C = S·N(d₁) − K·e−rT·N(d₂) · Greeks: Δ, Γ, Θ, ν
This simulator implements the Black-Scholes-Merton formula for European option pricing. It computes the theoretical price of call and put options and visualises how the price varies across a range of underlying asset prices. The payoff diagram overlays the intrinsic value at expiry against the current option price curve, revealing the time value component. Below, a Greeks chart shows how delta, gamma, theta and vega change with the spot price.
Fischer Black, Myron Scholes and Robert Merton published their options pricing model in 1973 — the same year the Chicago Board Options Exchange (CBOE) opened. Scholes and Merton received the Nobel Prize in Economics in 1997 for this work. The formula assumes constant volatility and log-normal returns, which doesn't hold in real markets — leading to the well-known volatility smile.
This simulator prices European call and put options using the Black-Scholes-Merton model. The call price follows C = S·N(d₁) − K·e−rT·N(d₂), where d₁ = [ln(S/K) + (r + ½σ²)T] / (σ√T) and d₂ = d₁ − σ√T. It plots the option price across a range of spot prices, overlays the intrinsic payoff at expiry, and charts the Greeks that measure the option's sensitivities.
The sliders set spot price S, strike K, time to expiry T (in years), annualised volatility σ and the risk-free rate r, while a toggle switches between call and put. Watching delta, gamma, theta and vega respond as you change inputs builds intuition for how traders hedge risk and value derivatives on exchanges such as the CBOE, where this 1973 model remains a foundational benchmark.
What does this simulator actually calculate?
It computes the theoretical fair value of European call and put options on a non-dividend-paying asset using the Black-Scholes-Merton formula. Alongside the price it shows the intrinsic value, the time value, and the five Greeks: delta, gamma, theta, vega and rho. Every quantity updates instantly as you move the sliders.
What is the Black-Scholes formula shown on the page?
For a call, C = S·N(d₁) − K·e−rT·N(d₂), where N is the standard normal cumulative distribution. Here d₁ = [ln(S/K) + (r + ½σ²)T] / (σ√T) and d₂ = d₁ − σ√T. The put price is obtained from the same d₁ and d₂ via put-call parity.
What do the five sliders control?
Spot Price S is the current price of the underlying asset; Strike K is the agreed exercise price; Time to Expiry T is the remaining life in years; Volatility σ is the annualised standard deviation of returns; and Risk-free Rate r is the continuously compounded interest rate. The Call/Put toggle selects which contract is priced and plotted.
Intrinsic value is what the option would be worth if exercised immediately: max(S − K, 0) for a call or max(K − S, 0) for a put. Time value is the remaining premium above that, reflecting the chance the option moves further into the money before expiry. The simulator displays both, and time value shrinks to zero as T approaches expiry.
Delta is the rate of change of price with respect to the spot, gamma is the rate of change of delta, theta is the daily time decay, vega is sensitivity to a one-point change in volatility, and rho is sensitivity to interest rates. The simulator reports theta per day and vega and rho per one-percent move, the conventions traders typically use.
The gap between the two curves is the time value of the option. Because there is always some probability the underlying moves favourably before expiry, an option is worth more than its immediate exercise value. As you reduce time to expiry or volatility, the price curve flattens toward the kinked intrinsic payoff line.
It is a useful idealisation rather than a perfect description. The model assumes constant volatility, log-normal returns, continuous trading and no transaction costs. Real markets show fat tails and the volatility smile, where implied volatility varies by strike. Traders still use Black-Scholes as a common quoting language while adjusting volatility inputs to fit observed prices.
Higher volatility raises both call and put prices, because a wider distribution of future outcomes increases the chance of large favourable moves while the downside stays capped at the premium. Move the volatility slider upward and you will see the entire price curve lift away from the intrinsic line, with vega quantifying that sensitivity.
Gamma measures how quickly delta itself changes as the spot moves. It peaks when the option is at the money, near the strike, because that is where a small price change has the greatest effect on the probability of finishing in the money. Far in or out of the money, delta is nearly fixed at 1 or 0, so gamma falls toward zero.
The cumulative normal N(d₁) and N(d₂) have no closed-form expression, so the simulator uses a rational polynomial approximation (the Abramowitz and Stegun formula with constants such as 0.254829592 and p = 0.3275911). This gives accuracy good to several decimal places, which is ample for visualising prices and Greeks in real time.
Market makers price and hedge portfolios by neutralising the Greeks: delta-hedging removes directional exposure, while gamma, vega and theta inform how often to rebalance and how decay erodes value. By exploring the sliders you can see why a position that looks neutral today can become risky as the spot, volatility or remaining time shifts.