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Audit the value. Calculate the theoretical price of European call and put options using the Black-Scholes-Merton model.
Step-by-step breakdown of the underlying equations.
Based on the Black-Scholes model, with a volatility of 20%, the fair value of a call option at strike $105 is $4.58. This assumes the stock price follows a geometric Brownian motion and that there are no transaction costs or dividends.
The Black-Scholes-Merton Options Pricing Model calculates the theoretical fair market value of European-style call and put options. By mathematically integrating the underlying spot stock price, strike price, time to expiration, risk-free interest rate, and annualized implied volatility, it provides the standard mathematical benchmark for modern derivatives valuation and risk management.
Worked Valuation Example:\nEvaluate a 6-month European call and put option (t = 0.5 years) where the underlying stock trades at $100 (S = 100), the strike price is $105 (K = 105), the risk-free rate is 5.0% (r = 0.05), and annualized volatility is 20.0% (ฯ = 0.20):\nโข d_1 = [ln(100/105) + (0.05 + 0.04/2) ร 0.5] รท [0.20 ร โ0.5] = [โ0.04879 + 0.035] รท 0.14142 = โ0.0975\nโข d_2 = โ0.0975 โ 0.14142 = โ0.2389\nโข N(d_1) = 0.4612; N(d_2) = 0.4056\nโข Theoretical Call Value: 100 ร 0.4612 โ 105 ร e^(โ0.025) ร 0.4056 = $4.57\nโข Theoretical Put Value: 105 ร e^(โ0.025) ร (1 โ 0.4056) โ 100 ร (1 โ 0.4612) = $6.98\nโข Put-Call Parity Verification: C โ P = 4.57 โ 6.98 = โ$2.41 = S โ Kรe^(โrt) (100 โ 102.41 = โ$2.41).
Compare theoretical model values against current exchange quotes on the Cboe. If market prices are significantly higher than model outputs, implied volatility (IV) exceeds your estimated historical volatility, suggesting options are relatively expensive (favorable for option sellers). Conversely, lower market prices suggest options are priced attractively for buyers.
European options can only be exercised on their exact expiration date, whereas American options can be exercised at any time prior to expiration. While European options are priced precisely using Black-Scholes, American options with early exercise features (especially on dividend-paying stocks) are typically valued using binomial trees or the Bjerksund-Stensland approximation.
Volatility represents the expected dispersion of future asset returns. Because option holders have limited downside (the premium paid) but theoretically unlimited upside (calls) or substantial upside (puts), higher volatility increases the statistical probability of reaching deep out-of-the-money profitability without expanding downside risk.
Put-Call Parity is a fundamental no-arbitrage principle stating that the value of a fiduciary call (call plus discounted cash equal to strike) must equal the value of a protective put (put plus underlying stock): C + K ร e^(โrt) = P + S. Any deviation creates risk-free arbitrage opportunities.
The classic 1973 model does not account for cash dividends. The Merton extension incorporates continuous dividend yield (q) by discounting the spot price: S ร e^(โqt), which lowers call prices and increases put prices.
Implied Volatility is the annualized standard deviation of future price returns that is implied by current market option prices. It is solved backward by plugging observed market premiums into the Black-Scholes formula and using numerical root-finding (such as the Newton-Raphson method).
Data verified: September 2026