Cox-Ross-Rubinstein Binomial Option Pricing & Lattice Lab
Price American and European derivatives with recombining binomial trees, backward induction, and early exercise premium modeling.
Option Contract Parameters
Cox-Ross-Rubinstein (CRR) Mathematics
p = [exp((r - q)·Δt) - d] / (u - d)
V_hold = exp(-r·Δt) · [p·V_up + (1-p)·V_down]
V_Amer = max(Intrinsic_Value, V_hold)
Δ = (V_u - V_d) / (S·u - S·d)
- Recombining Property: S·u·d = S·d·u = S, yielding N+1 terminal nodes instead of 2^N.
- Early Exercise Premium: Value(American) - Value(European). Positive when early cash collection dominates remaining time value.
- Risk-Neutral Pricing: Expected discounted payoff under martingale measure p.
Derivatives Valuation Comparison
Cox, Ross & Rubinstein (1979)| Pricing Model | Option Value ($) | % of Spot | Analytical Characteristics |
|---|---|---|---|
| CRR Binomial (Selected Style) | $0.00 | 0.0% | Discrete lattice backward induction |
| CRR Binomial (American Style) | $0.00 | 0.0% | Optimal early exercise accounted at every node |
| CRR Binomial (European Style) | $0.00 | 0.0% | Exercise restricted strictly to maturity T |
| Black-Scholes-Merton (European Continuous) | $0.00 | 0.0% | Continuous-time closed-form limit (N → ∞) |
| Early Exercise Optionality Premium | $0.00 | 0.0% | American Price minus European Price |
Recombining Binomial Lattice Tree Visualizer
Showing Nodes (N = 5)Displays asset price ($S) and option value ($V) at each node. Amber-highlighted nodes indicate where early exercise is strictly optimal over holding.
Quantitative Sensitivity Matrices
Matrix 1: Underlying Spot ($S) vs. Volatility (σ %) on Selected Option Value ($)
Highlights non-linear convexity (gamma) and volatility leverage (vega) across varying asset price regimes.
Matrix 2: Strike ($K) vs. Expiry (Years, T) on American Early Exercise Premium ($)
Quantifies where early exercise advantage peaks across deep moneyness and longer contract horizons.
Derivatives & Binomial Lattice Mastery Quiz
Understanding the Cox-Ross-Rubinstein Binomial Framework
The Cox-Ross-Rubinstein (CRR) model, published in 1979, transformed quantitative derivatives valuation by translating complex continuous stochastic calculus into an intuitive discrete-time lattice. Rather than solving partial differential equations, the binomial model slices the option's lifespan into $N$ discrete intervals $Delta t = T / N$.
During each interval, the underlying asset price moves up by factor $u = e^{sigma sqrt{Delta t}}$ or down by $d = e^{-sigma sqrt{Delta t}} = 1/u$. Because $u imes d = 1$, the tree recombines: an up move followed by a down move arrives at the exact same asset price as a down move followed by an up move ($S cdot u cdot d = S cdot d cdot u = S$). This prevents exponential tree explosion, allowing an $N$-step tree to have only $N+1$ terminal payoff nodes.
American Early Exercise & Economic Mechanics
- American Put Early Exercise: A deep-in-the-money put has an intrinsic value of $K - S$. If held to maturity, the holder earns no interest on $K$. Exercising early converts the position into immediate cash $K$, earning risk-free interest $r$ that outweighs the decaying extrinsic time value.
- Dividend Drag on Calls: American calls on non-dividend stocks should never be exercised early. However, when a stock pays a substantial continuous dividend yield $q > r$, holding the call forfeits dividend cash flow, making early exercise optimal immediately before dividends accrue.
- Dynamic Delta Hedging: At each node, the hedge ratio $Delta = rac{V_u - V_d}{S cdot u - S cdot d}$ represents the exact number of shares of stock needed to construct a riskless, self-financing replicating portfolio.