In the Black-Scholes formula C = S·N(d₁) − K·e^(−rT)·N(d₂), what does the term K·e^(−rT)·N(d₂) represent?
AThe current stock price weighted by the probability of exercise
BThe present value of the strike price weighted by the risk-neutral probability the option expires in the money
CThe implied volatility premium embedded in the option price
DThe delta of the option times the stock price
K·e^(−rT) is the present value of the strike payment discounted at the risk-free rate, and N(d₂) is the risk-neutral probability that the option expires in the money (S_T > K). So this term is the expected present-value outflow if exercise occurs. The full formula is the expected receipt (S·N(d₁)) minus the expected payment — a form of 'expected value of owning the asset upon exercise minus expected cost of exercising.'
Question 2 True / False
Implied volatility derived from Black-Scholes equals the realized (historical) volatility of the underlying asset.
TTrue
FFalse
Answer: False
Implied volatility is backed out from observed market option prices and reflects the market's forward-looking expectation of future volatility, plus a variance risk premium. Realized volatility measures actual past price fluctuations. The two diverge systematically — implied volatility tends to exceed realized volatility on average, a gap called the variance risk premium, because investors pay extra to hedge against volatility uncertainty.
Question 3 Short Answer
Why is the expected return of the underlying stock not an input in the Black-Scholes formula, even though a higher expected return seems like it would increase the value of a call option?
Think about your answer, then reveal below.
Model answer: Black-Scholes constructs a continuously rebalanced delta hedge that makes the combined portfolio of the option and a short stock position instantaneously riskless. In a riskless portfolio, the expected return is irrelevant — it must earn the risk-free rate by no-arbitrage. The hedge eliminates all dependence on the stock's drift, leaving only volatility (the random component) as the driver of option value.
This is the model's central insight. Intuition says 'higher expected stock return → higher call value,' but that intuition is wrong because you could also buy the stock directly to benefit from that return. What the option provides that the stock does not is asymmetric exposure to volatility. The risk-neutral pricing framework captures this: we price as if the stock grows at the risk-free rate (eliminating the expected return from the problem) and discount at the risk-free rate.