No-Arbitrage Bounds for Normal Implied Volatility and Rate Options
Summary
The document asks whether no-arbitrage conditions are known for Bachelier, or normal, implied volatility surfaces. Normal volatility is defined through the parameter that reproduces an option price under the Bachelier formula. The author contrasts this with work on Black-Scholes implied volatility, where conditions are obtained by mapping volatility surfaces into option prices and checking those prices for arbitrage consistency. The note proposes a similar transformation from arbitrage-free call-price conditions to implied-volatility conditions, but does not carry out the derivation.
The response points to research on displaced-lognormal volatility skews that provides some no-arbitrage bounds. It suggests this framework may be more useful for interest rates when a lower bound on rates matters. However, the response does not provide the bounds themselves or show that they characterize normal implied volatility directly. The document therefore identifies a related direction rather than giving a complete set of normal-volatility conditions or empirical validation.
Key ideas
- Bachelier implied volatility is the parameter that reproduces an option price under the normal model.
- No-arbitrage conditions can be studied by translating price restrictions into restrictions on implied volatility.
- The document asks for such conditions but does not derive them.
- Displaced-lognormal volatility research is cited as a related source of bounds that may suit rate applications.
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Full text
# No arbitrage conditions for normal implied volatility # No arbitrage conditions for normal implied volatility usually the term implied volatility refers to Black-Scholes implied volatility (also Log-Normal volatility): it is defined as a quantity which when plugged in the Black-Scholes formula returns the right price. In an article by Roper (2010) certain conditions are given which guarantee that the Black-Scholes implied Volatility Surface is consistent with No Arbitrage. When working with interest rates it has become convenient to use Bachelier implied volatility (or normal implied volatility) instead of Black-Scholes implied volatility. The Bachelier implied volatility is the parameter which when plugged into to the Bachelier option pricing formula returns the correct price. I was wondering whether there is an article which characterizes No Arbitrage conditions for normal implied volatility. I think that such conditions would differ from the conditions given in Roper, since he uses the Black-Scholes formula to transform the Black-Scholes implied volatility surface to the option price surface and checks for No-Arbitrage there. If not, I think a simple approach to obtain such conditions would be to transform the No-Arbitrage conditions for (Call)-option prices into conditions for implied volatility. References Roper, M. (2010). Arbitrage free implied volatility surfaces. preprint. ## Answer by user34971 (score 2) https://quant.stackexchange.com/a/51270 There is the following paper by Roger Lee and Dan Wang on Displaced Lognormal Diffusion implied volatility. They give some no arbitrage bounds. It could be more useful for interest rates than normal volatilities as you may actually want to bound your rates from below. Lee & Wang, Displaced Lognormal Volatility Skews, 2009
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