Option Pricing Calculator: Binomial, Black-76 & FX

About the Option Pricing Calculator: Binomial, Black-76 & FX

Value the options a plain Black–Scholes calculator cannot: American options that can be exercised early, on a Cox–Ross–Rubinstein binomial tree of up to 2,000 steps, with a dividend yield or cash dividends handled by Hull’s escrowed-dividend method; options on futures and forwards with Black-76; currency options with Garman–Kohlhagen, which uses both currencies’ interest rates; and the Bachelier (normal) model for prices near or below zero.

For each you get the value, the Greeks in stated units (delta and gamma per 1.00 of the underlying, theta per calendar day, vega per volatility point, rho per rate point), the early-exercise premium of an American option against the European one on the same tree, the implied volatility of a market price you type, a put-call parity check for the European models, and the value against the underlying’s price.

It works for share, index, commodity and currency options in any market — USD/INR, EUR/USD, MCX, Eurex or CME contracts alike — with the figures you enter. No market data is used.

Common uses

• Value an American put on a dividend-paying share, and see what early exercise is worth.
• Price an option on a futures contract (crude oil, gold, an index future) with Black-76.
• Price a USD/INR or EUR/USD option with both interest rates, and back out its implied volatility.
• Check an option on a price that is near or below zero with the Bachelier model.

The formulas

• Binomial tree (Cox, Ross and Rubinstein90015-1)): u = e^(σ√Δt), d = 1/u, p = (e^((r − q)Δt) − d) ÷ (u − d); each node is worth e^(−rΔt)[p × value up + (1 − p) × value down], or for an American option the value of exercising if that is larger. Δ and Γ come from the tree’s first two steps and Θ from the middle node two steps on; vega and rho from trees built again with σ and r moved (Hull, Options, Futures, and Other Derivatives, ch. 13 and 21).
• Cash dividends follow Hull’s escrowed-dividend method: the tree is built on S* = S − the present value of the dividends before expiry, and the price at a node is S* there plus the present value of the dividends still to come.
• Black-76 (Black90024-6)): c = e^(−rT)[F N(d₁) − K N(d₂)], p = e^(−rT)[K N(−d₂) − F N(−d₁)], d₁ = [ln(F/K) + σ²T/2] ÷ σ√T, d₂ = d₁ − σ√T.
• Garman–Kohlhagen (Garman and Kohlhagen80001-1)): Black–Scholes with the foreign rate in place of a dividend yield: c = S e^(−r_f T) N(d₁) − K e^(−r_d T) N(d₂).
• Bachelier (Choi, Kwak, Tee and Wang, eq. 1): c = e^(−rT)[(F − K) N(d) + σ_N√T n(d)], d = (F − K) ÷ σ_N√T, with a normal volatility σ_N in price units.
• Implied volatility is found with Brent’s method, which needs only prices, so it works for trees too.

Reading the results

• The early-exercise premium is what the right to exercise early adds: the American value less the European one on the same tree. An American call without dividends has none.
• Steps: a tree converges towards the exact value as the steps grow, wobbling as it does; 100 to 500 steps are usually close, and trees above 500 steps are worked out in the background so the page stays responsive.
• Put-call parity: for European options c − p must equal S e^(−qT) − K e^(−rT) (or e^(−rT)(F − K) for futures); the check shows the model’s figures meet it.
• Normal volatility (Bachelier) is in price units: 12 means a standard deviation of 12 over a year, whatever the price.

What it leaves out

Volatility that changes with the strike or over time (the smile and term structure); jumps; the bid–ask spread; and, for currency options, the exact settlement and day-count conventions of your contract. Exchange contracts have their own exercise rules (European or American, settlement in cash or the underlying): check them on the contract specification.

Limitations

• Constant volatility and interest rates over the option’s life.
• Trees from 1 to 2,000 steps; fewer than 10 are for checking textbook examples by hand.
• Up to 12 cash dividends; those after expiry are left out.
• It calculates from your numbers and uses no market data: it says nothing about where any price is going.

Privacy

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