Do Implied Volatility Market Models Preserve No-Arbitrage Conditions?

Summary

The document raises a modeling question about describing an option implied-volatility surface directly, in the style of Heath-Jarrow-Morton models, rather than modeling instantaneous volatility or forward variance swap prices. It refers to work by Ledoit and Santa-Clara, which derives a partial differential equation for implied volatilities from the option-price equation, and notes that later implied-volatility market models follow a related approach.

The central issue is whether deriving the implied-volatility equation within an arbitrage-free option-pricing framework guarantees that its solutions remain arbitrage-free over time. In particular, the author asks whether conditions such as Lee’s restrictions on implied-volatility behavior across strikes hold not just initially but at future dates as well. The document poses this as an open question and supplies no derivation, answer, or empirical evidence. It therefore motivates scrutiny of the link between an arbitrage-free price process and the constraints required of its implied-volatility representation; it does not establish that the PDE alone preserves those constraints.

Key ideas

• Some models describe the implied-volatility surface directly instead of modeling instantaneous volatility or forward variance.
• The cited approach derives a PDE for implied volatilities from the option-price equation.
• The question is whether an arbitrage-free price framework ensures future implied-volatility surfaces remain arbitrage-free.
• Lee-type restrictions are raised as an example of conditions that might need to hold through time.
• The document poses the issue but provides no answer or evidence resolving it.

Tags

# Market models of implied volatility and no arbitrage


# Market models of implied volatility and no arbitrage












Something has been bugging me for a while, and I can't really find an answer to it in papers. Maybe somebody can help me out.

In addition to modelling the instantaneous vol, or modelling forward variance swap prices, in order to explain the implied volatility surface some have tried to model implied volatility directly a la HJM.

One of the first to do this is afaik Ledoit and Santa Clara and they derive a complicated PDE that must be satisfied by all the implied vols. Other and subsequent market models for implied volatility, such as Schonbucher's, are in the spirit of Ledoit and Santa-Clara.

Now here is my question:

Since the PDE for the implied volatilities are derived from the option prices PDE, and as the latter is under an arbitrage free framework, does this mean that the IVs which are solutions to the PDE for the IVs arbitrage free as well, specifically will they satisfy for instance Lee's no arbitrage conditions not only today but also in the future?

I would think so, but is this really the case?

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