Option Pricing Formula

  • ​
    SS
    - spot price
  • ​
    Δ\Delta
    - option delta
  • ​
    Δoffset\Delta_{offset}
    - delta % offset
  • ​
    ϵ\epsilon
    - delta offset
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    KΔcall/putK_{\Delta}^{call/put}
    - delta strike computed using
    Δ\Delta
    ​
    • ​
      KΔcallK_{\Delta}^{call}
      - delta strike computed for a call option
    • ​
      KΔputK_{\Delta}^{put}
      - delta strike computed for a put option.
  • ​
    τ\tau
    - time to maturity
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    g:Δ→KΔput/callg : \Delta \rightarrow K_{\Delta}^{put/call}
    - delta strike function which maps the delta parameter
    Δ\Delta
    to the strike price
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    foracle:(S,K,τ)→σf_{oracle}: (S,K,\tau) \rightarrow \sigma
    - oracle function which maps the parameters
    S,K,τS,K,\tau
    to implied volatility
    σ\sigma
    ​
The price curve can be calculated as follows:
KΔput/call=g(Δ)K_{\Delta}^{put/call} = g(\Delta)
KΔ−ϵput/call=g(Δ−ϵ)K_{\Delta - \epsilon}^{put/call} = g(\Delta - \epsilon)
σmax⁡=foracle(S,KΔput/call,τ)\sigma_{\max} = f_{oracle}(S,K_{\Delta}^{put/call},\tau)
σmin⁡=σmax⁡±σmax⁡(Δoffset)\sigma_{\min} = \sigma_{\max} \pm \sigma_{\max}(\Delta_{offset})
tpercent=ti−t0ttotalt_{percent} = \frac{t_i - t_0}{t_{total}}
σmarket=σmax⁡−tpercent⋅(σmax⁡−σmin⁡)\sigma_{\text{market}} = \sigma_{\max} - t_{\text{percent}}\cdot(\sigma_{\max} - \sigma_{\min})