BThere exists an equivalent probability measure Q under which all discounted asset prices are martingales
CThe expected return of every asset equals the risk-free rate
DThe market contains at least as many tradeable assets as sources of randomness
The first FTAP (Harrison-Pliska theorem) establishes a deep equivalence: no-arbitrage is equivalent to the existence of an equivalent martingale measure (EMM) Q. Under Q, discounted asset prices are martingales — their expected returns equal the risk-free rate. Option C is a consequence of Q existing (under Q, expected returns are r), not the theorem itself. Option D relates to the second FTAP (completeness). The theorem is remarkable because it translates an economic condition (no free lunch) into a mathematical condition (existence of a measure).
Question 2 Multiple Choice
In the Black-Scholes model, a European call option with strike K and maturity T has price C = S₀Φ(d₁) − Ke^{-rT}Φ(d₂). The term Φ(d₂) represents:
AThe probability that the option expires in the money under the risk-neutral measure Q
BThe probability that the stock price exceeds K at maturity under the physical measure P
CThe delta of the option (number of shares in the replicating portfolio)
DThe expected payoff of the option, discounted at the risk-free rate
Under Q, ln(S_T) ~ N(ln S₀ + (r-σ²/2)T, σ²T), so Q(S_T > K) = Φ(d₂) where d₂ = (ln(S₀/K) + (r-σ²/2)T)/(σ√T). The term Ke^{-rT}Φ(d₂) is the present value of the strike times the exercise probability. The term S₀Φ(d₁) involves d₁ = d₂ + σ√T and represents E_Q[S_T · 1_{S_T>K}] discounted — the delta Φ(d₁) is the number of shares in the replicating portfolio (option C is close but refers to d₁, not d₂).
Question 3 Short Answer
Explain why the Black-Scholes option price does not depend on the stock's expected return μ.
Think about your answer, then reveal below.
Model answer: The option price is an expectation under the risk-neutral measure Q, not the physical measure P. Girsanov's theorem removes the physical drift μ and replaces it with the risk-free rate r. Under Q, dS = rS dt + σS dW̃ — the stock grows at rate r regardless of its real-world expected return. The reason μ drops out is the no-arbitrage argument: the option can be perfectly replicated by delta-hedging, and the hedging strategy's cost depends only on σ (which determines the hedging adjustments) and r (the financing cost), not on where the stock is expected to go. Two stocks with the same σ but different μ have the same option price.
This is one of the deepest insights of Black-Scholes theory. Risk preferences are irrelevant for derivative pricing (risk-neutral valuation) — you price options as if everyone were risk-neutral, even though they aren't. The physical drift μ affects the stock's expected return but not the cost of replicating the option, because the replicating portfolio is continuously adjusted to be instantaneously riskless.
Question 4 True / False
The Black-Scholes model assumes constant volatility σ. In practice, implied volatilities vary across strikes and maturities (the 'volatility smile'). This empirical fact:
TTrue
FFalse
Answer: True
The volatility smile/skew is one of the most well-documented departures from Black-Scholes. If the model were correct, all options on the same stock would produce the same implied volatility when inverted through the Black-Scholes formula. Instead, out-of-the-money puts typically have higher implied volatility than at-the-money options (the 'skew'), reflecting the market's pricing of tail risk and jump risk that GBM cannot capture. This motivates extensions: local volatility (Dupire), stochastic volatility (Heston), and jump-diffusion (Merton) models.