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We set out below backlinks to pages that contains analytical formulae for the payoffs, prices and option greeksof (European-style) vanilla set and phone possibilities and binary put and contact selections in a Black-Scholes world, see also e.g. Wilmott (2007). Thelinked pages also consist of even more hyperlinks to pages that allowusers to compute these costs and greeks both interactively (via direct input into ideal webpages) or programmatically e.g. within Microsoft Excel or equivalents (through the use of Net sent 'web services').


The input parameters employed are

K strike value

S cost of underlying

r interest price continuously compounded

q dividend produce continuously compounded

t time now

T time at maturity

sigma implied volatility (of cost of underlying)


Strictly speaking, the first Black-Scholes formulae use to vanilla European-style set and simply call choices that are not dividend bearing, i.e. have q . The formulae provided in the pages to which this knol inbound links refer to the Garman-Kohlhagen generalisations of the unique Black-Scholes formulae and to binary puts and calls as very well as to vanilla puts and calls.


See Notation for Black-Scholes Greeks for more notation applicable to the formulae granted under.


Vanilla Calls


Payoff, see MnBSCallPayoff

Selling price (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

Appeal (sensitivity forex software of delta to time), see MnBSCallCharm

Colour (sensitivity of gamma to time), see MnBSCallColour

Rho(curiosity) (sensitivity to fascination charge), 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 (value), see MnBSPutPrice

Delta (sensitivity to underlying), see MnBSPutDelta

Gamma (sensitivity of delta to underlying), see MnBSPutGamma

Speed (sensitivity of gamma to underlying), see MnBSPutSpeed

Theta (sensitivity to time), see MnBSPutTheta

Appeal (sensitivity of delta to time), see MnBSPutCharm

Colour (sensitivity of gamma to time), see MnBSPutColour

Rho(curiosity) (sensitivity to curiosity rate), see MnBSPutRhoInterest

Rho(dividend) (sensitivity to dividend yield), 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

Price (worth), 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

Allure (sensitivity of delta to time), see MnBSBinaryCallCharm

Color (sensitivity of gamma to time), see MnBSBinaryCallColour

Rho(fascination) (sensitivity to interest price), see MnBSBinaryCallRhoInterest

Rho(dividend) (sensitivity to dividend generate), see MnBSBinaryCallRhoDividend

Vega (sensitivity to volatility), see MnBSBinaryCallVega*

Vanna (sensitivity of forex market delta to volatility), see MnBSBinaryCallVanna*

Volga (or Vomma) (sensitivity of vega to volatility), see MnBSBinaryCallVolga*


Binary Puts


Payoff, see MnBSBinaryPutPayoff

Selling price (value), see MnBSBinaryPutPrice

Delta (sensitivity to underlying), see MnBSBinaryPutDelta

Gamma (sensitivity of delta to underlying), see MnBSBinaryPutGamma

Speed (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(curiosity) (sensitivity to interest fee), see MnBSBinaryPutRhoInterest

Rho(dividend) (sensitivity to dividend generate), 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, vanna and volga/vomma that entail partial differentials with respect to sigmaare in some feeling -invalid' in the context of Black-Scholes, due to the fact in its derivation we presume thatsigma is continuous. We may possibly interpret them alongside the lines of implementing to a product in which sigma was somewhat variable but or else was shut to frequent for all S, t, r, q and many others.. Vega,for instance, would then measure the sensitivity to changes in the mean amount of sigma. For some types of derivatives, e.g. binary puts and calls, it can then be very hard to interpret how these certain sensitivities ought to be recognized.

References


Wilmott, P. (2007). Regularly asked inquiries in quantitative finance. John Wiley & Sons, Ltd.
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