Options Pricing Calculator

The calculator gives the theoretical value of a European call or put: an option that can be exercised only at expiry. It uses the Black-Scholes-Merton model, which prices the option as the cost of a portfolio of stock and borrowing that reproduces the option's payoff. Because that replicating portfolio removes risk, the price does not depend on whether you expect the stock to rise or fall, only on the stock price, strike, time, interest rate, dividend yield and volatility.

Along with the price it reports the Greeks (how the price responds to each input), the split between intrinsic value and time value, the breakeven price at expiry, a profit and loss table, and a chart comparing the payoff at expiry with the value today. Enter a quoted market price and it also solves for the implied volatility, the one input you cannot observe directly.

c = S e^−qT N(d₁) − K e^−rT N(d₂)    p = K e^−rT N(−d₂) − S e^−qT N(−d₁)

d₁ = [ln(S ÷ K) + (r − q + σ² ÷ 2) T] ÷ (σ √T)    d₂ = d₁ − σ √T

S is the stock price, K the strike, T the time to expiry in years (days ÷ 365 here), r the continuously compounded risk-free rate, q the continuous dividend yield, σ the annual volatility and N the standard normal cumulative distribution. With q = 0 this is the original 1973 Black-Scholes formula. N(d₂) is the risk-neutral probability that a call finishes in the money.

• Delta is the change in option value for a $1 move in the stock.
• Gamma is the change in delta for a $1 move; it is largest near the strike and close to expiry.
• Theta is the change in value as one calendar day passes (the annual figure ÷ 365).
• Vega and rho are the changes for a one percentage point rise in volatility and in the interest rate.

The textbook call and put

Hull's standard example uses S = 42, K = 40, r = 10%, σ = 20% and six months to expiry. Here d₁ = 0.7693 and d₂ = 0.6278, giving a call worth $4.76 and a put worth $0.81, matching the published 4.76 and 0.81. The call is $2 in the money, so $2.76 of its price is time value. The two prices satisfy put-call parity: c − p = S − K e^−rT, so $4.76 − $0.81 = $3.95.

How time decay speeds up

An at-the-money call on a $100 stock at 25% volatility and a 4% rate is worth $6.35 with 120 days left and $3.02 with 30 days left. Its theta is −0.0290 per day at 120 days but −0.0530 per day at 30 days, about 1.8 times as fast, because time value shrinks roughly with the square root of the time remaining.

Backing out implied volatility

If that 30-day at-the-money call is quoted at $3.50 rather than its 25% value of $3.02, the volatility that reproduces the quote is 29.2%. Traders compare options by implied volatility rather than by price because it puts different strikes and expiries on one scale.

Stock at $100, 90 days to expiry, 25% volatility, 4% rate, no dividends. Calls lose value and puts gain value as the strike rises.

• American exercise. Most US single-stock options can be exercised early. Early exercise never pays for a call on a stock with no dividends, so the European value applies, but American puts and calls before a dividend can be worth more than shown. Those need a binomial or finite-difference model.
• Constant volatility. The model uses one volatility for every strike. Market prices imply different volatilities by strike and expiry (the volatility smile or skew), so one input cannot match a whole options chain.
• Discrete dividends. The dividend input is a continuous yield. A single large dividend before expiry is better handled by subtracting its present value from the stock price.
• Trading costs. Bid-ask spreads, commissions and assignment risk are not included. The breakeven and profit figures are per share before costs; a standard US equity contract covers 100 shares.