Positive Normal-Volatility Skew and Implied-Distribution Integration
Summary
The document describes an attempt to price from a market-implied distribution built from a linearly sloped normal-volatility smile. The author prices options with the Bachelier model across a strike grid, estimates the cumulative distribution from one-sided strike derivatives, then maps percentiles back to strikes for numerical integration. Interpolation changes did not resolve poor convergence when the positive smile slope became large.
The central concern is that an arbitrageable smile may produce invalid distribution properties, such as probabilities outside the admissible range, and thereby undermine the integration. The post asks whether a stated option-price integral identity holds despite problematic second derivatives and how far into the tails the integration should extend. It gives a concrete Excel workflow and observed convergence difficulty, but no answer or validated threshold; the chosen percentile range and strike spacing therefore remain unresolved methodological limits.
Key ideas
- A linear normal-volatility smile can be used to generate option prices across strikes with the Bachelier model.
- The author estimates cumulative probabilities from one-sided derivatives of call prices with respect to strike.
- A steep positive smile slope is associated with poor convergence in the described numerical procedure.
- Arbitrage violations can make the implied density or cumulative probabilities unsuitable as a probability distribution.
- The post leaves open how to select integration limits and whether interpolation can address the convergence problem.
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Full text
# basic numerical integration question related to case of high positive volatility skew # basic numerical integration question related to case of high positive volatility skew is the below equation true irrespective of if that 2nd derivative turns out to be negative or >1 , (ie even if theres an arbitrage) ? the reason i ask is that i am writing a single asset montecarlo (numerical integration) based on a market implied distribution. i want to see the affect of changing the vol smile slope on pricing. i am using linear normal vol smile. (i know that normal vols are arbitrageable). when i make slope more than 0.5 (change in vol per 1bp change in strike) , then my convergence to market price is very bad. What is the mathematical reason for that? I guess it may be because of probabilities outside 0-100% , but mathematically, i am not sure why thats not allowed, as all we are doing here really is a numeric integration , which is like reverse engineering. how do i know what limits to do my numerical integration - is it enough to go out to the 99.99% point? here is the equation i am talking about (copied from word, sorry about formatting) : $c_k=∫_k^∞(x-k) (d^2 c_k)/(dx^2 ) dx$ here are some more details of what i have done: i have done this all in excel. i could have used some VBA, but apparently, for this task , VBA would be slower than excel, perhaps because VBA is not 'native'. - i calc vol, simply at normal vol(K) = given atm vol + given slope * (K-F). 2, i use bachelier to get price. - i do this for small strike increments of 1bp apart for a big range (from delta% 0.01 - 99.99) - then i calc 1-sided dV/dK = CDF. - then i have a new table of %iles from 0.005 to 99.995 going up by 0.01% , where i lookup the CDF- and get value. - in step 5, i initially did with step function interp , then i did linear interp , that did not help at all
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