# Is homogeneity preserved under change of measure?

In a paper, Joshi proves that the call (or put) price function is homogeneous of degree 1 if the density of the terminal stock price is a function of $$S_T/S_t$$. In the paper I think Joshi is silently working under the $$\mathbb Q$$ measure. What happens to homogeneity, or equivalently the density, if we change measure to for example the share measure. Will it still be a function of $$S_T/S_t$$ (under the share measure)?

Recall that homogeneous of degree 1 means that $$\lambda C(S_t,K) = C(\lambda S_T, \lambda K)$$ where $$C$$ is the risk-neutral measure price and $$\lambda > 0$$. If the price of call option under the share measure is denoted by $$\tilde C(S_t,K)$$ then my question is does $$\lambda \tilde C(S_t,K) = \tilde C(\lambda S_T, \lambda K)$$ also hold?

Let's consider stochastic volatility models only, as local vol models are hardly if ever homogeneous.

EDIT: I think the answer is yes, but an extra pair of eyes looking at it won't hurt:

So let $$\Phi \left(\frac{S}{S_t} \right)$$ be the denisty of the (log) price under $$\mathbb Q$$. Then, following Joshi's proof (Theorem 2.1) in the paper, $$\tilde C(S_t,K) = \int (S - K)_+ \frac{S}{S_t} \Phi \left(\frac{S}{S_t} \right) \frac{dS}{S}$$ So \begin{align} \tilde C(\lambda S_t,\lambda K) &= \int (S - \lambda K)_+ \frac{S}{ \lambda S_t} \Phi \left(\frac{S}{\lambda S_t} \right) \frac{dS}{S} \\ &= \int (\lambda S' - \lambda K)_+ \frac{\lambda S'}{ \lambda S_t} \Phi \left(\frac{\lambda S'}{\lambda S_t} \right) \frac{d \lambda S'}{\lambda S'} \\ &= \int \lambda ( S' - K)_+ \frac{ S'}{ S_t} \Phi \left(\frac{ S'}{ S_t} \right) \frac{d S'}{S'} \\ &= \lambda \tilde C(S_t,K) \end{align} where the change of variable $$S = \lambda S'$$ has been performed.

Is this correct? Did I miss something?

• I find the question a bit confusing. Homogeneity is a property of functions. Like the BS option price function is homogeneous of degree one. What would measure change mean in this context?
– fes
Jun 29 at 8:32
• @fes For a while I thought homogeneity was only a property of the payoff until I read Joshi's paper some years ago where he shows that homogeneity is also determined by the form of the density. So for example the payoff of a call / put option is homogeneous, but it doesn't mean that the price function is homogeneous: for example under Bachelier dynamics it isn't homegeneous of degree 1 as defined above, but it is homogeneous of degree 1 under Black-Scholes dynamics. Jun 29 at 8:36
• @fes continued: so yes, the BS call price formula is homogeneous always. But it doesn't mean the model/market price is homogeneous even though you equate the two (via implied vol). So I was wondering if the SV model price of a call / put option price under the share measure retains homogeneity given that the SV model price under the pricing measure is homogeneous. Jun 29 at 8:37
• @fes One more thing, the BS call price formula in the presence of skew is actually not always homogeneous: it is only homogeneous of degree 1 if the IV is homogeneous of degree 0 (i.e. IV only depends on moneyness, and possibly instantaneous vol) Jun 29 at 8:42
• I know that option prices depend also on the stochastic process and hence this affects the homogeneity properties of the resulting price functions. But I still don't understand how a function could be homogenous under some measure and not under some or what this measure change even means in this context.
– fes
Jun 29 at 8:46