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Testing Option Price Homogeneity Under the Share Measure

Article Quant Q&A · Author: Frido
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Summary

The document asks whether degree-one homogeneity of call and put prices is preserved when changing from the risk-neutral measure to the share measure. It focuses on stochastic-volatility models and starts from the premise that, under the risk-neutral measure, the terminal-price density depends on the ratio of terminal to current price. The author proposes that the share-measure density is obtained by weighting that density by the terminal-to-current price ratio.

A change of variables in the option-pricing integral is then used to argue that scaling the current price and strike by the same positive factor scales the option value by that factor. This is a proposed derivation rather than a settled result: the document contains no replies or independent verification. Its conclusion depends on the density and measure-change assumptions stated, and it explicitly sets local-volatility models aside.

Key ideas

  • The question concerns whether degree-one option price homogeneity survives a change to the share measure.
  • The proposed derivation weights the risk-neutral density by the terminal-to-current price ratio.
  • A proportional rescaling of price and strike is shown in the proposed argument to scale the option value proportionally.
  • The derivation assumes the relevant density depends on the terminal-to-current price ratio and focuses on stochastic volatility.
  • The document presents a conjecture and calculation but no independent answer or verification.

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# Is homogeneity preserved under change of measure?


# 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?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.