Contract

$

Spot price (S)

$

Strike (K)

Volatility (σ)

Annualised, 0.20 = 20%

Time to maturity (T)

Years

Risk-free rate (r)

Annualised, 0.05 = 5%

Dividend yield (q)

Annualised, continuous

Call

Put

Exercise style

Try this

0 of 3 done

Find the oscillation

Open the convergence chart and compare the Cox-Ross-Rubinstein and Leisen-Reimer schemes from 1 to 50 steps.

What to look for: Cox-Ross-Rubinstein zig-zags above and below the closed form as the strike moves between lattice nodes. Leisen-Reimer sits on the closed form almost immediately.

See early exercise

Choose an American put with spot 80 and strike 100, then look at the tree diagram with a handful of steps.

What to look for: The tree price is above the European price and the early-exercise premium is positive. Nodes deep in the money are highlighted where exercising beats waiting.

An American call is a European call

Choose an American call with dividend yield 0. Then set the yield to 0.08 and price again.

What to look for: With no dividends the early-exercise premium is zero: it is never optimal to exercise a call early. A large dividend yield makes early exercise worthwhile and the premium appears.

Tree

Parameterisation

The classic tree: u = e^(σ√dt), d = 1/u, so the tree recombines around the spot. Simple, but the price oscillates as steps increase.

Steps

European call value

$10.43

Binomial (Cox-Ross-Rubinstein), 100 steps

Black-Scholes

$10.4506

Tree error

-0.0200

0.191% of the closed form

Tree parameters

Steps	100
Time step (dt)	0.010000
Up factor (u)	1.020201
Down factor (d)	0.980199
Up probability (p)	0.507502
Compute time (price and Greeks)	1.9 ms

Greeks

Raw values are partial derivatives. The practitioner column rescales them to the units traders quote: per 1 volatility point, per 1% of rate, or per calendar day. Delta, Gamma and Theta are read from the first two levels of the tree; the rest are bump-and-revalue.

Greek	Definition	Raw	Practitioner	Meaning
Delta Δ	`∂V/∂S`	0.63651	0.63651 per $1 of spot	How much the option price moves for a $1 move in the underlying. Also the number of shares needed to hedge one option.
Gamma Γ	`∂²V/∂S²`	0.018922	0.018922 per $1 of spot	How fast Delta changes as the underlying moves. High Gamma means the hedge must be rebalanced often.
Vega ν	`∂V/∂σ`	37.43	0.3743 per 1 vol point	Sensitivity to implied volatility. Long options gain value when volatility rises.
Theta Θ	`∂V/∂t`	-6.4453	-0.017658 per calendar day	Time decay: how much value the option loses as one day passes with everything else unchanged.
Rho ρ	`∂V/∂r`	53.221	0.53221 per 1% of rate	Sensitivity to the risk-free rate. Calls gain and puts lose when rates rise.
Epsilon ε	`∂V/∂q`	-63.651	-0.63651 per 1% of yield	Sensitivity to the dividend yield. Higher expected dividends lower the forward price, hurting calls and helping puts.

Second-order Greeks

Greek	Definition	Raw	Practitioner	Meaning
Vanna	`∂²V/∂S∂σ`	-0.28101	-0.0028101 per 1 vol point	How Delta changes when volatility changes, or equally how Vega changes when spot moves.
Vomma	`∂²V/∂σ²`	9.8339	0.00098339 per 1 vol point²	Convexity in volatility: how Vega itself changes as volatility moves.
Charm	`∂Δ/∂t`	-0.065537	-0.00017955 per calendar day	Delta decay: how the hedge ratio drifts as one day passes.
Speed	`∂³V/∂S³`	-4.4409E-15	-4.4409E-15 per $1 of spot	How Gamma changes as the underlying moves.
Zomma	`∂Γ/∂σ`	0.019667	0.00019667 per 1 vol point	How Gamma changes when volatility changes.
Color	`∂Γ/∂t`	0.0045865	1.2566E-05 per calendar day	Gamma decay: how Gamma changes as one day passes. Explodes for at-the-money options near expiry.

Watch the tree

The tree grows from today's price, one step at a time, until the last layer shows the possible prices at expiry and what the option pays at each. Then the value rolls back to today: each node is the discounted average of the two after it, weighted by the risk-neutral probability p.

Showing a 10-step tree, so that every node can be read; the price above uses 100 steps, so the two values differ a little.

Playback speed

Step moves one layer while the tree grows and one node while the value rolls back. Hover or focus a node for its numbers; the arrow keys move between nodes.

u = 1.0653 · d = 0.9387 · p = 0.5238 · Δt = 0.1000 · e^(−rΔt) = 0.995012

Pays off at expiry (in the money) Pays nothing at expiry (out of the money) Node being computedP: the risk-neutral probability of ending at that price, from the binomial distribution

Tree explorer

The same contract and parameterisation on a small tree, so every node can be read. Hover a node for its exact values.

Steps: 4

4

Root value with 4 steps: $9.9705

Hold: value is the discounted expectation of the next step Early exercise: payoff beats holding

Convergence

Price against step count for every parameterisation. The dashed line is the closed form. Cox-Ross-Rubinstein oscillates around it as the strike moves between nodes; Leisen-Reimer places a node on the strike and converges smoothly.

Binomial and trinomial trees