Deterministic Strike Volatility Curves for Option Pricing
Summary
The document asks whether an option model can represent volatility as a deterministic function of strike alone. It distinguishes this idea from local volatility, which the author understands as deriving a volatility surface from market prices with dependence on both the underlying price and time. For a single expiration, the proposed alternative is to fit an implied-volatility curve to quoted options and use interpolation or extrapolation to estimate volatility at unquoted strikes, then price those options with Black–Scholes.
The text poses the model-name and literature questions but provides no answer, evidence, or recommended curve-fitting method. It mentions cubic splines only as an example and raises hedging errors and model strengths and weaknesses as topics to investigate. A strike curve fitted to one maturity is limited to that expiry, and extrapolated prices may be unreliable; the document does not discuss arbitrage constraints or how to enforce them.
Key ideas
- The proposed model fits implied volatility as a function of strike for a single expiration.
- Interpolated or extrapolated volatility could be used to price options at unquoted strikes.
- The author contrasts this strike curve with local volatility, which also depends on time and underlying price.
- The document asks about curve choice and hedging but does not answer those questions or address arbitrage constraints.
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Full text
# Is there an arbitrage free option model that treats volatility as a deterministic function of strike? # Is there an arbitrage free option model that treats volatility as a deterministic function of strike? I am trying to get a good understanding of the different models out there, and thus be able to study hedging errors, and strengths and weaknesses. My understanding of the Local Volatility model in layman's terms is the following: > It requires market prices, from which you will derive a deterministic function for volatility. Volatility is dependent on underlying price and time. You basically derive a surface of implied volatilities by fitting to the market with whatever available option prices there are. Then, in order to price options with strikes/expirations for which the market does not have, you use the surface, and plug that implied volatility into the Black Scholes equation to derive an option price. Is this correct? If the above is correct, I was wondering if there was an even simpler model where volatility is a deterministic function of underlying price only. Such a model can only be used to price a single expiration of options. It would take current market option prices, fit an implied volatility curve to those prices (for example fit using a cubic spline). For strikes without market prices, it would use the implied volatility interpolated/extrapolated by the curve and return an option price. If so, what is this model called, and where can I find more literature on what curves should be used to fit option prices (cubic spline, parabola, etc...), hedging issues, etc... Thanks!
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