# Going from Stochastic Discount Factor / Risk Neutral Density -> Hedge Ratio

Assuming a probability distribution function is known in its entirety, what methods are available to construct a hedge ratio?

For guidance, I went to the canonical Empirical Pricing Kernels and found this section

3.4. Hedge ratio specification

We develop a hedge ratio estimation technique that does not depend on a particular specification of the state probability density or the pricing kernel.

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Therefore, we consider three possible one-day underlying price changes. The stock price could rise by one standard deviation to $S_t + \varepsilon$ remain constant at $S_t$; or fall by one standard deviation to $S_t + \varepsilon$. Each underlying price change results in a different date $t + 1$ put option price: $Put_{t+1|S_t+ \varepsilon}$, $Put_{t+1|S_t}$ or $Put_{t+1|S_t- \varepsilon}$.

This is using the EPK standard deviation when calculating the hedging units. I can convert from this EPK to the RND, so I believe this method actually does depend on the pricing kernel.

What methods are available using the risk-neutral density directly to produce a hedge ratio?

• Hi Jared. It is not really clear to me what you are asking. Any risk-neutral measure is constructed on the principle of a self-financing replicating portfolio (i.e. a hedging strategy) based on the traded assets of the economy under that model and this models' state variables (that can sometimes be directly mapped to the spot price of the traded assets in which case the model is said complete). – Quantuple Jul 11 '18 at 7:40