Valuation
Option Premium
$—
Reference (European, Black-Scholes)
BS price
—
Δ vs binomial
—
Greeks (from tree)
Δ
Delta
—
Γ
Gamma
—
Θ
Theta (per yr)
—
ν
Vega (bumped)
—
ρ
Rho (bumped)
—
—
Tree parameters · α = e(r−q)Δt
σ—
u—
d—
α—
p—
Δt—
EQ[ST] · martingale check
—
Tree parameters. u and d are the multiplicative factors for an up/down move at each step. p is the risk-neutral probability of an up move — not the real-world probability; it's whatever value makes the tree consistent with no-arbitrage pricing. α is the per-step forward growth factor (
Martingale check. Under the Q measure, expected terminal stock price must equal
e^((r−q)Δt)). Δt is the time per step. The whole tree is a discrete-time analog of geometric Brownian motion.Martingale check. Under the Q measure, expected terminal stock price must equal
S₀ · e^((r−q)T). The two numbers should always match exactly — if they ever didn't, the pricing engine would be broken.
Black-Scholes-Merton
Closed-form European-option price under continuous trading and lognormal returns. For American options the binomial price above is correct; BSM is shown as the European reference.
BSM Price (European)
$—
—
d₁
—
N(d₁)
—
d₂
—
N(d₂)
—
—
Interpretation
N(d₁) ≈ option delta (also: present-value-adjusted exercise weight)
N(d₂) = risk-neutral probability the option finishes ITM
For a call:
C = S·e−qT·N(d₁) − K·e−rT·N(d₂)
For a put:
P = K·e−rT·N(−d₂) − S·e−qT·N(−d₁)
Greeks (closed-form)
Δ
Delta
—
Γ
Gamma
—
Θ
Theta (per yr)
—
ν
Vega (per 1.0 σ)
—
ρ
Rho (per 1.0 r)
—
—
Implied Volatility
Newton-Raphson
The σ that makes BSM price equal the observed market price. What traders quote instead of price itself.
Reading the IV. The implied σ is the volatility that would make the BSM formula output exactly the market price you entered. If implied > current σ, the market is pricing more volatility than you assumed — options look "expensive" relative to your view (consider selling vol). If implied < current σ, options look "cheap" (consider buying vol). Traders quote options in IV rather than dollars because IV is comparable across strikes and expiries — a $5 option on stock A and a $50 option on stock B both quoted at "20 vol" tells you something meaningful.
Binomial vs BSM: binomial Greeks come from finite differences on the tree (path-dependent, exact for American). BSM Greeks are closed-form derivatives of the price formula (European only). They should match as N→∞ for European cases.
Convergence with N
Binomial price as the tree is refined. European case converges to Black-Scholes; American adds early-exercise premium.
Why the oscillation? The red line zig-zags because of the famous CRR even/odd-N parity effect: at even N, a path can return precisely to the strike K; at odd N it can't. This creates two interleaved convergence patterns. The average of consecutive N's is much smoother and is sometimes used as a quick smoothing trick. N = 30 is the lecture's "practice standard" — enough to be within a few cents of BSM for vanilla options. As N → ∞, the binomial price converges to the dashed BSM line.
Lattice Visualization
Stock price (top) and option value (bottom) at each node. Red dot = early exercise optimal.
Node
Early exercise (American)
In the money
How to read each node. Each box shows three numbers stacked: stock price at top, option value in the middle, local delta at the bottom (only for non-terminal nodes when N ≤ 4). Time flows left to right; each node branches into two prices on the next step. Red-outlined nodes are where early exercise is optimal for an American option — at those nodes, the intrinsic value (K−S for puts, S−K for calls) beats the value of waiting. Hover any node to see the intrinsic-vs-continuation breakdown.
Delta Hedging
Δ is the number of shares to hold against one option to create a (locally) riskless portfolio. Because Δ changes as S and t evolve, the hedge must be rebalanced periodically — and σ is what determines Δ.
Composition at t = 0
Short 1 call
Net cash (bond)
$0.00
σ used:
0.00%
Δ is highly sensitive to σ. A wrongly-priced σ produces a perfectly delta-hedged portfolio that still drifts in P&L — this is vega risk. Volatility forecasting is the actual edge.
The riskless portfolio identity. This is the core trick from Black-Scholes' derivation. The combined position — short option plus Δ shares — has zero instantaneous P&L regardless of which way the stock moves, because the option's value change (Δ · dS) exactly offsets the share P&L (Δ · dS, opposite sign). That's why such a portfolio must earn the risk-free rate — there's no risk to compensate for. The catch: Δ itself changes as S moves (that's gamma), so the hedge needs continuous rebalancing. In practice you rebalance discretely and accept the slippage.
Δ as a function of S — multiple times to expiry
Curves: Black-Scholes Δ at decreasing τ. Highlighted point: tree-computed Δ at current S₀. For American puts near the exercise boundary, tree Δ snaps to −1 while BS curve smooths.
Reading the Δ curves. The S-curve shape is the geometric signature of an option. Deep OTM (S far below K for calls, above K for puts): Δ ≈ 0, the option barely moves with the stock. ATM (S ≈ K): Δ ≈ 0.5 for calls, −0.5 for puts — the "coin flip" zone. Deep ITM: Δ ≈ 1 (calls) or −1 (puts), the option moves dollar-for-dollar with the stock. As τ → 0 (expiry approaches), curves steepen toward a step function at K — small price moves can flip Δ violently, which is why short-dated options have huge gamma. Higher σ flattens the curves: with lots of remaining vol, even ITM positions could still go OTM, so Δ stays moderate.