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