2026 · March

Portfolio Management:

• Alpha ($\alpha$): The "edge." How much an investment outperformed its benchmark

• Beta ($\beta$): Market Risk; Volatility relative to the market

• Sigma ($\sigma$): Standard Deviation; The total volatility of an asset. How much its price swings

• Lambda ($\lambda$): Leverage; In options to show % change in option price per 1% change in the stock.

Options:

• Delta ($\Delta$): Price Sensitivity. How much option's price changes for every $1 move in the underlying

• Gamma ($\Gamma$): Acceleration. The rate of change in Delta. It tells you how "stable" your Delta is.

• Theta ($\Theta$): Time Decay. How much value the option loses every day as it approaches expiration.

• Vega ($\nu$): Volatility. How much option's price changes based on a 1% change in implied volatility.

• Rho ($\rho$): Interest Rate. Sensitivity of the option price to changes in risk-free interest rates.

Lec 04

Arbitrage opportunity

Pice bounds for options

Put-call parity

Binomial model

Blackscholes

Lec 05

Pricing by arbitrage

We have multiple linear independent assets (think of polymarket contracts)

Arbitrage portfolio

Lec 06

Economic analysis of asset markets

• Pareto-optimal allocations

• Contract curve

• Market clearing condition: everything sold by one is bought by another

• Comotinicity

• Price responce to risk changes

Lec 07

Uncertain environments:

• compound lottery $λL+(1−λ)L′$

• Preference relation

  • Complete $for any L and L′∈L(Z ) we have L ⪰L′or L′⪰L$

  • Transitive $(L1 ⪰L2 and L2 ⪰L3) ⇒L1 ⪰L3$

• VNM (Von Neumann and Morgenstern) Expected Utility Theory

  • Independence

  • Continuity

Risk Averse

• Utility Index u

• Expected utility function v

• VNM preferences is risk averse if and only if her utility index is concave

Risk increase

• SOSD: Second-order stochastic dominance: if dominant then less risky than other

  • $EL[x] ≥EL′[x]$

  • $\int_{-\infty}^{y} F_L(x)dx \leq \int_{-\infty}^{y} F_{L'}(x)dx$ less weight on smaller returns

• Mean preserving spread -> more risk

• White noise: mean 0, can be seen as risk

Quantifying Risk Aversion

• Certainty Equivalant: uncertain lottery utility compared to a certain lottery

• Risk Premium: Expectation - certainty equivalant $\pi_L = E_L[x] - e_L$

• Risk Aversion coefficient

Exercise notes for Financial Engineering.

Exercise 1

• what financial engineering is
• no-arbitrage principle and Law of One Price
• present values and discount factors
• no-arbitrage forward price
• payoff diagrams for forwards, calls, and puts
• put-call parity
• simple static replicating portfolio

Exercise 2 ipynb

• one-period binomial model
• risk-neutral probability
• replicating portfolio
• multi-period (CRR) binomial tree
• Cox-Ross-Rubinstein parameterization
• converges to Black-Scholes

Continued taking Large Scale AI lectures.

Lecture 3

Supercomputer lecture by CSCS

Lecture 4

Communication overhead

Lecture 5

Model & Data Parallelism

Lecture 6

Mixture of Expert Models