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: the long-term mean of the variance
– ξ: the volatility
– ⍴: the correlation of the two Wiener processes or spot / vol correlation

The implied volatility surface is not flat under the Heston model, and the five parameters have different impacts on the shape of the volatility surface.

The initial variance ν0 and the long-term mean of the variance θ control the level of the implied volatility curve. They control the second moment, the variance, of the underlying asset return distribution implied by option prices.

The spot / vol correlation ⍴ controls the slope, the skew of the implied volatility curve. It controls the third moment, the skewness, of the underlying asset return distribution implied from option prices

The vol of vol ξ controls the smile of the implied volatility curve.
It controls the fourth moment, the kurtosis, of the return distribution implied from option prices.

The term structure of the expected annualised variance is a function of the initial variance ν0, the long-term mean of the variance θ, and the speed of mean-reversion κ.
When the long-term variance is lower (resp. higher) than the initial one, the curve is downward (resp. upward) sloping.
When we increase the speed of reversion, it increases the slope on the front-end of the curve as the volatility will converge to its long-term average in a faster way.

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List of Quant Next Courses

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.

A la carte:

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).

Finite Difference Methods for Option Pricing: numerical solving of the Black-Scholes equation, focus on the three main methods: explicit, implicit and Crank-Nicolson.

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.