Brownian Bridge™

1. In GBM dynamics dS = mu*S*dt + sigma*S*dW, the drift of ln(S_T) over [0, T] is:

mu*T(mu - sigma^2 / 2) * T(mu + sigma^2 / 2) * Tmu*T - sigma*sqrt(T)

2. The Black-Scholes PDE does not contain the drift mu of the underlying stock. The correct reason is:

The drift cancels out because calls and puts have opposite deltaThe delta-hedging argument eliminates the stochastic dW term and with it the dependence on mu; the option price depends only on sigma and rBlack and Scholes assumed mu = r in their original paperThe drift is absorbed into the dividend yield q

3. In the Black-Scholes formula C = S*exp(-qT)*N(d1) - K*exp(-rT)*N(d2), the term N(d2) represents:

The delta of the call optionThe risk-neutral probability that the option expires in the money, i.e., Q(S_T > K)The probability that S_T > K under the physical measure PThe ratio of the option price to the forward price

4. Put-call parity C - P = S*exp(-qT) - K*exp(-rT) is best described as:

A consequence of the Black-Scholes formula that holds only under lognormal dynamicsA model-independent no-arbitrage relationship that holds for any model with European calls and puts on the same underlying, same strike, and same maturityAn approximation that holds when volatility is lowA result that requires continuous dividend payments to hold exactly

5. For vanilla European calls and puts in the Black-Scholes model, gamma is:

Positive for calls, negative for putsNegative for calls, positive for putsAlways positive for both calls and putsPositive only when the option is in the money

6. A delta-hedged long call earns positive P&L over one trading day when:

The stock price moves upThe stock price moves downThe realised squared move (dS)^2 exceeds sigma_impl^2 * S^2 * dt, i.e., realised vol exceeds implied volThe option is in the money at the end of the day

7. The Black-Scholes call price is homogeneous of degree 1 in (S, K). This means:

Doubling sigma doubles the option priceThe option price scales linearly with time to maturityMultiplying both S and K by the same factor lambda multiplies the call price by lambdaThe option price is proportional to S for any strike

8. The implied volatility of a market option price is defined as:

The historical realised volatility of the underlying over the option's remaining lifeThe GARCH forecast of future volatility for the option's maturityThe value of sigma that, when substituted into the Black-Scholes formula, reproduces the observed market priceThe average of the bid and ask volatilities across all strikes and maturities

Quiz: The Black-Scholes Model

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