Finance

Chapter 7: Expectation and Variance (discrete)

Expectation
Variance: $\mathrm{Var}(X)=\mathbb{E}[(X-m)^2],\qquad m=\mathbb{E}[X].$

Chapter 8: Continuous distributions

Uniform distribution: $X\sim\mathcal{U}([a,b])$
Exponential distribution: $f(x)=\lambda e^{-\lambda x}$
Normal distribuiton: $f_{m,\sigma}(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp\!\left(-\frac{(x-m)^2}{2\sigma^2}\right).$

Chapter 9: Joints, Marginal and Asymptotics

Joint density: X,Y random variables and double integral for probability

Marginal density: We integrate over one variable so only the other is left

Independence: Joint heights factor into product of marginal heights

Chapter 10: Statistics

MLE
Hypothesis testing
Confidence intervals

Recap

17.08 finished recap and went through exam

Started recapping probability theory and statistics which I had in 4th semester of ETH.

Chapter 1: Probability spaces and Events

Chapter 2: Laplace Model and counting

Laplace means probability have same chance so problem turns into occurences / total set

Chapter 3 Conditional Probability & Bayes

Conditional Probability: $\mathbb{P}[A\mid B]=\frac{\mathbb{P}[A\cap B]}{\mathbb{P}[B]}.$
Law of total probability: $\mathbb{P}[A]=\sum_{i=1}^n \mathbb{P}[A\mid B_i]\,\mathbb{P}[B_i]$ when B are partition of $\Omega$.
Bayes: $\mathbb{P}[B_i\mid A]=\frac{\mathbb{P}[A\mid B_i]\,\mathbb{P}[B_i]}{\sum_{j=1}^n \mathbb{P}[A\mid B_j]\,\mathbb{P}[B_j]}.$ Given the outcome what is probability of hypothesis.

Chapter 4 Independence

$A$ and $B$ are independent when $\mathbb{P}[A\cap B]=\mathbb{P}[A]\,\mathbb{P}[B].$
Multiple events: Every subcollection needs to be independent
Random Variables: independent and identically distributed

Chapter 5 Random Variables

Measurable map, grouping of events with different payoff (function).

Continous:

• CDF (cumulative distribution function)
• PDF (probabiliry density function)

Discrete:

• CDF
• PMF (probabiity mass function)

Chapter 6 Discrete distributions

Bernouille: one yes/no trial
Binomial: n independent Bernouille trails
Geometric: Bernoille untill first success
Poisson: Approximation of Binomial for rare events, large n small p, $\lambda=np$, $\mathbb{P}[N=k]=e^{-\lambda}\frac{\lambda^k}{k!},\qquad k=0,1,2,\ldots.$

Level 9 Homework

Group A

Implementation of European Option Pricing with Black-Scholes Model
Put Call Parity relationships
Matrix Pricer to price multiple options
Option Sensitivities aka Greeks

Parameters

• $T$ expiry, $K$ strike, $\sigma$ volatility, $r$ risk-free rate, $S$ spot
• $b$ cost-of-carry: $b = r$ (stock), $b = r - q$ (dividend), $b = 0$ (futures), $b = r - R$ (FX)
• $N(x)$ standard normal CDF, $n(x)$ standard normal PDF

Black–Scholes (European call / put)

Put–call parity

(so $P = C + K e^{-rT} - S$, and $C = P + S - K e^{-rT}$)

Greeks (exact)

Greeks (divided differences)

Group B

Perpetual American Option Pricing

Perpetual American call

Perpetual American put

Group C Monte Carlo Simulation

GBM paths via Euler–Maruyama; European call/put by averaging discounted payoffs; sweep NT / NSim on Batches 1, 2, 4.

MC pipeline

1. Simulate a path of the underlying.
2. Get stock price at $t = T$.
3. Payoff at $T$; repeat $M$ times (NSim).
4. Average the payoffs.
5. Discount to $t = 0$.

Euler–Maruyama step

($k = T/N$, $N$ = time steps)

Takeaways

• NSim cuts statistical noise; NT cuts Euler time-discretization bias — both needed.
• Coarse mesh (N=10) leaves a bias floor even with millions of paths.
• Short maturity: NSim often matters more than N; long $T=30$ needs much larger NT and NSim.
• Realistic match to exact BS is order $10^{-3}$, not exact equality.

Group D Advanced Monte Carlo

Added generic SD / SE on the vector of discounted path payoffs (Statistics.hpp), wired into TestMC / RunMC; reported SD/SE vs NT / NSim on Batches 1–2.

($M$ = NSim)

Takeaways

• SD ≈ payoff dispersion for that option; stays roughly flat when N or NSim change.
• SE shrinks like $1/\sqrt{\mathrm{NSim}}$; NT barely affects SE.
• Once N is large enough, absolute error vs exact is often on the order of SE (MC noise, not Euler bias).
• Larger SD (e.g. Batch 2 call) → larger SE at the same NSim → harder to match exact.

Group E Excel Visualization

Ran Duffy Excel demos (TestSingleCurve, TestTwoCurve, TestMultiCurve) on Windows via Parallels + Office 365 (path / #import / Boost / IntDir fixes for Mac→VM). Then TestExactMeshExcel: mesh of $S$, exact BS prices from Group A into a vector, charted in Excel.

Takeaways

• Excel Automation (COM) is Windows-only.
• Pattern: mesh in $S$ → price each spot → send vectors to ExcelDriver → chart.

Group F Finite Difference Methods

Explicit Euler on the Black–Scholes PDE (European put); TestBSPDE1 + Excel. Batches 1–4 with mesh $S_{\max}=2K$, $J=2K$, $N=9999$.

Main pieces

• ParabolicPDE.hpp — IBVP callbacks ($\sigma$, $\mu$, $b$, $f$, BCL, BCR, IC)
• Mesher.hpp — uniform grids in $S$ and $t$
• FDM.hpp — one explicit Euler time step
• FDMDirector.hpp — meshes + IC + time loop to $T$
• Excel helpers — plot the solution

Takeaways

• Batches 1–3: FDM puts within ~$10^{-3}$ of exact BS puts.
• Batch 4 ($T=30$): same $N$ → explicit Euler unstable (time step too large vs domain); need larger $N$ (or a stabler scheme) for a useful price.

Level 9:

Option Pricing and exact solutions

• exact solutions for perpetual American options
• Affine interest rate models (interest rate derivatives)
  • Stochastic differential equations (with brownian motion)

get it working:

• Clewlow
• Strickland
  get it right:
• adaptable
• Kienitz/Duffy
  get it optimized:
• Monte Carlo Engines

Monte Carlo Simulations

• easy to program and always get an answer, we simulate many paths and take the average

FDA (Finite Difference Approximation)

• Euler (Explicite and Implicite)
• Milstein (Ito)
• Runge-Kutta

Finite Difference Methods

Lattice methods

• Binomial methods

Lecture 1

• Introduction to ODE
• First-order ODE
• Separable ODE
• First-order linear ODE
• Default probabilities and hazard rate
• Exact ODEs

Interview:
Sanity check at the end is useful, you can tell the interviewer that you will do a sanity check

Did more but not going to put it here for now, as I already have it on my edu site

2 Weeks of IMC Prosperity 4 Challenge

Analyse the Problem & Data clearly first

• Bot behaviour

• Price movements etc

Generate clear logic first, then a plan on how to implement it and only then generate the program

Pay attention to small details

• Sometimes how they scale, or had to adjust challenges might reveal information on the data

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

• In investing you need to be able to endure. Follow your own strategy, endure others earning more than you, earning while you aren't

• A strategy that works for one is best (everyone can have their own strategy), but they need to follow it and not get emotional

• GuoWen strategy is to find a good business and look at valuation, he'll only buy if it is low enough for him

• He was "lucky" to have chosen the small PE firm instead of going to the big one (foreign investments came)

• It is important who your teacher is, which project is happening, more than just a small employee in the biggest company

• A company with only money is usually not touched by investors, as it is hard to find a good business

• USA could be a good place to learn, but it is like a powder keg

• I could get into investment banking in Hong Kong and then PE, instead of the us