We set out below back links to pages that contains analytical formulae for the payoffs, rates and alternative greeksof (European-design) vanilla put and phone alternatives and binary set and simply call options in a Black-Scholes entire world, see also e.g. Wilmott (2007). Thelinked pages also contain additional hyperlinks to pages that allowusers to calculate these prices and greeks either interactively (through direct input into suitable webpages) or programmatically e.g. inside Microsoft Excel or equivalents (via the use of Internet delivered 'web services'). The input parameters utilized are K strike price tag S selling price of underlying r interest charge repeatedly compounded q dividend generate repeatedly compounded t time now T time at maturity sigma implied volatility (of price of buy stocks underlying) Strictly talking, the first Black-Scholes formulae use to vanilla European-design put and contact possibilities that are not dividend bearing, i.e. have q . The formulae presented in the pages to which this knol hyperlinks refer to the Garman-Kohlhagen generalisations of the original Black-Scholes formulae and to binary puts and calls as properly as to vanilla puts and calls. See Notation for Black-Scholes Greeks for even more notation pertinent to the formulae presented under. Vanilla Calls Payoff, see MnBSCallPayoff Cost (price), see MnBSCallPrice Delta (sensitivity to underlying), see MnBSCallDelta Gamma (sensitivity of delta to underlying), see MnBSCallGamma Velocity (sensitivity of gamma to underlying), see MnBSCallSpeed Theta (sensitivity to time), see MnBSCallTheta Allure (sensitivity penny stocks of delta to time), see MnBSCallCharm Colour (sensitivity of gamma to time), see MnBSCallColour Rho(fascination) (sensitivity to fascination fee), see MnBSCallRhoInterest Rho(dividend) (sensitivity to dividend produce), see MnBSCallRhoDividend Vega (sensitivity to volatility), see MnBSCallVega* Vanna (sensitivity of delta to volatility), see MnBSCallVanna* Volga (or Vomma) (sensitivity of vega to volatility), see MnBSCallVolga* Vanilla Puts Payoff, see MnBSPutPayoff Price tag (worth), see MnBSPutPrice Delta (sensitivity to underlying), see MnBSPutDelta Gamma (sensitivity of delta to underlying), see MnBSPutGamma Velocity (sensitivity of gamma to underlying), see MnBSPutSpeed Theta (sensitivity to time), see MnBSPutTheta Attraction (sensitivity of delta to time), see MnBSPutCharm Colour (sensitivity of gamma to time), see MnBSPutColour Rho(curiosity) (sensitivity to curiosity cedar finance amount), see MnBSPutRhoInterest Rho(dividend) (sensitivity to dividend produce), see MnBSPutRhoDividend Vega (sensitivity to volatility), see MnBSPutVega* Vanna (sensitivity of delta to volatility), see MnBSPutVanna* Volga (or Vomma) (sensitivity of vega to volatility), see MnBSPutVolga* Binary Calls Payoff, see MnBSBinaryCallPayoff Cost (price), see MnBSBinaryCallPrice Delta (sensitivity to underlying), see MnBSBinaryCallDelta Gamma (sensitivity of delta to underlying), see MnBSBinaryCallGamma Pace (sensitivity of gamma to underlying), see MnBSBinaryCallSpeed Theta (sensitivity to time), see MnBSBinaryCallTheta Appeal (sensitivity of delta to time), see MnBSBinaryCallCharm Color (sensitivity of gamma to time), see MnBSBinaryCallColour Rho(curiosity) (sensitivity to fascination amount), see MnBSBinaryCallRhoInterest Rho(dividend) (sensitivity to dividend generate), see MnBSBinaryCallRhoDividend Vega (sensitivity to volatility), see MnBSBinaryCallVega* Vanna (sensitivity commodity prices of delta to volatility), see MnBSBinaryCallVanna* Volga (or Vomma) (sensitivity of vega to volatility), see MnBSBinaryCallVolga* Binary Puts Payoff, see MnBSBinaryPutPayoff Price (value), see MnBSBinaryPutPrice Delta (sensitivity to underlying), see MnBSBinaryPutDelta Gamma (sensitivity of delta to underlying), see MnBSBinaryPutGamma Pace (sensitivity of gamma to underlying), see MnBSBinaryPutSpeed Theta (sensitivity to time), see MnBSBinaryPutTheta Attraction (sensitivity of delta to time), see MnBSBinaryPutCharm Colour (sensitivity of gamma to time), see MnBSBinaryPutColour Rho(interest) (sensitivity to interest fee), see MnBSBinaryPutRhoInterest Rho(dividend) (sensitivity to dividend yield), see MnBSBinaryPutRhoDividend Vega (sensitivity to volatility), see MnBSBinaryPutVega* Vanna (sensitivity of delta to volatility), see MnBSBinaryPutVanna* Volga (or Vomma) (sensitivity of vega to volatility), see MnBSBinaryPutVolga* * Greeks like vega, etfs vanna and volga/vomma that include partial differentials with respect to sigmaare in some sense -invalid' in the context of Black-Scholes, considering that in its derivation we presume thatsigma is consistent. We may interpret them alongside the lines of making use of to a design in which sigma was a bit variable but normally was near to continual for all S, t, r, q etcetera.. Vega,for instance, would then measure the sensitivity to adjustments in the indicate degree of sigma. For some forms of derivatives, e.g. binary puts and calls, it can then be very difficult to interpret how these unique sensitivities should be recognized. References Wilmott, P. (2007). Regularly asked concerns in quantitative finance. John Wiley & Sons, Ltd.