# Blog

You will find here articles, guides, tutorials related to quantitative finance.

### 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

### 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

### 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

### 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

### 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

### 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.

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### Quant and Trading Interview Questions about Option Greeks and Risk Management

It is crucial to have a strong understanding of the main option Greeks for quant or trading positions. You will find below several questions you

### 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:

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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,