> For the complete documentation index, see [llms.txt](https://docs.knox.premia.finance/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.knox.premia.finance/appendix/option-pricing-formula.md).

# Option Pricing Formula

* $$S$$ - spot price
* $$\Delta$$ - option delta
* $$\Delta\_{offset}$$ - delta % offset
* $$\epsilon$$ - delta offset
* $$K\_{\Delta}^{call/put}$$ - delta strike computed using $$\Delta$$
  * $$K\_{\Delta}^{call}$$ - delta strike computed for a call option
  * $$K\_{\Delta}^{put}$$ - delta strike computed for a put option.
* $$\tau$$ - time to maturity
* $$g : \Delta \rightarrow K\_{\Delta}^{put/call}$$ - delta strike function which maps the delta parameter $$\Delta$$ to the strike price
* $$f\_{oracle}: (S,K,\tau) \rightarrow \sigma$$ - oracle function which maps the parameters $$S,K,\tau$$ to implied volatility $$\sigma$$

The price curve can be calculated as follows:

$$
K\_{\Delta}^{put/call} = g(\Delta)
$$

$$
K\_{\Delta - \epsilon}^{put/call} = g(\Delta - \epsilon)
$$

$$
\sigma\_{\max} = f\_{oracle}(S,K\_{\Delta}^{put/call},\tau)
$$

$$
\sigma\_{\min} = \sigma\_{\max} \pm \sigma\_{\max}(\Delta\_{offset})
$$

$$
t\_{percent} = \frac{t\_i - t\_0}{t\_{total}}
$$

$$
\sigma\_{\text{market}} = \sigma\_{\max} - t\_{\text{percent}}\cdot(\sigma\_{\max} - \sigma\_{\min})
$$