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:

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

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

$$\sigma_{\max} = f_{oracle}(S,K_{\Delta}^{put/call},\tau)$$

$$\sigma_{\min} = \sigma_{\max} \pm \sigma_{\max}(\Delta_{offset})$$

$$t_{percent} = \frac{t_i - t_0}{t_{total}}$$

$$\sigma_{\text{market}} = \sigma_{\max} - t_{\text{percent}}\cdot(\sigma_{\max} - \sigma_{\min})$$