What is the Black-Scholes price of a European call option with underlying price S_t, strike K, risk-free rate r, volatility sigma, and time to maturity T-t?

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Multiple Choice

What is the Black-Scholes price of a European call option with underlying price S_t, strike K, risk-free rate r, volatility sigma, and time to maturity T-t?

The price of a European call in Black-Scholes comes from risk-neutral valuation of the payoff max(S_T − K, 0). Under a risk-free rate r and a lognormal stock process (no dividends assumed here), the present value is given by S_t times the probability-weighted factor Φ(d1) minus the present value of the strike times Φ(d2). This yields the closed-form: S_t Φ(d1) − K e^{−r (T−t)} Φ(d2), where Φ is the standard normal CDF. The terms d1 and d2 reflect the distributions of S_T under the risk-neutral measure: d1 = [ln(S_t/K) + (r + ½ σ^2)(T−t)] / [σ √(T−t)], and d2 = d1 − σ √(T−t). This structure captures how much of the option’s value comes from the current stock price (the S_t Φ(d1) part) and how much is the discounted strike adjusted by the probability the option finishes in the money (the K e^{−r (T−t)} Φ(d2) part). Other forms with incorrect time to maturity (such as e^{−r t} instead of e^{−r (T−t)}) or with reversed or negated Φ arguments do not align with the risk-neutral valuation for a call, so they do not produce the correct price.

The price of a European call in Black-Scholes comes from risk-neutral valuation of the payoff max(S_T − K, 0). Under a risk-free rate r and a lognormal stock process (no dividends assumed here), the present value is given by S_t times the probability-weighted factor Φ(d1) minus the present value of the strike times Φ(d2). This yields the closed-form: S_t Φ(d1) − K e^{−r (T−t)} Φ(d2), where Φ is the standard normal CDF.

The terms d1 and d2 reflect the distributions of S_T under the risk-neutral measure: d1 = [ln(S_t/K) + (r + ½ σ^2)(T−t)] / [σ √(T−t)], and d2 = d1 − σ √(T−t). This structure captures how much of the option’s value comes from the current stock price (the S_t Φ(d1) part) and how much is the discounted strike adjusted by the probability the option finishes in the money (the K e^{−r (T−t)} Φ(d2) part).

Other forms with incorrect time to maturity (such as e^{−r t} instead of e^{−r (T−t)}) or with reversed or negated Φ arguments do not align with the risk-neutral valuation for a call, so they do not produce the correct price.

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