Binomial Tree Models for DeFi Option Pricing

Introduction

Decentralized finance (Volatility Modeling in Decentralized Finance) has reshaped how markets operate by removing intermediaries and enabling self‑service contracts on public blockchains. Among the most powerful tools that DeFi has borrowed from traditional finance are derivatives, particularly options (Mastering Option Pricing with Binomial Trees). Options give traders the right, but not the obligation, to buy or sell an asset at a predetermined price. Pricing these instruments accurately is essential for risk management, arbitrage, and liquidity provision.

While Black‑Scholes and other continuous‑time models dominate academic literature, the discrete nature of blockchain transactions and the prevalence of gas‑limited execution make binomial tree models especially appealing. A binomial tree allows us to approximate the evolution of an asset price in a series of discrete steps, each step representing a period of time such as a block or a day. In a DeFi setting, each node of the tree can be associated with a smart contract execution, making the model inherently compatible with on‑chain logic (Practical Binomial Tree Models for Smart Contract Options).

This article walks through the theory and practice of binomial tree models for DeFi option pricing. We explore the fundamentals, demonstrate how to construct the tree, incorporate DeFi‑specific features such as impermanent loss and liquidity incentives, and illustrate the approach with a step‑by‑step implementation example in Solidity and Python.

Key Concepts

1. Asset Dynamics in a Discrete World

In classical finance, asset prices follow a continuous stochastic process, often modeled as geometric Brownian motion. In a DeFi context, price updates occur at discrete intervals defined by block times or oracle feeds. A binomial tree captures this by letting the price either go up or down at each step, with a probability that matches the expected return and volatility over that period.

The price at node i is given by

S_i = S_0 * u^k * d^(i−k)

where u is the up factor, d is the down factor, and k is the number of up moves among the i steps. The tree is constructed so that the expected value of the underlying after one step equals the risk‑neutral growth factor:

E[S_{i+1}] = r * S_i

with r the risk‑neutral discount factor (usually 1 + risk‑free rate per period).

2. Risk‑Neutral Valuation

DeFi does not have a central bank, but a risk‑neutral measure can still be defined by the growth of a collateralized pool or a stable‑coin that serves as the “numeraire.” In a binomial tree, risk‑neutral probabilities p and 1–p are computed as

p = (r – d) / (u – d)

ensuring that the discounted expected payoff equals the current option price (Building a DeFi Option Pricing Engine with Volatility Forecasting).

3. Payoff Functions for Common Options

The tree can price European and American options. Payoffs are computed at the terminal nodes and then discounted backward. For a European call on a token X with strike K and maturity T,

payoff = max(S_T – K, 0)

For an American option, at each node we compare the intrinsic value with the discounted continuation value and take the maximum.

Building the Tree

Step 1: Choose Time Horizon and Step Size

Decide on the option maturity in blocks or days. Suppose we want a 30‑day option and the network averages 15‑second blocks; that gives about 3,840 steps. For tractability we often aggregate several blocks into a single time step. In DeFi, a practical approach is to use the oracle’s update frequency (e.g., 12‑hour price feeds), giving 60 steps for 30 days.

Step 2: Estimate Volatility

Volatility in DeFi can be derived from on‑chain historical price data or implied volatility from liquidity pools. Because of the discrete nature of blockchain, use the standard deviation of log returns over the chosen step size:

σ = sqrt(Var[ln(S_{t+Δt}/S_t))] / Δt

Adjust σ to match the oracle’s volatility if necessary.

Step 3: Compute Up and Down Factors

The classic Cox‑Ross‑Rubinstein (CRR) model defines:

u = exp(σ * sqrt(Δt))
d = 1 / u

This ensures that the variance over one step equals σ²Δt.

Step 4: Compute Risk‑Neutral Probability

Using the risk‑free rate r per step (often zero in a pure DeFi environment unless a stable‑coin is used), compute

p = (exp(r * Δt) – d) / (u – d)

If r is zero, the formula simplifies.

Step 5: Populate the Tree

Iterate through steps, filling each node’s asset price. Because the tree is binary, the number of nodes doubles at each level, but only n+1 unique prices exist at step n (one for each number of up moves). This compression dramatically reduces memory usage.

Step 6: Apply Payoff at Terminal Nodes

At maturity, calculate the payoff for each node using the chosen option’s payoff function.

Pricing Logic

Backward Induction

Start at maturity and move backward:

1. European Options: Discount each node’s payoff by p and 1–p, then multiply by the discount factor exp(–rΔt).
2. American Options: At each node, compute the intrinsic value (max(S–K, 0) for a call). Compare it with the discounted continuation value. The option’s value at that node is the maximum of the two.

Repeat until reaching the root, which yields the option price today.

Handling DeFi‑Specific Constraints

Impermanent Loss

In liquidity pools, the token pair’s value can shift, causing impermanent loss for liquidity providers. To price an option that protects against this loss, adjust the payoff to reflect the pool’s net value after accounting for pool‑specific dynamics (Hands‑On Volatility Modeling for Decentralized Derivatives).

Gas Costs

Execution cost is an essential factor in DeFi. When modeling an American option that can be exercised early, incorporate the expected gas fee into the intrinsic value calculation:

intrinsic = max(S – K – gas_fee, 0)

where gas_fee is expressed in the underlying token’s denomination.

Stochastic Interest Rates

Some DeFi protocols expose on‑chain interest rates that vary over time. If significant, treat the risk‑free rate as a stochastic process and embed it into the tree’s probability computation or use a two‑dimensional tree (price and interest rate).

Practical Implementation

Below is a concise example of building a binomial tree in Python, then deploying a simple contract in Solidity that can compute the option price on chain using pre‑computed parameters.

Python Script: Building the Tree

import math
import numpy as np

# Parameters
S0 = 100.0        # Spot price
K = 105.0         # Strike
T = 30            # Days to maturity
dt = 1            # One‑day steps
N = int(T/dt)     # Number of steps
sigma = 0.20      # Annualized volatility
r = 0.0           # Risk‑free rate (assuming stable‑coin)
dt_yr = dt/365.0  # Convert to years

u = math.exp(sigma * math.sqrt(dt_yr))
d = 1 / u
p = (math.exp(r*dt_yr) - d) / (u - d)

# Asset prices at maturity
S_T = np.array([S0 * (u**k) * (d**(N-k)) for k in range(N+1)])
payoff = np.maximum(S_T - K, 0)  # European call

# Backward induction
value = payoff
for step in range(N, 0, -1):
    value = np.exp(-r*dt_yr) * (p * value[1:] + (1-p) * value[:-1])

print(f"Option price: {value[0]:.4f}")

Run this script to obtain the price. The same u, d, and p values can be passed to the smart contract.

Solidity Contract: On‑Chain Pricing

// SPDX-License-Identifier: MIT
pragma solidity ^0.8.20;

contract BinomialOption {
    uint256 public S0;      // Spot price (in wei)
    uint256 public K;       // Strike price
    uint256 public u;       // Up factor * 1e18
    uint256 public d;       // Down factor * 1e18
    uint256 public p;       // Risk‑neutral probability * 1e18
    uint256 public N;       // Number of steps

    uint256 constant ONE = 1e18;

    constructor(
        uint256 _S0,
        uint256 _K,
        uint256 _u,
        uint256 _d,
        uint256 _p,
        uint256 _N
    ) {
        S0 = _S0;
        K = _K;
        u = _u;
        d = _d;
        p = _p;
        N = _N;
    }

    function price() external view returns (uint256) {
        // Terminal asset prices and payoffs
        uint256[] memory value = new uint256[](N + 1);
        for (uint256 k = 0; k <= N; k++) {
            uint256 price = S0 * powApprox(u, k) / powApprox(ONE, k);
            price = price * powApprox(d, N - k) / powApprox(ONE, N - k);
            uint256 payoff = price > K ? price - K : 0;
            value[k] = payoff;
        }

        // Backward induction
        for (uint256 step = N; step > 0; step--) {
            for (uint256 k = 0; k < step; k++) {
                uint256 cont = (p * value[k + 1] + (ONE - p) * value[k]) / ONE;
                value[k] = cont;
            }
        }
        return value[0];
    }

    // Fast exponentiation for fixed‑point numbers
    function powApprox(uint256 base, uint256 exp) internal pure returns (uint256) {
        uint256 result = ONE;
        while (exp > 0) {
            if (exp & 1 == 1) {
                result = (result * base) / ONE;
            }
            base = (base * base) / ONE;
            exp >>= 1;
        }
        return result;
    }
}

Explanation

• The contract stores the parameters as fixed‑point numbers scaled by 1e18.
• powApprox implements fast exponentiation while keeping numbers in fixed‑point format.
• The price function calculates terminal payoffs and then runs backward induction.

This contract can be called by off‑chain services or used directly within DeFi protocols for on‑chain pricing.

Case Study: Pricing a Call on Wrapped ETH

Suppose a DeFi platform wants to offer a 30‑day call option on wETH with a strike of 1.5 wETH. Historical on‑chain data over the past month shows an annualized volatility of 25 %. The platform uses an oracle that updates every 6 hours, giving 120 steps for the 30‑day horizon.

1. Compute Parameters
   Δt = 6 h = 0.25 days → 0.000684 years.
   u = exp(0.25 × sqrt(0.000684)) ≈ 1.016.
   d = 1/1.016 ≈ 0.984.
   p ≈ (1 – d)/(u – d) ≈ 0.50 (since r ≈ 0).

2. Construct Tree
   Python script yields an option price of 0.18 wETH.

3. Deploy on Chain
   Parameters are passed to the Solidity contract. Users can call price() to see the fair premium and decide whether to purchase the option.

4. Dynamic Re‑pricing
   If the oracle updates volatility or price, the platform can adjust the contract’s parameters by creating a new instance or by using an upgradeable proxy pattern.

DeFi‑Specific Enhancements

1. Liquidity‑Add Incentives

Many protocols offer rebates for early option exercise to encourage liquidity provision. To model this, add a rebate term R that depends on the time to exercise:

rebate = max(0, R * (T - t) / T)

Incorporate rebate into the intrinsic value at each node.

2. Multi‑Asset Options

DeFi introduces exotic options such as token baskets or liquidity‑pool weighted options. Extend the tree to multiple dimensions or use a Monte‑Carlo approximation for high dimensionality.

3. On‑Chain Verification

When the option is exercised, the contract must verify that the holder’s position meets the intrinsic value. This can be achieved by integrating with the same oracle that built the tree, ensuring consistency and preventing manipulation.

Performance Considerations

1. Gas Efficiency

The Solidity implementation above is intentionally simple; for production, consider:

• Pre‑computed tables: Store discounted payoffs at each node to avoid on‑chain recomputation.
• Sparse representation: Only keep necessary nodes, reducing storage.
• Batch execution: Use call to off‑chain services that return the price, then store it on chain.

2. Tree Size Trade‑Off

A finer time step improves accuracy but increases tree size exponentially. In practice, 50–100 steps are sufficient for many DeFi options, balancing precision and gas cost.

Conclusion

Binomial tree models bridge the gap between traditional option pricing and the unique requirements of decentralized finance. By integrating volatility forecasting, handling impermanent loss, and accounting for gas costs, practitioners can build robust pricing engines that operate entirely on smart contracts (Practical Binomial Tree Models for Smart Contract Options) and remain responsive to on‑chain market dynamics. The methods outlined here—along with the companion resources in DeFi—provide a solid foundation for building the next generation of automated, transparent, and efficient option markets.