## 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 used in finance to simulate the price of the underlying asset and then to value options. They are able to explain with a few additional parameters why the Black-Scholes implied […]

## 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 while the buyer of a European Option has the right to exercise the option at the maturity date only. We assume that there is no arbitrage and we place ourself […]

## 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. It consists of a discretization of both spatial (underlying price in our case) and time intervals to obtain a finite grid. We find solutions for a differential equation by approximating […]

## 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 we can recover the initial function from its frequency domain representation by applying the Fourier inversion theorem. Why is it Useful for the Pricing of Options? The characteristic function of […]

## 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 a second order Taylor expansion of the option value over a short time interval, we can decompose the change of the price of the option with first order greeks on […]

## 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 a mean-reverting Cox-Ingersoll Ross process, and it is correlated with the asset price. The Heston model has five unknown parameters:– ν0: the initial variance– κ: the speed of reversion– θ: […]

## 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 and Myron Scholes in 1973 and later refined by Robert Merton. The model assumes that the financial market has two assets, one risky asset such as a stock price and […]

## 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 model relating the option price to the hedging strategy consisting of buying or selling the underlying asset to eliminate the risk. It is well known that some of the assumptions […]

## 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 could be asked in an interview for a quant or a trading role, with possible answers. Save 10% on All Quant Next Courses with the Coupon Code: QuantNextBlog10 For students […]

## 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 for a large number of models such as Black-Scholes, Merton’s jump diffusion, Variance Gamma, Heston, … As detailed in the presentation below, when you differentiate both sides of the equation […]