Trinomial Tree (Boyle)
Prices an option with the Boyle trinomial tree, an evolution of the binomial where the price, at each step, can move up, down or stay flat. That third path gives the tree more flexibility and faster, more stable convergence than the binomial for the same number of steps. It's widely used for options whose features demand numerical precision. Choose call or put and enter the price, strike, rate, volatility, term and number of steps.
Result
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Trinomial Tree (Boyle)
Prices an option with the Boyle trinomial tree, an evolution of the binomial where the price, at each step, can move up, down or stay flat. That third path gives the tree more flexibility and faster, more stable convergence than the binomial for the same number of steps. It's widely used for options whose features demand numerical precision. Choose call or put and enter the price, strike, rate, volatility, term and number of steps.
Three paths at each step
Phelim Boyle's trinomial tree improves on the binomial with a simple but powerful addition: at each step, the price can move up, down or stay flat. That third path, the middle one, gives the model more degrees of freedom to represent the price distribution, which translates into faster, more stable convergence to the right number.
In practice, a trinomial tree with half the steps tends to be as accurate as a binomial with twice as many. That's why it's preferred in situations demanding fine numerical precision or when you need to fit the tree to barriers and other features the binomial represents poorly. The probabilities of the three branches are calibrated to reproduce the correct mean and variance at each step.
Choose call or put and enter the spot price, the strike, the rate, the volatility, the term and the number of steps. The tool returns the European price by the trinomial tree. Compare with Black-Scholes to see the convergence: few steps already get very close to the analytical value.
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Binomial Tree (Cox-Ross-Rubinstein)
Prices an option with the Cox-Ross-Rubinstein binomial tree, the most teachable numerical method for options. At each step, the price moves up or down by factors calibrated to the volatility, and the option value is computed backward, from expiry to today. Unlike Black-Scholes, the tree prices American options, checking early exercise at each node. Choose call or put, European or American, and enter the price, strike, rate, volatility, term and number of steps.
Forward-Start Option
Computes the price of a forward-start call: an option granted now, but whose strike is only set on a future date, usually as a proportion of the price at that moment. It's the structure behind employee option plans and cliquets, where new at-the-money options are issued periodically. Since the strike tracks the future price, the value doesn't depend on the current level in a trivial way. Enter the spot price, the moneyness, the rate, the volatility, the grant date and the expiry.
Option Theta (Black-Scholes)
Computes the theta of a European call option under Black-Scholes, the Greek that measures how much premium the option bleeds with each unit of time that passes. The formula pairs the decay of extrinsic value, −S·φ(d1)·σ/(2√T), with the strike-discount effect, −r·K·e^(−rT)·N(d2). For a call with no dividends theta is always negative: time works against the buyer. The result comes as annual theta and per day (÷365). Enter the spot price, the strike, the interest rate, the term in years and the volatility.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.