The Poisson Process

The Poisson process N(t) is a counting process used to describe the occurrence of events in a time interval of length t. It satisfies the following conditions:

1. N(0) = 0
2.  Increments are independents
3. The number of occurrences in any interval of length t follows a Poisson(λ.t) distribution with λ>0 a fixed parameter

The waiting time between two consecutive events occurring in a Poisson process is exponentially distributed with parameter λ.

The average waiting time between two consecutive events is 1 / λ.

The Python code is available below.

Presentation

To Go Further

The Merton Jump Diffusion Model (link)

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We summarize below quantitative finance training courses proposed by Quant Next. Courses are 100% digital, they are composed of many videos, quizzes, applications and tutorials in Python.

Complete training program:

Options, Pricing, and Risk Management Part I: introduction to derivatives, arbitrage free pricing, Black-Scholes model, option Greeks and risk management.

Options, Pricing, and Risk Management Part II: numerical methods for option pricing (Monte Carlo simulations, finite difference methods), replication and risk management of exotic options.

Options, Pricing, and Risk Management Part III:  modelling of the volatility surface,  parametric models with a focus on the SVI model, and stochastic volatility models with a focus on the Heston and the SABR models.

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Monte Carlo Simulations for Option Pricing: introduction to Monte Carlo simulations, applications to price options, methods to accelerate computation speed (quasi-Monte Carlo, variance reduction, code optimisation).

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Replication and Risk Management of Exotic Options: dynamic and static replication methods of exotic options with several concrete examples.

Volatility Surface Parameterization: the SVI Model: introduction on the modelling of the volatility surface implied by option prices, focus on the parametric methods, and particularly on the Stochastic Volatility Inspired (SVI) model and some of its extensions.

The SABR Model: deep dive on on the SABR (Stochastic Alpha Beta Rho) model, one popular stochastic volatility model developed to model the dynamic of the forward price and to price options.

The Heston Model for Option Pricing: deep dive on the Heston model, one of the most popular stochastic volatility model for the pricing of options.

Python Code

import numpy as np
plt.style.use('ggplot')
import matplotlib.pyplot as plt

#Simulation of a Poisson process
def poisson_process(rate, time):

    t = 0
    event_count = 0
    event_times = []
    event_times.append(0)
    while t < time:
        t = t + np.random.exponential(1/rate)
        if t < time:
            event_count += 1
            event_times.append(t)

    return event_count, event_times

#Plot Poisson Process
def plot_poisson_process(rate, time):
    event_count, event_times = poisson_process(rate, time)

    plt.figure(figsize=(10, 5))
    plt.step(event_times, range(0, event_count + 1), where='post')
    plt.xlabel('Time')
    plt.ylabel('Number of Occurrences')
    plt.title('Poisson Process Simulation')
    plt.grid(True)
    plt.show()

# Example:
rate = 1  # Rate of the Poisson process
time = 25  # Time
plot_poisson_process(rate, time)