Directly Fitting a Smooth Option Premium Surface
Summary
The document considers how to interpolate sparse, noisy market premiums for out-of-the-money puts and calls without relying on a complex pricing model. It proposes fitting a parameterized skewed Student-t distribution to market option prices, with the distribution’s parameters varying across maturity under monotonicity and curvature constraints. The fitted distribution can then generate premiums through expected option payoffs, but requires numerical integration for each price and entails a number of parameters to tune.
As an alternative, the author asks whether a functional form can be fitted directly to the premium surface, avoiding the distribution fit and repeated integration. The current baseline is two-dimensional linear interpolation, and no direct surface formula, fitted results, or comparison is supplied. The document is therefore a modeling proposal and open question, not a validated interpolation method. It also limits its stated scope to out-of-the-money options and does not discuss arbitrage constraints or performance on held-out quotes.
Key ideas
- The goal is to interpolate noisy option premiums across strike and maturity.
- A skewed Student-t distribution with maturity-dependent parameters can generate option prices through expected payoffs.
- Fitting the distribution involves constrained parameters and numerical integration.
- The author seeks a direct premium-surface fit as a simpler alternative to distribution-based pricing.
- No direct fitting method or empirical comparison is presented.
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Full text
# Fit Option Premium Surface directly
# Fit Option Premium Surface directly
Goal - smooth surface to interpolate option premiums, over sparse and noisy market data. OTM puts and calls only, ITM ignored.
It's possible to approximate it indirectly, by choosing underlying distribution as
$$\text{PDF}(S, t) \sim \text{SkewStudentT}(\mu(t), \sigma(t), \lambda(t), v(t))$$
Where parameter functions have following structure and monotonicity and curvature constraints via first and second derivatives.
$$f(t|a,b) = a + b(log|sqrt|pow)(t)$$
And fit it to market (full algorithm described below), there will be only 8 params, additionally constrained with 1 and 2 derivatives. And even less, 6, if we fix the degree of freedom.
Then we can use it to calculate price of any option. But it feels a bit complicated, and slow (requires numerical integration for every point).
I wonder if there's a direct approach? Maybe some analytical formula for the premium surface, something like $\text{SkewStudentT}(\mu(t), \sigma(t), \lambda(t), v(t))$ but for the premium surface itself?
There're also Heston, SVJ and other models, but it feels even more complicated for the task of interpolation.
I'm currently using simple 2D linear interpolation, and looking for something better that not required very complicated model tuning.
### Fitting SkewStudentT to market
Then assume option price as european call premiums could be calculated as (and similar for puts):
$$C(K, T|\theta) = E[(S-K)^+]$$
And fit it to market data (each point would require recalculating numerical integral for $E[(S-K)^+]$):
$$\min L^2 \frac{C_{\text{model}}(K_i, T_i \mid \theta) - C_{\text{mkt}}(K_i, T_i)}{C_{\text{mkt}}(K_i, T_i)}$$
Then any option could be interpolated as
$$C(K, T|\theta) = E[(S-K)^+]$$Shown 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.