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AnalyticApproach

AnalyticEuropean
Corresponds to the QuantLib AnalyticEuropean Engine.
It uses the Black-Scholes analytical formula for pricing european options. Web reference available
here
AnalyticDividendEuropean
Corresponds to the QuantLib AnalyticDividendEuropean Engine.
BlenmanClark
Minimum required license: Student
Arbitrage-free analytical formula for pricing primarily European options with payoff
Payoff::Payoff Type::RSO
The call RSO price is given by
C(S,t,r,σ,δ;α,λ,β,K) = βexp(-δt)SN(d1) - λexp(-αδt)YN(d2)
The put RSO price is given by
P(S,t,r,σ,δ;α,λ,β,K) = - βexp(-δt)S[1-N(d1)] + λexp(-αδt)Y[1-N(d2)]
where
S is the initial underlying price
t is the time to option expiry in annual units
r is the effective flat continuously compounded interest rate.
σ is the flat lognormal volatility of the price of the option's underlying.
δ is the flat continuously compounded dividend yield of the option's underlying.
d1 = [(1-α)ln(S/K) - lnλ + lnβ + (1-α)(r-δ+σ²/2)t] / [(1-α)σt^½]
d2 = [(1-α)ln(S/K) - lnλ + lnβ + (1-α)(r-δ+σ²(α-½))t] / [(1-α)σt^½]
Y = [K^(1-α)](S^α)exp[(α-1)(r+ασ²/2)t]
and N(.) denotes the cumulative standard normal distribution function.

The greeks can be calculated by the following formulas, where n(.) denotes the standard normal density function:

call delta = dC/dS = C/S + (1-α)λexp(-αδt)(Y/S)N(d2) always > 0
put delta = dP/dS = P/S - (1-α)λexp(-αδt)(Y/S)[1-N(d2)] always < 0

call gamma = d²C/dS² = βn(d1)exp(-δt)/(Sσt^½) - λα(α-1)exp(-αδt)(Y/S²)N(d2) - αβn(d1)exp(-δt)/(Sσt^½) always > 0
put gamma = d²P/dS² = βn(d1)exp(-δt)/(Sσt^½) + λα(α-1)exp(-αδt)(Y/S²)[1-N(d2)] - αβn(d1)exp(-δt)/(Sσt^½) may be positive or negative

call vega = dC/dσ = (1-α)t^½βSexp(-δt)n(d1) - λα(α-1)σtexp(αδt)YN(d2) always > 0
put vega = dP/dσ = (1-α)t^½βSexp(-δt)n(d1) + λα(α-1)σtexp(αδt)Y[1-N(d2)] may be positive or negative

call theta = dC/dt = -δβSexp(-δt)N(d1) + αλδYexp(-αδt)N(d2) + λ(1-α)(r+½ασ²)Yexp(-αδt)N(d2) + βS(1-α)σn(d1)exp(-δt)/(2t^½)
put theta = dP/dt = δβSexp(-δt)[1-N(d1)] - αλδYexp(-αδt)[1-N(d2)] - λ(1-α)(r+½ασ²)Yexp(-αδt)[1-N(d2)] + βS(1-α)σn(d1)exp(-δt)/(2t^½)

call rho = dC/dr = λ(1-α)texp(-αδt)YN(d2) always > 0
put rho = dP/dr = -λ(1-α)texp(-αδt)Y[1-N(d2)] always < 0

call warp = dC/dα = δλtexp(-αδt)YN(d2) - λSexp(-αδt)σt^½Yn(d2) - YN(d2)λexp(-αδt)[ln(S/K)+(r+ασ²-½σ²)t] always < 0
put warp = dP/dα = -δλtexp(-αδt)Y[1-N(d2)] - βSexp(-δt)σt^½n(d1) - [1-N(d2)]λexp(-αδt)[Yln(S/K)+(r+ασ²-½σ²)t] always < 0

call lambda gearing = dC/dλ = -exp(-αδt)YN(d2) always < 0
put lambda gearing = dP/dλ = exp(-αδt)Y[1-N(d2)] always > 0

call beta gearing = dC/dβ = Sexp(-δt)N(d1) always > 0
put beta gearing = dP/dβ = -Sexp(-δt)[1-N(d1)] always < 0

call K gearing = dC/dK = -λ(1-α)exp(-αδt)(Y/K)N(d2) always < 0
put K gearing = dP/dK = λ(1-α)exp(-αδt)(Y/K)[1-N(d2)] always > 0

The following features are currently not supported:
American exercise, barriers, discrete dividends/storage costs.
AnalyticBSMHullWhite
Minimum required license: Student
Corresponds to the QuantLib AnalyticBSMHullWhite Engine.
2-factor model, with a closed-form solution, driven by stochastic underlying price and interest rates.
In particular, the underlying price is modelled to follow a Black-Scholes type, lognormal diffusion whereas the interest rate is modelled according to
Hull White Model
The two processes are correlated with a given flat correlation number.
AnalyticHestonHullWhite
Minimum required license: Standard
Corresponds to the QuantLib AnalyticHestonHullWhite Engine.
3-factor model, with a semi closed-form solution for call options, driven by stochastic underlying price, volatility and interest rates.
In particular, the underlying price is modelled to follow a Heston stochastic volatility process as in
Heston Model, whereas the interest rate is modelled according to Hull White Model
References: Karel in't Hout, Joris Bierkens, Antoine von der Ploeg, Joe in't Panhuis.
AnalyticDigitalAmerican
Minimum required license: Basic
Corresponds to the QuantLib AnalyticDigitalAmerican Engine.
AnalyticBarrier
Minimum required license: Basic
Corresponds to the QuantLib AnalyticBarrier Engine.
AnalyticHeston
Minimum required license: Student
Corresponds to the QuantLib AnalyticHeston Engine.
2-factor model, with a closed-form solution, driven by stochastic underlying price and volatility.
The underlying price process is modelled according to
Heston Model
Allows the specification of:
Model[Vanilla Option]::Complex Log Formula
Model[Vanilla Option]::Integration
and also of the absolute/relative tolerance, maximum number of evaluations and the Andersen Piterbarg epsilon wrt the Fourier integration.
AnalyticPTDHeston
Minimum required license: Standard
Corresponds to the QuantLib AnalyticPTDHeston Engine, which is the Piecewise Time Dependent version of the regular AnalyticHeston Engine.
2-factor model, with a semi closed-form solution, driven by stochastic underlying price and volatility.
The underlying price process is modelled according to
PTD Heston Model
Bates
Minimum required license: Student
Corresponds to the QuantLib Bates Engine.
4-factor model, with a closed-form solution, driven by stochastic underlying price, volatility and jumps with random occurence and size.
The underlying price process is modelled according to
Bates Model
Integral
Minimum required license: Standard
Corresponds to the QuantLib Integral Engine.
It prices european options through numerical computation of the integral of the payoff function over all possible stock price states at expiry. Web reference available
here
Corresponds to the QuantLib BaroneAdesiWhaleyApproximation Engine.
It uses an approximating semi-analytical formula for pricing american options. Web reference available
here
BjerksundStenslandApprox
Minimum required license: Student
Corresponds to the QuantLib BjerksundStenslandApproximation Engine.
It uses an approximating semi-analytical formula for pricing american options. Web reference available
here
Minimum required license: Standard
Corresponds to the QuantLib JuQuadraticApproximation Engine.
It uses an approximating semi-analytical formula for pricing american options. Web reference available
here
Warning: Barone-Adesi-Whaley critical commodity price calculation is used.
It has not been modified to see whether the method of Ju is faster.
Ju does not say how he solves the equation for the critical stock price, e.g. Newton method.
He just gives the solution.
The method of BAW gives answers to the same accuracy as in Ju (1999).
AnalyticGJRGARCH
Minimum required license: Standard
Corresponds to the QuantLib AnalyticGJRGARCH Engine.
The underlying price is modelled according to
GJRGARCH Model
JumpDiffusion
Minimum required license: Student
Corresponds to the QuantLib JumpDiffusion Engine.
3-factor model, with a semi closed-form solution, driven by stochastic underlying price and jumps with random occurence and size.
The underlying price process is modelled according to
Merton76 Model