2
$\begingroup$

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?

Improve this question
$\endgroup$
13
  • $\begingroup$ 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
    Jun 29 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
    Jun 29 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
    Jun 29 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
    Jun 29 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
    Jun 29 at 8:46

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.