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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?

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  • $\begingroup$ 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? $\endgroup$
    – fes
    Commented Jun 29, 2023 at 8:32
  • $\begingroup$ @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. $\endgroup$
    – Frido
    Commented Jun 29, 2023 at 8:36
  • $\begingroup$ @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. $\endgroup$
    – Frido
    Commented Jun 29, 2023 at 8:37
  • $\begingroup$ @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) $\endgroup$
    – Frido
    Commented Jun 29, 2023 at 8:42
  • $\begingroup$ 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. $\endgroup$
    – fes
    Commented Jun 29, 2023 at 8:46

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