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中文题目A gambler starts with $\$4$ and repeatedly bets $\$1$ on a biased coin that lands heads with probability $p = 0.55$. She gains $\$1$ on heads and loses $\$1$ on tails. The game ends when her fortune reaches $\$0$ or $\$10$. Using the martingale $(q/p)^{X_n}$ and the optional stop
打开 →Let W_t be standard Brownian motion. For what constant c is the process M_t = W_t^2 - c t a martingale?
打开 →In the martingale decomposition of a call, the discounted share-side term is 38.2 and the call value is 6.8. What discounted strike-side term must be subtracted?
打开 →M_n = S_n/(n+1), where S_n = X_1+...+X_n for iid symmetric ±1 increments. Is (M_n) a martingale?
打开 →M_n = X_(n+1)-p for iid Bernoulli(p) variables with natural filtration F_n. Is (M_n) a martingale?
打开 →M_n = S_n^2 - n/2 for a symmetric ±1 random walk. Is (M_n) a martingale?
打开 →M_n = 2^n S_n for a symmetric ±1 random walk. Is (M_n) a martingale?
打开 →Before carrying out any lognormal integral in the martingale derivation, what object should you isolate first?
打开 →Before invoking any theorem-level pricing statement in the martingale route, what should you identify first?
打开 →Before interpreting d2 economically in the martingale derivation, what should you know first?
打开 →Why do d1 and d2 appear as two different normal arguments in the martingale derivation of a Black-Scholes call?
打开 →Why is discounted pricing central in the martingale derivation rather than a cosmetic last step?
打开 →Why should the martingale route and the PDE route give the same Black-Scholes price?
打开 →Why does a continuous dividend yield enter the martingale derivation before the final formula is written down?
打开 →Spot is 100, maturity is 1 year, and the prepaid forward price is 97.0446. What continuous dividend yield q is implied?
打开 →Spot is 80, maturity is 1 year, the forward price is 82.4364, and the risk-free rate is 5%. What continuous dividend yield q is implied?
打开 →A stock has spot 90 and 1-year discounted risk-neutral expectation E[e^{-rT}S_T] = 88.218. What continuous dividend yield q is implied?
打开 →Under risk-neutral pricing, spot is 120, risk-free rate is 4%, dividend yield is 1.5%, and maturity is 2 years. What is E[S_T]?
打开 →A 1-year prepaid forward on a stock is 96, and the corresponding forward price is 100.9221. What risk-free rate r is implied?
打开 →A Markov chain on $\{0, 1, 2, 3\}$ has transition probabilities: $$p(1, 0) = \tfrac{1}{3}, \quad p(1, 2) = \tfrac{2}{3}, \quad p(2, 1) = \tfrac{1}{2}, \quad p(2, 3) = \tfrac{1}{2}.$$ State $0$ is absorbing and state $3$ is reflecting: $p(3, 2) = 1$. Let $T = \inf\{n \ge 0 : X_n
打开 →A call is worth 6.2, a put is worth 4.7, and the discounted strike is 51.5. What prepaid forward price is implied by put-call parity?
打开 →A call price is 5.4 and the discounted strike-side term K e^{-rT} N(d2) is 24.2. What discounted share-side term S0 e^{-qT} N(d1) is implied?
打开 →The prepaid forward is 48.5, the call price is 7.3, and the discounted strike is 50.1. What put price follows from put-call parity?
打开 →In a call decomposition, the discounted asset-side tail expectation is 44.1 and the discounted strike-side tail expectation is 35.7. What call value follows?
打开 →What should you write down first before evaluating any tail expectation for a call payoff?
打开 →Let M_0=0, M_1, M_2, ... be a martingale whose increments satisfy |M_k - M_{k-1}| <= 1 for all k. Using the Azuma-Hoeffding inequality, give the best upper bound it provides on P(M_100 >= 20).
打开 →In an election candidate A receives 7 votes and candidate B receives 3 votes; the 10 votes are counted in a uniformly random order. Using a martingale / optional stopping argument, find the probability that A is strictly ahead of B throughout the entire count.
打开 →For standard Brownian motion, choose $c$ so that $X_t=W_t^3-c\int_0^t W_s\,ds$ is a local martingale.
打开 →Choose a so that X_t = e^{a t} cos W_t is a martingale.
打开 →Choose a so that X_t = exp(aW_t - 2t) is a martingale.
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