The Black and Scholes model is intended to be used to value

Question

BlackTom AI · Accepted Answer

The Black–Scholes model is a theoretical framework for pricing European-style options on underlying assets, so "options" is the correct choice.

1) Relevant knowledge points and concepts
- Black–Scholes (1973) is an option-pricing model built on no-arbitrage, continuous trading, lognormally distributed underlying prices, constant volatility (σ), and a constant risk-free rate (r). The original model applies to European call and put options on non-dividend-paying stocks.
- Key formula components: current stock price S0, strike K, time to maturity T, volatility σ, risk-free rate r, and the cumulative normal distribution Φ.

2) Complete reasoning process (step by step)
- The model derives a replicating portfolio of stock and bonds to eliminate risk and hence price an option uniquely.
- For a European call, Black–Scholes gives the option price as:
  C = S0 * Φ(d1) − K * e^(−rT) * Φ(d2)
  where
  d1 = [ln(S0/K) + (r + σ^2/2)T] / (σ * sqrt(T))
  d2 = d1 − σ * sqrt(T)
- The formula computes the present value of expected payoff under the risk-neutral measure, adjusted by probabilities Φ(d1) and Φ(d2).

3) Intermediate steps shown
- Compute d1 using ln(S0/K), add (r + σ^2/2)T, divide by σ√T.
- Compute d2 by subtracting σ√T from d1.
- Evaluate Φ(d1) and Φ(d2) from the standard normal CDF.
- Plug into C = S0Φ(d1) − K e^(−rT) Φ(d2).

4) Why the correct answer is correct
- Black–Scholes was designed specifically to value options (European calls and puts). Its assumptions and derivation directly produce option prices, not prices of primary securities.

5) Why other options are incorrect
- Futures contracts: priced by cost-of-carry and forward pricing relations; while modified Black-type formulas exist for options on futures (Black 1976), the original Black–Scholes model targets options, not futures themselves.
- Bonds: bond valuation uses discounting of cash flows and yield curves, not Black–Scholes.
- Stocks: stock prices are underlying inputs (S0) to the model, not outputs the model values.

Conclusion: Therefore, the Black–Scholes model is intended to value options, matching the provided answer "options."

The Black and Scholes model is intended to be used to value单项选择题

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