This repository stores my solutions to computational problems in financial modelings

Python 3.10 (Numpy, Pandas, Scipy, Seaborn, Matplotlib), Jupyter Notebook

The repository contains four folders: A1~A4.

Each folder covers the following topics:

• Binomial Lattice
  • Properties of Binomial Lattice
• Risk Neutral property
  • Risk neutral probability q*
• Fair value of european options
• Properties of standard Wiener Process (standard Brownian Motion)
  • Ito's Lemma

Non-programming questions; only hand-written answers

• Simulating drifting Binomial Lattice
  • Convergence rate of Binomial Lattice method
• Binomial Lattice with underlying with dividends (European Options)
  • Interpolation between missing underlying (stock) prices
• Monte Carlo simulation for European Options
  • Down-and-out call option
  • Time discretization error of MC simulation

• Numerical schemes for SDEs (Stochastic Differential Equations)
  • Euler-Maruyama method
  • Milstein Method
• Numerical schemes for correlated SDEs
  • Cholesky matrix and decomposition
• Delta hedging from a Binomial Lattice
  • European Butterfly Option
  • Interpolation between underlying (stock) prices
  • VaR (Value at Risk), cVaR (cumulative Value at Risk), and P&L (Profit and Loss) calculation from MC (monte carlo) simulation
• Trading simulation using monte carlo simulation
  • Use option value from Black-Scholes equation
  • Set various rebalancing strategy and evaluate the possibility of an arbitrage
• Jump Diffusion model
  • Double exponential jump process (probability distribution function is doubly exponential)
  • Add random jump process in the simulation
  • Compare observed implied volatility

• Finite Difference Method (FDM)
  • Apply fully-implicit, Crank-Nicolson, and CN-Rannacher method
  • Use sparse matrix to facilitate calculation (Scipy)
  • Convergence rate between the each method (linear / quadratic)
  • Plot graphs for delta and gamma of options (Are the gamma graph smooth?)
• FDM for American options, with variable timestepping
  • Apply fully-implicit, Crank-Nicolson, and CN-Rannacher method with variable timestepping
  • Is it never optimal to exercise early for American call options?
  • Plots of delta and gamma
• Model Calibration
  • Find implied volatility that explains the option price
  • Using simple feed forward neural network as the local volatility function (LVF)
  • Non-linear least square method using Jacobian Matrix, Levenberg-Marquardt method
  • Verify the model using 3d plot
• Optimal static hedging
  • Least squares optimization problems with conditions
  • Monte Carlo simulation to derive conditions
  • Lagrangian multipliers with Karush–Kuhn–Tucker (KKT) conditions
• cVaR and Convexity
  • Convexity test
  • cVaR optimization problem