OptionEdge : Advanced Option Pricing and Hedging Strategies

Preface

This repository stores my solutions to computational problems in financial modelings

Languages and technologies used

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

Topics

The repository contains four folders: A1~A4.

Each folder covers the following topics:

A1

• 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

A2

• 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

A3

• 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

A4

• 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