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 with respect to λ you can show quickly that in the Black-Scholes framework:
1) N(d1) is nothing else than the delta of the option.
2) There is a direct relationship between the risk-neutral density of the asset price and the second derivative of the call price with respect to the strike price. This is a special case of the Breeden-Litzenberger formula which is true not only in the Black-Scholes model.

Presentation

Save 25% on All Quant Next Courses with the Coupon Code: QuantNextBlog25

For students and graduates: We offer a 50% discount on all courses, please contact us if you are interested: contact@quant-next.com

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.