binomial-pricing

Binomial Tree Pricing (CRR)

Price American calls and puts via a Cox-Ross-Rubinstein binomial tree. Recover the European price as a sanity check and report the early-exercise premium = American − European.

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Step 1: Extract Inputs

Same as Black-Scholes (S, K, T years, r, q, σ, opt_type) plus:

Field	Notes	Default
n_steps	Tree depth	200
style	"american" or "european"	american

For discrete dividends, use the escrowed-dividend trick (subtract PV of dividends from S before building the tree).

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Step 2: Build the Tree

dt = T / n
u  = exp(σ · √dt)
d  = 1 / u
p  = (exp((r − q)·dt) − d) / (u − d)
disc = exp(−r·dt)

If p ∉ [0, 1], the tree is ill-conditioned (e.g., σ too small relative to drift over dt) — stop and report. Increase n_steps or reduce |r − q|·dt.

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Step 3: Terminal Payoff

At node (i, j) with i = 0..n and j = number of up-moves:

S_ij = S · u^j · d^(i − j)
payoff_call_i = max(S_ni − K, 0)
payoff_put_i  = max(K − S_ni, 0)

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Step 4: Backward Induction

For each time step i from n−1 down to 0, for each node j ≤ i:

S_ij = S · u^j · d^(i − j)
continuation = disc · (p · V_(i+1, j+1) + (1 − p) · V_(i+1, j))
if style == "american":
    V_ij = max(continuation, max(S_ij − K, 0))   # call
    V_ij = max(continuation, max(K − S_ij, 0))   # put
else:
    V_ij = continuation

V_00 is the option price.

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Step 5: Sanity Checks

1. Re-run with style = "european" and compare to your Black-Scholes price → residual must be < 1% for n ≥ 200 (converges as O(1/n)).
2. Early-exercise premium ≥ 0 always. If American < European, your tree has a bug.
3. Doubling n_steps should roughly halve the BS-vs-tree residual.

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Step 6: When Is Early Exercise Optimal?

• American call on non-dividend stock: never optimal — premium = 0 (Merton 1973). If your tree shows premium > 0 with q = 0, the tree is buggy.
• American call on dividend-paying stock: optimal just before ex-div date if dividend > time value of the remaining call.
• American put: often optimal when deep ITM and far from expiry (you can earn r on K immediately).

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Step 7: Respond to User

Report:

1. American price, European price, early-exercise premium (absolute and as % of American).
2. The optimal exercise boundary at t=0 if requested (the strike level above/below which immediate exercise dominates continuation).
3. Convergence check note: how the price changed when you doubled n_steps.

For pure European pricing, recommend black-scholes (faster, exact).