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This graduate-level troubleshooting guide supports students across the Los Angeles MSA & San Francisco Bay Area MSA, including UCLA, USC, UC Berkeley, UC Irvine, UC Davis, UC Santa Cruz, UC Riverside & all CSU campuses. It covers advanced Math Theory and Finance Theory topics found in graduate programs, including calculus, linear algebra, optimization, probability, asset pricing, portfolio theory, derivatives, and financial mathematics.
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This troubleshooting guide isolates the exact theoretical gaps that cause confusion in graduate math and finance coursework. Each question targets a specific conceptual failure point—optimization conditions, eigenvalues, stochastic processes, arbitrage logic, pricing kernels, or risk-neutral valuation—so you can diagnose errors in your reasoning. The goal is to understand why models behave the way they do, how assumptions shape results, and how to evaluate whether your mathematical or financial derivations are theoretically valid.
Math & Finance Theory FAQ (50 Graduate-Level Questions)
1. What is the difference between a function and a functional?
A function maps numbers to numbers; a functional maps functions to numbers.
2. What is a gradient?
The gradient is the vector of partial derivatives of a multivariable function.
3. What is the Hessian matrix?
The Hessian is the matrix of second-order partial derivatives used to determine curvature.
4. How do I check convexity?
A twice-differentiable function is convex if its Hessian is positive semidefinite.
5. What is a positive definite matrix?
A matrix is positive definite if x’Ax > 0 for all nonzero x.
6. What are eigenvalues?
Eigenvalues are scalars λ such that Ax = λx for some nonzero vector x.
7. What is the spectral decomposition?
It expresses a symmetric matrix as A = QΛQ’.
8. What is the difference between L1 and L2 norms?
L1 sums absolute values; L2 sums squared values under a square root.
9. What is the chain rule for multivariable functions?
It expresses derivatives of composite functions using Jacobians.
10. What is the Lagrangian method?
It incorporates constraints into optimization using multipliers.
11. What are first-order conditions?
Set the gradient of the Lagrangian equal to zero.
12. What are second-order conditions?
Use the bordered Hessian to determine maxima or minima.
13. What is a Taylor expansion?
A Taylor expansion approximates a function using derivatives at a point.
14. What is a differential equation?
An equation involving derivatives of an unknown function.
15. What is a stochastic process?
A collection of random variables indexed by time.
16. What is Brownian motion?
A continuous-time stochastic process with independent, normally distributed increments.
17. What is Ito’s Lemma?
It gives the differential of a function of a stochastic process.
18. What is a martingale?
A process whose conditional expectation equals its current value.
19. What is the difference between ordinary and partial differential equations?
ODEs involve one variable; PDEs involve multiple variables.
20. What is the Laplacian operator?
The Laplacian is the divergence of the gradient, often written as ∇²f.
21. What is the determinant of a matrix?
The determinant measures volume scaling and invertibility.
22. What is the rank of a matrix?
Rank is the dimension of the column space.
23. What is the Moore–Penrose pseudoinverse?
It generalizes matrix inversion for non-square matrices.
24. What is a fixed point?
A point x such that f(x) = x.
25. What is the contraction mapping theorem?
It guarantees a unique fixed point for contraction mappings.
26. What is the no-arbitrage principle?
No-arbitrage states that identical cash flows must have identical prices.
27. What is a pricing kernel?
A pricing kernel maps future payoffs to present values.
28. What is risk-neutral valuation?
Risk-neutral valuation prices assets by discounting expected payoffs under the risk-neutral measure.
29. What is the fundamental theorem of asset pricing?
No-arbitrage is equivalent to the existence of a risk-neutral measure.
30. What is the CAPM equation?
CAPM states:
E[R_i] = R_f + β_i (E[R_m] - R_f)
31. What is the beta of an asset?
Beta measures sensitivity to market returns.
32. What is the APT model?
APT expresses returns as a linear function of multiple risk factors.
33. What is the difference between systematic and idiosyncratic risk?
Systematic risk affects all assets; idiosyncratic risk is asset-specific.
34. What is duration?
Duration measures interest rate sensitivity of a bond.
35. What is convexity?
Convexity measures curvature in the price–yield relationship.
36. What is the Black–Scholes PDE?
The PDE is:
∂V/∂t + (1/2)σ²S² ∂²V/∂S² + rS ∂V/∂S - rV = 0
37. What is delta?
Delta measures sensitivity of option value to the underlying price.
38. What is gamma?
Gamma measures curvature of option value with respect to the underlying.
39. What is vega?
Vega measures sensitivity to volatility.
40. What is rho?
Rho measures sensitivity to interest rates.
41. What is the risk-neutral drift of a stock price?
Under risk-neutral measure, drift equals the risk-free rate.
42. What is the difference between forward and futures pricing?
Futures incorporate daily settlement; forwards do not.
43. What is put–call parity?
Put–call parity states:
C - P = S - K e^{-rT}
44. What is the binomial option pricing model?
It prices options using up/down movements and risk-neutral probabilities.
45. What is the Sharpe ratio?
Sharpe ratio measures excess return per unit of risk.
46. What is the efficient frontier?
The efficient frontier is the set of portfolios with maximum return for a given risk.
47. What is mean–variance optimization?
It chooses portfolios that minimize variance for a given expected return.
48. What is the stochastic discount factor?
The SDF prices assets by weighting payoffs by marginal utility.
49. What is the difference between real and nominal interest rates?
Real rates adjust for inflation; nominal rates do not.
50. What is the Vasicek interest rate model?
It models interest rates as mean-reverting stochastic processes.
Related Pages
- Financial Math
- Financial Modeling
- Finance
- Economics
- Microeconomics
- Macroeconomics
- Statistics
- Mathematical Statistics
- Calculus & Higher Math
- Calculus & Linear Algebra
- Linear Programming
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Call or Text: (510) 398-0006
Email: tutor@californiagraduatetutor.com