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Homogeneity of Option Prices, What Does It Tell Us?
Category :
November 27, 2023
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
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