Volatility & Derivatives
Binary Options: Pricing, Replication and Skew Sensitivity
The final payoff of a binary (or digital) call option is either one if the asset price is above the strike price at the expiry
November 10, 2024
Blog
You will find here articles, guides, tutorials related to quantitative finance.
Credit Risk Modelling and Management
Pricing of a Defaultable Bond with a Reduced-Form Model (Part I & II)
In this post, we will see how to price a risky bond with a reduced-form model for default risk. We assume in this first part
October 21, 2024
Volatility & Derivatives
Introduction to Stochastic Volatility Models
We will introduce in this post stochastic volatility models. They assume that the asset price but also its variance follow stochastic processes. Such models are
August 29, 2024
Credit Risk Modelling and Management
An Introduction to Reduced-Form Credit Risk Models
There are two main families of default models. Structural models, such as the Merton model based on the firm’s assets and liabilities and reduced-form models
July 22, 2024
Probability & Statistics
Quasi-Monte Carlo Methods
Monte Carlo is a very flexible numerical method which can model and price complex instruments when other methods can not. But it has the strong
July 16, 2024
Volatility & Derivatives
American Option Pricing with Binomial Tree
The buyer of an American Option has the right to exercise the option at any time before and including the maturity date of the option
July 2, 2024
Credit Risk Modelling and Management
Credit Risk Modelling: the Default Time Distribution
We will focus here on the default time distribution. We will see the relationship between the cumulative and the marginal default rates and how these
June 24, 2024
Volatility & Derivatives
Introduction to Finite Difference Methods for Option Pricing
Finite-difference methods is the generic term for a large number of techniques that can be used for solving differential equations, approximating derivatives with finite differences.
June 18, 2024
Credit Risk Modelling and Management
Credit Risk Modelling: the Probability of Default
We will focus here on the probability of default, one of the key measure of credit risk, introducing different ways to measure it. What is
June 10, 2024
Volatility & Derivatives
Fourier Transform and Applications for Option Pricing
The Fourier Transform The spectral decomposition of a period function via Fourier series can be generalised to any integrale function via the Fourier transform. And
June 3, 2024
Credit Risk Modelling and Management
Credit Risk: an Introduction
We will give an introduction to credit risk, presenting the main types of credit risk, the key components and measures of credit risk, discussing the
May 29, 2024
Volatility & Derivatives
Why is it Key to Understand Vanna and Volga Risks?
P&L Attribution & Greeks: Vanna and Volga Risks Greeks are used to understand and manage the different dimensions of risk involved when trading options. With
May 12, 2024
Volatility & Derivatives
How the Heston Parameters Control the Implied Volatility Surface
In the Heston model, both the dynamic of the asset price and its instantaneous variance nu are stochastic. The model assumes that the variance follows
May 10, 2024
Volatility & Derivatives
From Black-Scholes to Heat Equation
The Black-Scholes Model The Black-Scholes model is a pricing model used to determine the theoretical price of options contracts. It was developed by Fischer Black
April 27, 2024
Volatility & Derivatives
Why Does the Black-Scholes Model Remain so Popular?
The Black-Scholes model was developed by Fischer Black and Myron Scholes in 1973 and later refined by Robert Merton. It was the first arbitrage free
April 23, 2024
Statistics & Machine Learning
Principal Component Analysis in Finance
The Principal Component Analysis or PCA is a statistical technique for reducing the dimension of a large dataset. It does it by transforming the dataset
April 16, 2024
Test Your Knowledge
Quant and Trading Interview Questions about Option Greeks and Risk Management
It is crucial for quant or trading positions to have a strong understanding of the main option Greeks and how they can be used for
April 15, 2024
Probability & Statistics
The Merton Jump Diffusion Model
Merton Jump Diffusion Model The Merton Jump Diffusion model proposed by Merton in 1976 is an extension of the Black-Scholes model (link). It contains: μ:
April 3, 2024
Probability & Statistics
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
March 11, 2024
Probability & Statistics
The Cox-Ingersoll-Ross (CIR) Model
The Cox-Ingersoll-Ross (CIR) model is a stochastic interest rate model used in finance to describe the evolution of interest rates. The model was introduced in
February 24, 2024
Probability & Statistics
Calibration of the Vasicek Model to Historical Data with Python Code
We present here two methods for calibrating the Vasicek model (link) to historical data: The Python code is available below. Presentation Save 10% on All
January 29, 2024
Probability & Statistics
The Vasicek Model
The Vasicek model is a mathematical model used in finance to describe the movement of interest rates over time. It was developed by Oldrich Vasicek
January 16, 2024
Probability & Statistics
The Brownian Motion: an Introduction
The Brownian motion is one of the most famous and important stochastic process. Let’s start with a bit of history. Robert Brown discovered the Brownian
December 11, 2023
Volatility & Derivatives
Homogeneity of Option Prices, What Does It Tell Us?
The option price C(S0, K, T) is homogeneous in order 1 in (S0, K) when: C(λ.S0, λ.K, T) = λ.C(S0, K, T) This is true
November 27, 2023
Test Your Knowledge
Test Your Knowledge: Options, Pricing and Risk Management (Series 1)
You will find in this document the results of a series of questions posted on LinkedIn to test your knowledge on Options, Pricing and Risk
November 13, 2023
Volatility & Derivatives
Options, Greeks and P&L Decomposition (Part 3)
In previous articles (see Options, Greeks and P&L Decomposition (Part 1) and Options, Greeks and P&L Decomposition (Part 2)) we decomposed the P&L of simple
August 28, 2023
Volatility & Derivatives
Implied Volatility Calculation with Newton-Raphson Algorithm
In this article, we will present the Newton-Raphson method for calculating the implied volatility from option prices. Black-Scholes Price vs Volatility In Black-Scholes model, the
July 3, 2023
Statistics & Machine Learning
Artificial Neural Network for Option Pricing with Python Code
The valuation and the risk management of options can be quickly complex. It depends on the option’s feature and the pricing model. There is often
June 20, 2023
Volatility & Derivatives
Options, Greeks and P&L Decomposition (Part 2)
In a previous article (see Options, Greeks and P&L Decomposition (Part 1)) we analysed the decomposition of the P&L of an option strategy in a
May 16, 2023
Volatility & Derivatives
The Black-Scholes Model
The Black-Scholes model is a pricing model used to determine the theoretical price of options contracts. It was developed by Fischer Black and Myron Scholes
April 26, 2023
Volatility & Derivatives
Option Greeks and P&L Decomposition (Part 1)
Understanding option sensitivities and greeks is crucial to be successful in trading and risk management of options. In this post we will see how to
April 17, 2023
Statistics & Machine Learning
Principal Component Analysis – An Introduction
What is it? How does it work? Principal Component Analysis (PCA) is a statistical method for reducing the dimension of a dataset. It is a
March 28, 2023
Volatility & Derivatives
Why Does Volatility Smile and Smirk?
The Volatility is Constant in the Black-Scholes Model… In the Black-Scholes model, the volatility of a stock price is assumed to be constant, independent of
March 21, 2023
Probability & Statistics
Introduction to Stochastic Calculus
Foundations of Stochastic Calculus Stochastic Calculus is a branch of mathematics that deals with random processes. Beyond probabilities it also has links with differential equations,
March 14, 2023
