The lattice model for option pricing values an option by splitting the time to expiration into discrete steps, mapping every price the underlying asset could reach along the way, and working backward from expiration to arrive at a value today. At each node in the resulting tree, the model can check whether exercising beats holding, which is why it handles American-style options, employee awards, and path-dependent payoffs that a single-formula approach like Black-Scholes cannot capture in one pass.
When the Lattice Model Is the Right Tool
For a plain-vanilla European option on a non-dividend-paying stock, Black-Scholes produces a fast, accurate price and there is no reason to build a tree. The lattice earns its keep when the contract has features that a closed-form equation cannot absorb cleanly. American options, which can be exercised any time before expiration, are the classic case: the tree lets you test exercise against continuation at every node.
Employee stock options under ASC 718 are where the model does its most distinctive work. These awards carry vesting periods, short post-termination exercise windows, and employees who tend to exercise once the stock reaches some multiple of the strike price. A lattice absorbs all of that into the tree instead of compressing it into a single expected-term input the way Black-Scholes does. ASC 718-10-55-17 through 55-18 recognize that lattice models may produce a more accurate fair value for employee awards for exactly this reason. Awards with market conditions, payoff caps, or restrictions on exercisability generally require either a lattice or a Monte Carlo simulation.
The flip side is worth stating plainly. For at-the-money options with only a service-based vesting condition and thin exercise history, a Black-Scholes valuation built on a well-developed expected-term assumption will stand up to audit scrutiny, and the extra precision of a lattice rarely justifies the heavier build.
The Inputs You Need
Every lattice starts from the same handful of market-observable and contract-specific variables. Getting these right matters more than the number of steps in your tree; precise arithmetic on bad inputs still produces a wrong price.
- Current stock price, which anchors the first node.
- Strike price written into the contract.
- Time to expiration, stated in years so it aligns with annualized volatility and interest rates.
- Expected annualized volatility of the stock’s continuously compounded returns over the option’s life.
- Risk-free interest rate, typically drawn from U.S. Treasury yields matching the option’s duration.
- Expected dividend yield, which adjusts the expected growth of the stock inside the tree.
Both FASB ASC 718 and IFRS 2 require each assumption to reflect observable market data free from the bias of any specific party. IFRS 2 specifically lists the exercise price, option life, current share price, expected volatility, expected dividends, and risk-free rate as the minimum inputs any option pricing model must incorporate.1IFRS Foundation. IFRS 2 Share-Based Payment Stale or unjustified data invites audit adjustments and potential restatement of compensation expense.
Historical Versus Implied Volatility
Volatility is the single most influential input, and practitioners have two sources. Historical volatility looks back at what the stock actually did over a period roughly matching the option’s expected term. Implied volatility extracts what the market is currently forecasting from the prices of traded options on the same stock.
The SEC does not mandate one method over the other. The goal is the volatility assumption that marketplace participants would use. A company with actively traded options on its shares can rely exclusively on implied volatility as long as those options are near-the-money, have at least one year of remaining maturity, and are measured on a date close to the grant date. Exclusive reliance on historical volatility is equally acceptable where the company has no reason to expect future volatility to differ from the past and uses a sequential period at least as long as the option’s expected term.2U.S. Securities and Exchange Commission. Staff Accounting Bulletin No. 107 A blend is fine, provided the company documents why that blend best reflects expected future conditions.
How Many Time Steps
The time-step size, written Δt, is the total time to expiration divided by the number of periods in your tree. More steps produce a finer grid and better accuracy, but each one multiplies the nodes the model has to evaluate. For financial reporting, practitioners commonly use 30 to 100 steps. At roughly 50 steps, a well-constructed binomial tree closely approximates the Black-Scholes price for a European option; beyond 100 steps, additional accuracy is negligible for most contracts.3Spreadsheets in Education. Convergence of the Cox-Ross-Rubinstein Binomial Option Pricing Model to the Black-Scholes-Merton Version
Building the Price Tree
Two multipliers control how far the stock moves at each step. The up factor equals e raised to the power of volatility times the square root of Δt. The down factor is the reciprocal of the up factor, which keeps the tree recombining: an up move followed by a down move lands at the same price as a down followed by an up.3Spreadsheets in Education. Convergence of the Cox-Ross-Rubinstein Binomial Option Pricing Model to the Black-Scholes-Merton Version
From the current stock price at the first node, the tree branches forward. The upper node is the price times the up factor; the lower node is the price times the down factor. At the second step each branches again, but because the tree recombines you get three unique prices instead of four. An N-period tree holds N+1 terminal prices arranged in a geometric progression from the lowest (all down moves) to the highest (all up moves).
The recombining structure is what makes the model computationally practical. A non-recombining tree would double its nodes at every step and reach over a million unique endpoints in just 20 periods. Recombining keeps the node count manageable and lets you build trees of 50 or 100 steps in an ordinary spreadsheet.
Risk-Neutral Probabilities
Once the price tree is built, the model assigns a probability to each branch. These are not real-world forecasts; they are mathematical weights chosen so that no arbitrage is possible. Under this risk-neutral framework, the expected return on the stock equals the risk-free rate, regardless of how investors actually feel about risk. The upward probability equals the continuously compounded risk-free growth factor minus the down factor, divided by the difference between the up and down factors. The downward probability is one minus the upward probability.
These probabilities exist for one reason: when you weight future option values by them and discount at the risk-free rate, you get a price consistent with the current market price of the stock. No opinion about risk premiums gets baked in. That is what makes the output arbitrage-free and lets it qualify as fair value under ASC 718 and IFRS 2.
Backward Induction
Valuation starts at the end of the tree and works backward. At each terminal node, the option’s value is its intrinsic payoff, meaning what you would pocket by exercising at that price. For a call, that is the stock price minus the strike, floored at zero. For a put, it is the strike minus the stock price, floored at zero. Any node where the option finishes out of the money is worth zero.
With the terminal values set, step one period back. At each earlier node, take the probability-weighted average of the two values it leads to (up-branch value times upward probability plus down-branch value times downward probability), then discount that average by one period at the risk-free rate. Repeat for every node in that column, then step back again. The process peels through the tree one column at a time until it reaches the single starting node at time zero. That node’s value is the model’s fair value estimate today.
For a European option, that pass is the entire calculation, and the result converges toward the Black-Scholes price as the number of steps grows.
Pricing American Options and Early Exercise
American-style options add one comparison at every interior node. After computing the hold value (the discounted expected value of continuing to the next period), the model also calculates the intrinsic value of exercising immediately. Whichever is higher becomes the node’s value. No closed-form equation can replicate that node-by-node comparison efficiently.
Early exercise clusters in predictable spots. Deep in-the-money American calls on dividend-paying stocks can be worth exercising just before an ex-dividend date. Deep in-the-money American puts can be worth exercising early when interest earned on the strike outweighs remaining time value. At nodes where exercise is optimal, intrinsic value dominates the hold value and the model locks in the higher figure.4The Journal of Derivatives. American Option Pricing: An Accelerated Lattice Model with Intelligent Lattice Search
Employee Stock Options Under ASC 718
Employee options behave differently from exchange-traded options because employees are not purely rational economic actors. They exercise early to lock in gains, they leave the company and forfeit unvested awards, and their post-termination exercise windows run in weeks rather than years. A lattice built for ASC 718 compliance has to carry all of this.
Suboptimal Exercise
Most employees exercise once the stock reaches a multiple of the strike price, typically somewhere between 1.5 and 2.5 times, though the ratio varies by company. In the tree, you trigger exercise at any node after vesting where the stock hits that multiple. A purely rational holder would wait longer, which is why the behavior is called suboptimal. Companies develop this assumption from their own exercise history rather than generic industry averages.
Post-Vesting Termination
At each node after vesting, the model applies a probability that the employee leaves during that period. If the option is in the money at departure, the model assumes exercise. If it is out of the money, the option expires worthless. Many plans give departing employees 30 to 90 days to exercise vested options, compressing the remaining time value sharply.
Layering these behavioral assumptions into the tree is the main reason a lattice produces a different expected term than Black-Scholes. Instead of taking expected term as an input, the lattice generates it as an output: the probability-weighted average of the periods at which exercise or forfeiture occurs across the tree. That number can be disclosed as the expected term in footnotes and even plugged back into Black-Scholes as a cross-check.
Adjusting for Dividends
Dividends reduce the stock price, which changes when early exercise becomes attractive, so the tree needs an adjustment.
For stock indices, currencies, or stocks with frequent small dividends, the standard approach treats dividends as a continuous yield. The fix is a single substitution: replace the risk-free growth factor in the probability formula with the risk-free rate minus the dividend yield. Up and down step sizes stay the same. The entire tree shifts downward slightly at each step, reflecting the steady drain of value.
When a stock pays a known dollar dividend on a specific date, the continuous shortcut breaks down. The common fix splits the stock price into a clean price that follows the normal geometric branching and a deterministic piece representing the present value of upcoming dividends. At each ex-dividend date, the tree drops the stock price by the dividend amount. Without that adjustment, the model overstates the stock price after the payment date and miscalculates the early-exercise boundary for American calls.
Trinomial Trees
A binomial tree allows two moves per step. A trinomial tree adds a third: the price can stay roughly unchanged. The up factor is typically calibrated as e raised to the power of volatility times the square root of 3Δt, with the down factor as its reciprocal and the middle branch at a multiplier of one.
The extra branch produces a finer grid and converges to the Black-Scholes price faster than a binomial tree with the same number of steps. It also handles barrier options more gracefully, because denser node spacing makes it easier to align a layer of nodes exactly on the barrier. The trade-off is computational: each step generates three branches instead of two, and extending the approach to multi-asset problems multiplies the node count quickly.
Barrier and Path-Dependent Options
Barrier options activate or deactivate when the stock crosses a predetermined level during the option’s life. A knock-out call, for instance, becomes worthless the moment the stock drops below the barrier, regardless of where it finishes. In a lattice framework, you set the option value to zero at any node where the stock has breached the barrier and run backward induction through the surviving nodes.
The difficulty is that a standard recombining tree rarely places nodes exactly on the barrier. When the barrier falls between two layers of nodes, the model either over-counts or under-counts the paths that breach it, producing oscillating and sometimes wildly inaccurate prices. Two well-known fixes exist: one repositions the tree so a row of nodes sits precisely on the barrier, and the other adjusts transition probabilities at nodes near the barrier to compensate for the grid mismatch. Both converge to the correct continuous-time price much faster than a naive lattice.
Disclosure and Audit Considerations
Picking a lattice is not just a modeling choice; it triggers specific disclosure and audit requirements.
The SEC does not require any particular valuation model. A company is free to use Black-Scholes, a lattice, or Monte Carlo, as long as the chosen method is grounded in established financial theory and captures all material features of the award. If a company switches models, the basis for the change must be disclosed in the footnotes. Regardless of model choice, footnotes must describe the method used, the significant assumptions (especially the method for estimating expected volatility), and how the current stock price was determined.2U.S. Securities and Exchange Commission. Staff Accounting Bulletin No. 107
For companies reporting under international standards, IFRS 2 requires fair value measurement at the grant date using market prices where available or a valuation technique where they are not. Expected volatility must be estimated by considering implied volatility from traded instruments, historical price data, and any known reasons future volatility might differ from the past.1IFRS Foundation. IFRS 2 Share-Based Payment Non-market vesting conditions, such as remaining employed for three years, are not built into the model’s fair value; they instead adjust the number of awards expected to vest.
Auditors evaluating a proprietary lattice follow the PCAOB’s framework for testing internal controls over financial reporting. Management must implement controls over the model’s significant assumptions and the IT systems that run it. Auditors test both design and operating effectiveness, and the evidence they require scales with the complexity of the model and the materiality of the compensation expense. In practice they probe the suboptimal exercise factor, the post-vesting termination rate, the volatility methodology, and the dividend assumption, which are the inputs where judgment is highest and the margin for error is widest.5Public Company Accounting Oversight Board. AS 2201 – An Audit of Internal Control Over Financial Reporting