Estimating Risk-Neutral Probabilities from Option Price Differences

Summary

The document asks how to recover a risk-neutral distribution for an underlying asset from option prices using the Breeden-Litzenberger relationship. It describes estimating the probability density at a strike from the second derivative of call prices, approximated with a centered finite difference. This second difference corresponds to the value of a narrow butterfly spread, scaled by the square of the strike interval.

It then asks whether a first difference of call prices can estimate a cumulative tail probability and whether the approximation can be interpreted as a normalized spread of calls. The question highlights practical concerns for swaption data, including strike liquidity and the choice of strike spacing, and asks how to bound estimation error. No answer, empirical comparison, or error bound is provided. In application, discrete and noisy prices, discounting conventions, and the distinction between a tail probability and its complementary CDF all affect interpretation; the document leaves these points unresolved.

Key ideas

• The second strike derivative of call prices is used to infer a risk-neutral density.
• A centered second difference approximates that derivative and corresponds to a scaled butterfly spread.
• The author asks whether a first difference can estimate a cumulative tail probability.
• Strike spacing, liquidity, and approximation error are open practical concerns, especially for swaption prices.

Tags

# methodology confirmation for computing implied risk-neutral CDF from option prices


# methodology confirmation for computing implied risk-neutral CDF from option prices












In this question, the risk-neutral probability distribution $q(S_T=s)$ for the underlying at time $t = T$ is given by the Breeden-Litzenberger identity as:

$$ \frac{1}{P(0,T)} \frac{ \partial^2 C }{\partial K^2} (K=s,T) $$

In practice, this can be computed numerically for a given $K = s$ with a centered, second-order finite-difference approximation:

$$ \frac{ \partial^2 C }{\partial K^2} (K=s,T) \approx \frac{ C(K=s-\Delta K,T) - 2 C(K=s,T) + C(K=s+\Delta K,T)}{ (\Delta K)^2 } $$

i.e., the payoff for a butterfly spread around $K = s$ normalized for its width.

If instead we are interested in the CDF $Q(S_T \geq s)$, can we look at the first order finite-difference approximation:

$$ \frac{ \partial C }{\partial K} (K=s,T) \approx \frac{ C(K=s-\Delta K,T) - C(K=s+\Delta K,T)}{ (2 \Delta K) } $$

to determine the implied risk-neutral CDF? And similar to the above, is it appropriate to think of this as a normalized payer spread option (buying a call at $K = s + \Delta K$ and selling a call at $K = s - \Delta K$)?

In particular, if the underlying is a swap rate and the market data are swaption prices, what sort of issues (and remedies?) arise from this interpretation (i.e., potential liquidity constraints, or discretization complications with $\Delta K$)? Finally, is there a standard method for constructing a meaningful error bound on the cumulative probability implied by a certain strike?

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