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中文题目A one-step CRR binomial tree prices an at-the-money one-year European call at 9.95, while the Black-Scholes value with the same spot, strike, rate and volatility is 8.43. By how much does the coarse tree overprice the option, and what single change to the tree would most directly
打开 →Determine the radius of convergence of the power series sum_(n=1)^inf (3^n / n) x^n.
打开 →Near a simple root r with f'(r) != 0, what local rate does Newton's method typically enjoy?
打开 →For which of the following exponents p does the series sum_(n=1)^inf 1/n^p converge: p = 1/2, p = 1, p = 3/2, p = 2? List all values for which it converges.
打开 →A trader is long spot and short one futures contract. Initially spot is 100 and futures is 104, so the cash basis is -4. Later the basis changes by 1.5. Ignoring financing, what is the trade's marked-to-market PnL change?
打开 →A trader is long spot and short one futures contract. Initially spot is 80 and futures is 76, so the cash basis is 4. Later the basis changes by -2. Ignoring financing, what is the trade's marked-to-market PnL change?
打开 →A trader is long spot and short one futures contract. Initially spot is 52 and futures is 52.8, so the cash basis is -0.8. Later the basis changes by 0.6. Ignoring financing, what is the trade's marked-to-market PnL change?
打开 →A trader is long spot and short one futures contract. Initially spot is 120 and futures is 123, so the cash basis is -3. Later the basis changes by 2. Ignoring financing, what is the trade's marked-to-market PnL change?
打开 →Let $X_n \sim \text{Binomial}(n, \lambda/n)$ for fixed $\lambda > 0$ and $n > \lambda$. (a) Write down the characteristic function $\varphi_{X_n}(t) = E[e^{itX_n}]$ in closed form. (b) Show that $\lim_{n \to \infty} \varphi_{X_n}(t) = e^{\lambda(e^{it} - 1)}$ for every $t \in \
打开 →If the risk-free rate rises while the state payoffs stay the same, which piece of the one-step replicating portfolio is directly affected even before the hedge ratio changes?
打开 →A second-order finite-difference scheme uses Richardson extrapolation E_R = (4E_fine - E_coarse)/3. If E_R=10.6 and E_fine=10.4, what coarse-grid estimate E_coarse is implied?
打开 →A second-order finite-difference scheme uses Richardson extrapolation E_R = (4E_fine - E_coarse)/3. If E_R=2.8 and E_coarse=2.4, what fine-grid estimate E_fine is implied?
打开 →Let $N_\lambda\sim\mathrm{Poisson}(\lambda)$. Show that \[ \frac{N_\lambda-\lambda}{\sqrt{\lambda}}\Rightarrow N(0,1) \quad\text{as }\lambda\to\infty \] by working directly with characteristic functions.
打开 →Let $X_1,X_2,\dots$ be i.i.d. with $P(X_i=1)=P(X_i=-1)=1/2$. Show, using characteristic functions, that \[ \frac{X_1+\cdots+X_n}{\sqrt n}\Rightarrow N(0,1). \]
打开 →Let $X_n\sim\mathrm{Binomial}(n,\lambda/n)$ with fixed $\lambda>0$. Use characteristic functions to show that $X_n\Rightarrow \mathrm{Poisson}(\lambda)$.
打开 →Why does making the binomial time step smaller make the replication route feel closer to continuous Black-Scholes hedging?
打开 →Why does adding more binomial steps usually make backward induction more informative about dynamic hedging rather than less?
打开 →Why does the cash basis $S-F$ become negative in contango markets?
打开 →Why does the cash basis $S-F$ become positive in backwardation?
打开 →Why can a cash-futures hedge still leave risk even if the futures notional matches the cash position perfectly?
打开 →Why does basis often shrink as a futures contract approaches expiry?
打开 →Why is basis worth monitoring as its own state variable rather than only watching spot and futures separately?
打开 →Why is a simple root f'(r) != 0 the standard clean setting for fast Newton convergence?
打开 →Why do exact gradient descent convergence and the normal-equation solution agree for OLS?
打开 →An urn starts with 1 red and 2 blue balls. Each step a ball is drawn uniformly at random, observed, and returned together with one additional ball of the same color. Let R_n/T_n be the fraction of red balls after n draws. This fraction is a bounded martingale converging to a limi
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