Derman-Kani Implied Volatility Tree

Local Vol

Discrete approximation to Dupire local volatility

A binomial tree model calibrated to match an implied volatility surface exactly at each node. Uses forward induction to construct Arrow-Debreu state prices and extract local volatilities. The tree provides discrete-time approximation to local volatility dynamics while ensuring no-arbitrage through risk-neutral transition probabilities.

Mathematical Formulation

S_{i, j} = S_{0} \cdot u^{j} \cdot d^{i - j}

λ_{i, j} = k \sum λ_{i - 1, k} \cdot p_{k \to j} \cdot e^{- r Δ t}

p_{up} = \frac{F - S _{d}}{S _{u} - S _{d}}, F = S \cdot e^{r Δ t}

Parameters

Symbol	Description	Constraint
$S_{i, j}$	Stock price at node (i, j)	$S > 0$
$λ_{i, j}$	Arrow-Debreu state price	$λ \geq 0$
$p_{up}$	Risk-neutral up probability	$p \in (0, 1)$
$u, d$	Up and down multipliers	$u > 1 > d > 0$
$r$	Risk-free rate	$r \in R$
$σ_{loc}$	Local volatility at each node	$(calibrated)$

Key Assumptions

•Recombining binomial tree: computationally efficient
•Calibrated to match market implied volatility surface
•Arrow-Debreu prices enable probability-weighted discounting
•Discrete approximation to continuous local volatility
•No-arbitrage ensured via risk-neutral probabilities

Reference

Derman, E., Kani, I. (1994). "Riding on a Smile." Risk, 7(2), 32-39.

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Derman-Kani Implied Volatility Tree

Mathematical Formulation

Parameters

Key Assumptions

Reference

Model Hierarchy

Dupire

GBM

Empirical Context (SPY Returns)

What DK Tree captures

What it cannot capture

Estimation Data

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Dupire

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