stochastic — Quantitative Finance Practice Questions

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1. Under Girsanov's Theorem, what does a change of probability measure primarily alter in a stochastic process dr
2. In the context of the HJM framework, what is the primary lesson regarding the drift of the forward rate curve?
3. In the Vasicek short-rate model dr_t = κ(θ - r_t) dt + σ dW_t, what happens to the drift when the current rate
4. For a standard Brownian motion W_t, what is the expected value of W_t^2?
5. Under the geometric Brownian motion model with μ = 0.12, σ = 0.30, and S_0 = 100, what is the median stock pri
6. According to the lognormal property, what is the expected stock price at time T = 1 year if the initial price
7. A stock follows geometric Brownian motion dS = μ S dt + σ S dW. Using Itô's Lemma, find the volatility of the
8. Under Girsanov's theorem, if a stock follows dS_t = μ S_t dt + σ S_t dW_t under the real-world measure mathbbP
9. What is the expected price of the stock in one year, E[S_1]?
10. What is the expected value of an Itô integral mathbbE[int_0^T H_s dW_s] for any adapted process H_s?
11. In the Girsanov theorem framework, what is the 'market price of risk' θ for an asset with real-world expected
12. If W_t is a standard Brownian motion, what is the expected value of the process X_t = W_t^2 - t at any time t
13. If the risk-free rate is 2%, what is the market price of risk (θ) required for a Girsanov change to the risk-n
14. Under the Geometric Brownian Motion model with drift μ = 0.12 and volatility σ = 0.30, which of the following
15. Which of the following describes the 'volatility drag' effect in Geometric Brownian Motion?
16. If a stock price S_t follows Geometric Brownian Motion with drift μ = 12% and volatility σ = 30%, what is the
17. If the asset's true real-world expected return is μ = 10%, what is the market price of risk θ used in Girsanov
18. In the Heath-Jarrow-Morton (HJM) framework for modeling the term structure of interest rates, what is the sign
19. According to Girsanov's theorem, when transitioning from the real-world probability measure P to the risk-neut
20. An It^o integral of the form I_t = int_0^t H_s dW_s is alway… — What is the primary reason for this?
21. Consider an It^o process dX_t = μ_t dt + σ_t dW_t. Under what condition is this process a martingale?
22. If M_t is a martingale and h is a bounded, predictable strategy, why is the 'stochastic integral' (h · M)_t =
23. If we change the numéraire from the money-market account to a different asset, how does the resulting derivati
24. In a simple one-period binomial model, the risk-neutral probability p^* of an up-move is calculated using whic
25. In Girsanov's theorem, we change the probability measure fro… — What happens to the volatility σ during this c
26. In the context of stochastic calculus and the Itô multiplication table, what is the value of the product (dW_t
27. In the context of the Itô multiplication table, how is d[t, W]_t (the cross-variation of time and Brownian mot
28. In the discrete version of Brownian motion (a random walk)… — What is the square of this step, and how does it
29. Standard Brownian motion W_t is characterized by having incr… — What does this property imply for non-overlapp
30. How is it defined for a stock with drift μ, volatility σ, and risk-free rate r?