Testing Option-Implied Distributions by Repricing Calls
Summary
The document explores whether an option-implied risk-neutral distribution can be checked by integrating its density twice and comparing the reconstructed call prices with observed option prices. It grounds the idea in the Breeden-Litzenberger relationship, which links the curvature of call prices across strikes to the risk-neutral density at expiration. Because market option prices are available only at discrete strikes, the document notes that interpolation is needed to estimate a smooth price curve before differentiating.
A lognormal density is proposed as a theoretical example, with numerical cumulative integrations used to illustrate the reconstruction. The author recognizes that the simple double integration alone produces prices with the wrong monotonic direction for calls, and suggests adding a linear function of strike with constants to be determined. The document presents this as an exploratory question, not a validated procedure: it gives no worked calibration, boundary conditions, error analysis, or empirical comparison. Those choices would matter to whether reconstructed prices offer a meaningful diagnostic.
Key ideas
- Breeden-Litzenberger relates the second strike derivative of call prices to the risk-neutral density.
- Discrete option quotes require a smooth estimate of prices across strikes before numerical differentiation.
- Integrating a candidate density twice can produce a call-price curve for comparison with market prices.
- The integration constants and boundary conditions must be handled to recover the appropriate call-price shape.
- The proposed check is exploratory and has no empirical validation in the document.
Tags
Full text
# PDF of the lognormal distribution # Validating an option-implied risk-neutral distribution by integrating it twice and comparing the resulting "prices" with the original ones From Breeden-Litzenberger, we know that the second derivative of a European call option's price with respect to the strike price is equal to the risk-neutral probability density function of the underlying asset's price at the option's expiration. However, options_price(strike_price) is not a continuous function, so additional calculations are needed, such as interpolating in IV space and converting back to price space. One thing that I've not managed to find in the literature is analysis regarding the possibility of validating such a generated risk-neutral distribution by integrating it twice, and therefore, by Breeden-Litzenberger, arriving at prices-according-to-the-RND, and then comparing said prices with the original prices. Here's a theoretical example where the risk-neutral-distribution to be validated is just a log-normal distribution. The y-value of the resulting second numerical integration would be compared to the initial set of prices, and conclusions would be drawn: ``` # PDF of the lognormal distribution # In a real context, we would receive this from the PDF-generating model, # which takes in option prices and strike prices pdf_lognorm = lognorm.pdf(x_ln, sigma_ln, scale=scale_ln) # Numerical Integration of the PDF integral_pdf = np.cumsum(pdf_lognorm) * (x_ln[1] - x_ln[0]) # Second Integration of the PDF second_integral_pdf = np.cumsum(integral_pdf) * (x_ln[1] - x_ln[0]) ``` Would this be at least the beginning of a theoretically-valid validation method? Thank you. EDIT: Of course, the implementation of the double integration method above is incomplete. It would always generate a monotonically increasing set of option prices - but the options are calls, which ought to have monotonically decreasing prices. The correct calculation would be option_price(strike_price)=the_incorrect_double_integration_i_have_above(strike_price)+ C1*strike_price+C2, where C1 and C2 are constants to be determined.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.