题目3591 · 随机过程
GBM Drift for a Median Target Over Two Years
A geometric Brownian motion satisfies dS_t = mu S_t dt + 0.3 S_t dW_t with S_0 = 100. If the median of S_2 should equal 128, what drift mu is required?
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中文题目题目3594 · 随机过程
GBM Drift Implied by a Mild Median Uplift
A geometric Brownian motion satisfies dS_t = mu S_t dt + 0.15 S_t dW_t with S_0 = 60. If the median of S_1.5 should equal 66, what drift mu is required?
打开 →题目3574 · 随机过程
Dynamics of the Reciprocal of GBM
A GBM satisfies dS_t = 0.05 S_t dt + 0.3 S_t dW_t. If Y_t = 1 / S_t, what are the drift and diffusion coefficients of Y_t?
打开 →题目3575 · 随机过程
Dynamics of the Square Root of GBM
A GBM satisfies dS_t = 0.02 S_t dt + 0.4 S_t dW_t. If Z_t = S_t^{1/2}, what are the drift and diffusion coefficients of Z_t?
打开 →题目6045 · 随机过程
Expected Value of a GBM
A stock follows dS_t = 0.1 S_t dt + 0.4 S_t dW_t with S_0 = 50. Compute E[S_3].
打开 →题目3611 · 随机过程
Discounted GBM Dynamics
A GBM satisfies dS_t = 0.05 S_t dt + 0.2 S_t dW_t. Define Y_t = e^{-0.05 t} S_t. What SDE does Y_t satisfy?
打开 →题目3612 · 随机过程
Log of GBM
A GBM satisfies dS_t = 0.08 S_t dt + 0.3 S_t dW_t. If Y_t = log S_t, what drift and diffusion coefficients does Y_t have?
打开 →题目3587 · 随机过程
Why Log GBM Loses Half Sigma Squared
Why does $d\log S_t$ for GBM contain the drift adjustment $-\sigma^2/2$?
打开 →题目3572 · 随机过程
Discount Rate Making e^{-gamma t} S_t^2 Driftless
A GBM satisfies dS_t = 0.03 S_t dt + 0.2 S_t dW_t. What gamma makes e^{-gamma t} S_t^2 a local martingale?
打开 →题目3571 · 随机过程
Nonzero Power Making S_t^p Driftless
A GBM satisfies dS_t = 0.04 S_t dt + 0.2 S_t dW_t. Find the nonzero p such that S_t^p is a local martingale.
打开 →题目3599 · 随机过程
Short-Horizon Volatility from an Upper Quantile Ratio
For a GBM, the ratio of the 90th percentile of S_0.5 to its median is observed to be 1.3. What volatility sigma does this imply?
打开 →题目3573 · 随机过程
Time Shift Making log S_t Driftless
A GBM satisfies dS_t = 0.07 S_t dt + 0.3 S_t dW_t. What c makes log S_t + c t a local martingale?
打开 →题目6047 · 随机过程
Variance of a Geometric Brownian Motion
A GBM satisfies dS_t = 0.06 S_t dt + 0.25 S_t dW_t with S_0 = 1. Compute Var(S_2).
打开 →题目3596 · 随机过程
Volatility from a 95th-to-Median Ratio in One Year
For a GBM, the ratio of the 95th percentile of S_1 to its median is observed to be 1.9. What volatility sigma does this imply?
打开 →题目3598 · 随机过程
Volatility Matching a 97.5th-to-Median Spread
For a GBM, the ratio of the 97.5th percentile of S_1.5 to its median is observed to be 2.2. What volatility sigma does this imply?
打开 →题目3721 · 随机过程
Choosing a Measure for Expected Realized Dividend Yield
An equity team wants the expected dividend yield that will actually be realized over the next year. Should that expectation be taken under P or Q?
打开 →题目6050 · 随机过程
Closed-Form Solution of a Linear Multiplicative SDE
Solve the SDE dX_t = a X_t dt + b X_t dW_t with X_0 given, where a and b are constants. Write the explicit closed-form expression for X_t.
打开 →题目3722 · 随机过程
Explaining Why Q Is Not 'What the Market Thinks Will Happen'
A candidate says Q is simply 'the market's true belief about future prices.' Why is that statement too crude?
打开 →题目3701 · 随机过程
Linear Desk PnL Drift Target from a Shared Driver
Under P, dX_t = 0.8dt + 0.4dW_t and dY_t = 0.3dt + 0.2dW_t share the same Brownian motion. Let L_t = 2X_t + 1Y_t. If Q is chosen so that L has drift 0.6 under Q, what theta is required and what is the resulting Q-drift of X?
打开 →题目3725 · 随机过程
Monte Carlo for CVA Price Versus Monte Carlo for Expected Exposure Forecast
If one Monte Carlo is being run for CVA pricing and another for a forecast of realized counterparty exposure paths, should they naturally live under the same measure?
打开 →题目3601 · 随机过程
OU Long-Run Mean Implied by a Conditional Mean Target 1
An OU process satisfies dX_t = 0.8(theta - X_t)dt + sigma dW_t with X_0 = 7. If E[X_1] should equal 5.2, what theta is implied?
打开 →题目3603 · 随机过程
OU Long-Run Mean Implied by a Conditional Mean Target 3
An OU process satisfies dX_t = 0.6(theta - X_t)dt + sigma dW_t with X_0 = 10. If E[X_0.5] should equal 8.9, what theta is implied?
打开 →题目3724 · 随机过程
Scenario Analysis for a Funding Desk
A funding desk wants a scenario distribution for next week's funding spread moves so it can size liquidity buffers. Which measure is appropriate?
打开 →题目3702 · 随机过程
Spread Drift Target via One Girsanov Tilt
Under P, dX_t = 0.7dt + 0.5dW_t and dY_t = 0.2dt + 0.4dW_t share the same Brownian motion. Let L_t = 1X_t + -1Y_t. If Q is chosen so that L has drift -0.1 under Q, what theta is required and what is the resulting Q-drift of X?
打开 →题目3723 · 随机过程
Why P Is Not Enough to Price a Replicable Claim
Why is a physical-measure expectation not enough on its own to pin down the fair price of a replicable derivative?
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