 22, S_d = 18.

Δ = (1 − 0) / (22 − 18) = 0.25.

Replicating portfolio: hold 0.25 shares, borrow B such that 0.25 × 22 − Be^(r Δt) = 1 and 0.25 × 18 − Be^(r Δt) = 0 → B = e^(−r Δt) × (0.25 × 18) ≈ 4.39 financed by shares worth 5.00, net option ≈ 0.61—matches.

Adapting for Dividends, Indices, FX, and Futures

For a futures option, build the tree on the futures price F with p* = (1 − d)/(u − d) when the expected futures growth is zero under the risk-neutral measure used with futures as the state variable, then discount option payoffs at r. For currency options, replace q with r_f. For indices, use q equal to the dividend yield.

Worked index tweak

S0 = 1000 (index), q = 2%, r = 5%, Δt = 0.25, u = 1.10, d = 0.90.

p* = (e^((0.05−0.02)×0.25) − 0.90) / (1.10 − 0.90) = (e^0.0075 − 0.90) / 0.20 ≈ (1.0075 − 0.90) / 0.20 = 0.5377.

Compared with q = 0 (where e^(rΔt)≈1.0126 and p*≈0.563), dividends lower p*, which lowers calls and raises puts—same directional story as continuous-yield Black–Scholes.

Synthesis

Memorize the backward-induction recipe, the American max(exercise, continuation) rule, CRR u/d, delta from adjacent nodes, and the carry adjustment inside p*. Worked one- and two-step numerics are heavily examined; show p*, intermediate node values, and the early-exercise decision explicitly.