Leisen-Reimer Option Pricing Model

First introduced in 1996 by Dietmar Leisen and Matthias Reimer in their paper “Binomial Models for Option Valuation—Examining and Improving Convergence,” the Leisen-Reimer (LR) model is an advanced, high-precision binomial tree framework designed to overcome the primary structural flaw of traditional lattice models: slow and erratic convergence.

While the classic Cox-Ross-Rubinstein (CRR) model introduced an intuitive lattice approach to option pricing, its convergence toward the continuous-time Black-Scholes solution exhibits severe oscillations (sawtooth patterns) as the number of time steps increases. The LR model fixes this by altering how tree probabilities and node spaces are calculated, producing faster, smoother, and more stable option pricing—especially for options near or at the money.

1. The Core Problem: CRR Sawtooth Oscillations

In standard binomial models like CRR, increasing the number of steps () does not guarantee a smooth, monotonic approach to the continuous Black-Scholes price. Instead, the option price bounces above and below the true value depending on whether is even or odd, or where the strike price falls relative to the terminal nodes.

Option Price
  ▲
  │     / \     / \     / \              <-- CRR Oscillating Convergence
  │    /   \   /   \   /   \
  │───/─────\─/─────\─/─────\─────────── <-- True Black-Scholes Value
  │  /       V       V       \
  │
  └───────────────────────────────────► Number of Steps (n)

Why Oscillations Occur?

1. Strike Placement Discontinuity: In a standard CRR tree, small changes in step count cause the strike price to shift back and forth between lying directly on a terminal node or falling between two nodes. This alters the calculated intrinsic payoff distribution at maturity.
2. Probability Approximation Error: Standard models use simple central-limit approximations for the probability terms and , which struggle to capture the curvature of the cumulative normal distribution at low step counts.

2. Mathematical Design of the Leisen-Reimer Model

The breakthrough of the Leisen-Reimer framework lies in replacing traditional probability parameters with inversion formulas for the cumulative normal distribution, directly leveraging the continuous and terms from the Black-Scholes formula.

Step 1: Black-Scholes and Parameters

Recall the standard Black-Scholes terms for time step :

Step 2: Peizer-Pratt Inversion Formulas

To transform these continuous parameters into discrete binomial probabilities, Leisen and Reimer utilized the Peizer-Pratt second inversion formula, an exceptionally accurate approximation of the normal distribution:

Where returns if , if , and if .

Using this function, LR defines two transformed probabilities, and :

Step 3: Calibrating Tree Parameters (, , and )

Unlike CRR, where and are set independently of and , the LR model forces the tree parameters to align with and :

By forcing and through Peizer-Pratt inversions, the tree’s center node aligns with the strike price , smoothing out the payoff curve at maturity and eliminating discrete strike jumps.

Constraint: To maximize the efficiency of the center-node alignment, the step count in a Leisen-Reimer tree must always be an odd integer ().

3. Comparative Performance Analysis

Feature	Cox-Ross-Rubinstein (CRR)	Leisen-Reimer (LR)
Convergence Trajectory	Slow, highly oscillatory (sawtooth)	Exceptionally smooth and monotonic
Step Requirement	can be even or odd	must be an odd integer
Steps Needed for High Accuracy	Often requires to steps	Achieves equal accuracy at to steps
Greeks Calculation ()	Unstable near maturity due to node jumps	Highly stable, continuous derivative estimates
Primary Use Cases	Teaching, basic option pricing models	Quantitative desks, risk engines, complex American derivatives

4. Real-World Institutional Example: Energy & Commodity Options

To understand why quantitative risk management teams choose Leisen-Reimer over CRR, consider an institutional trading desk pricing complex American put options on crude oil futures or major energy equities like ExxonMobil (XOM).

Practical Scenario: High-Frequency Volatility Surface Calibration

During periods of market volatility, trading engines must continuously recalibrate thousands of options across strike prices and maturities to recalculate portfolio risk metrics (Delta, Gamma, Vega).

• Asset: ExxonMobil (XOM) American Put Option
• Underlying Spot Price (): USD115.00
• Strike Price (): USD115.00 (At-the-Money)
• Risk-Free Rate (): 4.5%
• Volatility (): 28%
• Time to Maturity (): 30 days ( years)

Valuation Speed vs. Accuracy Comparison
----------------------------------------------------------------------
Model                 Steps (n)   Calculated Put Price   Time / Compute
----------------------------------------------------------------------
CRR (Binomial)        n = 31      USD3.12 (High Error)     Low
CRR (Binomial)        n = 501     USD3.24 (Oscillating)    Medium
CRR (Binomial)        n = 2,001   USD3.251 (Converged)     High
----------------------------------------------------------------------
Leisen-Reimer (LR)    n = 31      USD3.251 (Converged)     Very Low
----------------------------------------------------------------------

Institutional Takeaway

1. Computational Efficiency: The LR model achieves the same target accuracy at steps that the CRR model reaches only after steps.
2. Speed Advantage: Because lattice processing time grows at , running 31 steps instead of 2,000 steps cuts processing time per option from milliseconds to microseconds.
3. Risk Management Reliability: Because sensitivity metrics (Greeks) depend on precise node differentials, the smooth convergence of LR avoids false early-exercise triggers that frequently corrupt CRR risk reports.

5. Applications in Modern Financial Engineering

1. American Options on Dividend-Paying Assets: LR is widely implemented in pricing software for American options, where discrete cash dividends require flexible, high-speed tree architectures.
2. Real Options Valuation: Corporate strategy groups use LR models to value multi-stage capital expenditure investments (e.g., pharmaceutical drug pipelines, mining extraction rights) where decision points are discrete and early abandonment options exist.
3. Implied Volatility Trees: Quantitative desks build implied binomial trees (Rubinstein/Derman-Kani models) using LR parameterizations to ensure smooth fitting to market volatility smiles and skews.