Weekly exercises of the Stochastic Methods for Finance course A.Y. 2021/2022.
Prof. Martino Grasselli
The assignment consists in the task of pricing a call option with a Binomial Recursive Tree Model, for two different underlying assets (GOOG and BNTX), using historical data of prices found on Yahoo Finance, to estimate a call option with maturity 3 months.
The results are reported both in Excel and in a Jupyter-Notebook written in Julia 1.7.0.
In the excel I used a Binomial Tree with 1 step and one of 30 steps, while in Julia I wrote down a function which allows to estimate the prices for a arbitrary number of steps, which allows to appreciate the convergence for the number of steps that goes to infinite.
The assignment consists in the task of pricing the dividend, first computing the discount rate using a Box-Spread strategy, and then through the pricing of a forward contract, pricing the dividend.
The results are reported both in Excel and in a Jupyter-Notebook written in Julia 1.7.0.
In the excel I computed the results for 1 month maturity, while in Julia I wrote down a function which allows to estimate the discount rate and the dividend for each maturity.
The assignment consists in the task of prooving the convergence of the Binomial Recursive Tree model expectation for the Call option price to the Black-Scholes formula.
The results are reported both in Excel and in a Jupyter-Notebook written in Julia 1.7.0. In particular, while for the Binomial Model I used the function already used in W1.Binomial-Model, for the Black-Scholes price I used the function blsprice of FinancialToolBox.
The assignment consists in the task of computing the greeks surfaces for a call option and verify the presence of the volatility smile.
The results are reported in a Jupyter-Notebook written in Julia 1.7.0. The Data are downloaded using the python package 'yfinance'. The computations are done using FinancialToolBox.
The assignment consists in the task of computing the Value at Risk with several methods:
- Parametric method
- Parametric method with EWMA volatility
- Monte Carlo simulation
- Historical method
- Historical simulation
The results are reported in a Jupyter-Notebook framework written in Julia 1.7.2.
The assignment consists in the task of generating a number of simulations of a Geometric Brownian Motion and, using them, computing the price of several types of options, such as:
- Vanilla Options
- Asian Options with fixed strike
- Asian Options with floating strike
- Lookback Options
- Barrier Options up-and-in
- Barrier Options up-and-out
- Barrier Options down-and-in
- Barrier Options down-and-out
- Double Barrier Options
The results are reported in a Jupyter-Notebook framework written in Julia 1.7.2. We used both user provided function and the package FinancialMonteCarlo.jl.