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BevisE1955's Profile
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We set out below backlinks to pages containing analytical formulae for the payoffs, selling prices and option greeksof (European-fashion) vanilla place and get in touch with options and binary put and simply call possibilities in a Black-Scholes planet, see also e.g. Wilmott (2007). Thelinked pages also include further links to pages that allowusers to determine these costs and greeks either interactively (through direct input into acceptable webpages) or programmatically e.g. within Microsoft Excel or equivalents (by way of the use of World wide web sent 'web services').
The input parameters used are
K strike cost
S price tag of underlying
r curiosity charge continuously compounded
q dividend deliver continually compounded
t time now
T time at maturity
sigma implied volatility (of price of underlying)
Strictly talking, the authentic Black-Scholes formulae implement to vanilla European-fashion place and simply call alternatives that are not dividend bearing, i.e. have q . The formulae presented in the pages to which this knol links refer to the Garman-Kohlhagen generalisations of the original Black-Scholes formulae and to binary puts and calls as nicely as to vanilla puts and calls.
See Notation for Black-Scholes Greeks for further notation related to the formulae provided below.
Vanilla Calls
Payoff, see MnBSCallPayoff
Value (worth), see MnBSCallPrice
Delta (sensitivity to underlying), see MnBSCallDelta
Gamma (sensitivity of delta to underlying), see MnBSCallGamma
Speed (sensitivity of gamma to underlying), see MnBSCallSpeed
Theta (sensitivity to time), see MnBSCallTheta
Allure (sensitivity how options work of delta to time), see MnBSCallCharm
Color (sensitivity of gamma to time), see MnBSCallColour
Rho(fascination) (sensitivity to curiosity fee), see MnBSCallRhoInterest
Rho(dividend) (sensitivity to dividend deliver), 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
Charm (sensitivity of delta to time), see MnBSPutCharm
Colour (sensitivity of gamma to time), see MnBSPutColour
Rho(interest) (sensitivity to interest rate), see MnBSPutRhoInterest
Rho(dividend) (sensitivity to dividend generate), 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
Colour (sensitivity of gamma to time), see MnBSBinaryCallColour
Rho(curiosity) (sensitivity to interest fee), see MnBSBinaryCallRhoInterest
Rho(dividend) (sensitivity to dividend generate), see MnBSBinaryCallRhoDividend
Vega (sensitivity to volatility), see MnBSBinaryCallVega*
Vanna (sensitivity of options group 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
Charm (sensitivity of delta to time), see MnBSBinaryPutCharm
Color (sensitivity of gamma to time), see MnBSBinaryPutColour
Rho(interest) (sensitivity to fascination amount), see MnBSBinaryPutRhoInterest
Rho(dividend) (sensitivity to dividend deliver), 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 sensation -invalid' in the context of Black-Scholes, since in its derivation we suppose thatsigma is frequent. We could interpret them alongside the lines of implementing to a model in which sigma was a bit variable but otherwise was close to continuous for all S, t, r, q and many others.. Vega,for case in point, would then measure the sensitivity to alterations in the imply degree of sigma. For some types of derivatives, e.g. binary puts and calls, it can then be extremely challenging to interpret how these particular sensitivities need to be comprehended.
References
Wilmott, P. (2007). Usually asked concerns in quantitative finance. John Wiley & Sons, Ltd.
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