Estimating Risk-Neutral Probabilities by Minimizing Relative Entropy

Summary

The document introduces a question about adjusting a historical stock-price distribution to obtain a risk-neutral distribution, motivated by strike-adjusted option spreads and skew. The question proposes a simple three-outcome price distribution and asks how to apply a minimum-relative-entropy approach, but does not provide a worked calculation or answer.

The central concept is to treat the historical distribution as a starting estimate and find a risk-neutral distribution that stays close to it under a relative-entropy measure while satisfying pricing constraints. Those constraints are essential: in a basic no-arbitrage setting, the adjusted probabilities must make discounted expected asset prices consistent with market prices, including the underlying and relevant traded claims. The example alone does not determine a unique adjustment because the current price, discounting assumptions, and constraints are unspecified. The document is therefore a useful statement of the problem, rather than a complete recipe or evidence that the method identifies options as cheap or expensive.

Key ideas

• The question concerns converting a historical price distribution into a risk-neutral distribution.
• Relative entropy is proposed as a way to keep the adjusted distribution close to the historical estimate.
• Risk-neutral probabilities must also satisfy market pricing constraints to be useful for option valuation.
• The example lacks the pricing inputs and constraints needed to calculate a unique adjusted distribution.

Tags

# Trying to understand Strike Adjusted Spread, can someone explain using a simple example?


# Trying to understand Strike Adjusted Spread, can someone explain using a simple example?












I should start by saying that I am not a quant, I am someone interested in options but I perhaps lack the mathematics background to always follow along. I recently stumbled upon a terrific article about trying to adjust for skew and basically trying to find which options were cheap/expensive on a skew adjusted basis: http://www.emanuelderman.com/media/strike_adjusted_spread.pdf

However, this where I got lost:

> Instead, we will estimate the current risk-neutral return distribution Q( ) for a stock from its historical distribution P( ) by assuming that the latter is a plausible estimate for the former, and then requiring that the relative entropy S(P,Q) between the distributions is minimized.

How does one do this? I tried reading in the appendix and had a bit of a hard time following and I'm hoping one of you guys could perhaps explain?

Let's say I have a very simple stock with the following distribution:

$110: 25% of the time

$100: 50% of the time

$90: 25% of the time

And that is the entirety of the stock's possible distribution of prices.

How would I go about coercing this "historical" probability distribution into a risk-neutral probability distribution? If the exact math is too complicated perhaps a truncated/intuitive guesstimation? That would work for me as well.

If alternatively someone could provide me with a place where i could go to learn/read more (a paper where they did a step by step example would be best I suppose).

I appreciate any and all help.

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