Why Positive Local Volatility Models Produce Arbitrage-Free Option Prices
Summary
The document asks whether linearly interpolating local volatility across strikes preserves no-arbitrage conditions, given that call prices must be convex in strike and implied volatility cannot in general be interpolated linearly without care. The answer distinguishes local volatility as a model input from call prices or implied volatilities as market quote surfaces.
It states that a positive local volatility surface specifies a consistent Dupire model and, by construction, generates mutually consistent arbitrage-free prices. By contrast, a surface of call prices or implied volatilities does not itself define a unique arbitrage-free model; price surfaces must satisfy conditions such as being above intrinsic value, decreasing and convex in strike, and increasing with maturity. The explanation is conceptual and assumes positive local volatilities and a valid model setup. It does not give interpolation procedures, numerical checks, or a proof of how a particular interpolation behaves at boundaries or across maturities.
Key ideas
- A positive local volatility surface specifies a Dupire model that produces consistent, arbitrage-free prices.
- A call-price or implied-volatility surface does not automatically define an arbitrage-free model.
- Call prices must satisfy strike and maturity consistency conditions, including convexity in strike.
- The explanation assumes positive local volatility and does not provide a practical interpolation algorithm.
Tags
Full text
# Linear interpolation of local vol no arbitrage
# Linear interpolation of local vol no arbitrage
We already know the equivalence between `local vol`, `implied vol` and `option price` and there are one-one maps between pairs: $$(\sigma_{local},K),\ (\sigma_{implid},K),\ (C,K)$$ here $K$ is the strike and $C$ is the call price. And the one-one maps are determined by Black-Scholes formula.
And we know that one of the no-arbitrage condition is that call price $C$ should be a convex function against the strike $K.$
So, given some market equivalent sample points $$(\sigma^i_{local},K^i),\ (\sigma^i_{implid},K^i),\ (C^i,K^i)$$ assume they meet the no arbitrage condition as precondition, then we know that linear interpolation of implied vol $\sigma^i_{implid}$ is not allowed.
My question is whether linear interpolation of local vol $\sigma^i_{local}$ is allowed? Since linear interpolation of market price $C^i$ seems allowed.
If linear interpolation of local vol $\sigma^i_{local}$ is allowed, what's the reason? It seems related to the convexity of the maps between $(\sigma^i_{local},K^i)\ (C^i,K^i)$ and $(\sigma^i_{implid},K^i),\ (C^i,K^i).$
## Answer by Antoine Savine (score 2)
https://quant.stackexchange.com/a/43248
Any specification of the local volatility surface produces non-arbitrageable prices (as long as it is guaranteed that resulting local volatilities are always positive). The reason is that any (positive) lv surface specifies a unique, consistent, arbitrage-free (Dupire) model, which, by construction, produces mutually consistent, arbitrage-free prices. By contrast, the surface of call prices (or equivalently of implied volatilities) does not constitute the parameter set of a unique, arbitrage-free model, therefore any (positive) surface of prices or ivs is not guaranteed arbitrage-free. It is well known that call prices must be above intrinsic, increasing in maturity, decreasing and convex in strike, which translates in non-trivial ways in terms of iv. May I refer to my volatility lectures for further details? https://www.slideshare.net/AntoineSavine/lecture-notes-from-volatility-modelling-lectures-at-copenhagen-universityShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.