I have a finite difference pricing model and would like to factor in a volatility surface for each underlying equity. However, I have limited data. Essentially I'm just pulling a few implied volatilities from Bloomberg for each underlying equity. As such, I would like to postulate a functional form and simply fit it to my data. I was thinking of something like $$ \sigma(S_t/K, t) = \alpha + \beta \exp(-\gamma S_t/K) - \delta t. $$ I'm not familiar with any sort of theory for the functional form, so I don't know if this is the right approach.
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2$\begingroup$ Directly interpolating carries risk of arbitrage $\endgroup$– ArshdeepCommented Apr 2 at 22:24
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$\begingroup$ @Arshdeep Yeah, a functional form probably also invites arbitrage. However, they're probably both better than a constant volatility assumption. $\endgroup$– Charles0349Commented Apr 3 at 0:40
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1$\begingroup$ What about Gatheral's SVI arbitrage-free implied volatility parameterization? $\endgroup$– KevinCommented Apr 4 at 22:07
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$\begingroup$ @Kevin Maybe. I should read the paper. $\endgroup$– Charles0349Commented Apr 8 at 16:17
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