AnalyticApproach
List of valid values:
AnalyticEuropean
Subtype of Pricing Method
Corresponds to the QuantLib AnalyticEuropeanEngine.
It uses the Black-Scholes analytical formula for pricing european options. Web reference available here
AnalyticDividendEuropean
Subtype of Pricing Method
Corresponds to the QuantLib AnalyticDividendEuropeanEngine.
BlenmanClark
Subtype of Pricing Method
Arbitrage-free analytical formula for pricing primarily European options with payoff RSO
The call RSO price is given by
C(S,t,r,σ,δ;α,λ,β,K) = βe⁻ᵟᵗSN(d₁) - λe⁻ᵅᵟᵗYN(d₂)
The put RSO price is given by
P(S,t,r,σ,δ;α,λ,β,K) = - βe⁻ᵟᵗS[1-N(d₁)] + λe⁻ᵅᵟᵗY[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)σt¹ᐟ²]
d₂ = [(1-α)ln(S/K) - lnλ + lnβ + (1-α)(r-δ+σ²(α-½))t] / [(1-α)σt¹ᐟ²]
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)λe⁻ᵅᵟᵗ(Y/S)N(d₂) always > 0
put delta = dP/dS = P/S - (1-α)λe⁻ᵅᵟᵗ(Y/S)[1-N(d₂)] always < 0
call gamma = d²C/dS² = βn(d₁)e⁻ᵟᵗ/(Sσt¹ᐟ²) - λα(α-1)e⁻ᵅᵟᵗ(Y/S²)N(d₂) - αβn(d₁)e⁻ᵟᵗ/(Sσt¹ᐟ²) always > 0
put gamma = d²P/dS² = βn(d₁)e⁻ᵟᵗ/(Sσt¹ᐟ²) + λα(α-1)e⁻ᵅᵟᵗ(Y/S²)[1-N(d₂)] - αβn(d₁)e⁻ᵟᵗ/(Sσt¹ᐟ²) may be positive or negative
call vega = dC/dσ = (1-α)t¹ᐟ²βSe⁻ᵟᵗn(d₁) - λα(α-1)σteᵅᵟᵗYN(d₂) always > 0
put vega = dP/dσ = (1-α)t¹ᐟ²βSe⁻ᵟᵗn(d₁) + λα(α-1)σteᵅᵟᵗY[1-N(d₂)] may be positive or negative
call theta = dC/dt = -δβSe⁻ᵟᵗN(d₁) + αλδYe⁻ᵅᵟᵗN(d₂) + λ(1-α)(r+½ασ²)Ye⁻ᵅᵟᵗN(d₂) + βS(1-α)σn(d₁)e⁻ᵟᵗ/(2t¹ᐟ²)
put theta = dP/dt = δβSe⁻ᵟᵗ[1-N(d₁)] - αλδYe⁻ᵅᵟᵗ[1-N(d₂)] - λ(1-α)(r+½ασ²)Ye⁻ᵅᵟᵗ[1-N(d₂)] + βS(1-α)σn(d₁)e⁻ᵟᵗ/(2t¹ᐟ²)
call rho = dC/dr = λ(1-α)te⁻ᵅᵟᵗYN(d₂) always > 0
put rho = dP/dr = -λ(1-α)te⁻ᵅᵟᵗY[1-N(d₂)] always < 0
call warp = dC/dα = δλte⁻ᵅᵟᵗYN(d₂) - λSe⁻ᵅᵟᵗσt¹ᐟ²Yn(d₂) - YN(d₂)λe⁻ᵅᵟᵗ[ln(S/K)+(r+ασ²-½σ²)t] always < 0
put warp = dP/dα = -δλte⁻ᵅᵟᵗY[1-N(d₂)] - βSe⁻ᵟᵗσt¹ᐟ²n(d₁) - [1-N(d₂)]λe⁻ᵅᵟᵗ[Yln(S/K)+(r+ασ²-½σ²)t] always < 0
call lambda gearing = dC/dλ = -e⁻ᵅᵟᵗYN(d₂) always < 0
put lambda gearing = dP/dλ = e⁻ᵅᵟᵗY[1-N(d₂)] always > 0
call beta gearing = dC/dβ = Se⁻ᵟᵗN(d₁) always > 0
put beta gearing = dP/dβ = -Se⁻ᵟᵗ[1-N(d₁)] always < 0
call K gearing = dC/dK = -λ(1-α)e⁻ᵅᵟᵗ(Y/K)N(d₂) always < 0
put K gearing = dP/dK = λ(1-α)e⁻ᵅᵟᵗ(Y/K)[1-N(d₂)] always > 0
The following features are currently not supported:
American exercise, barriers, discrete dividends/storage costs.
AnalyticBSMHullWhite
Subtype of Pricing Method
Corresponds to the QuantLib AnalyticBSMHullWhiteEngine.
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
Subtype of Pricing Method
Corresponds to the QuantLib AnalyticHestonHullWhiteEngine.
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
Subtype of Pricing Method
Corresponds to the QuantLib AnalyticDigitalAmericanEngine.
AnalyticBarrier
Subtype of Pricing Method
Corresponds to the QuantLib AnalyticBarrierEngine.
AnalyticHeston
Subtype of Pricing Method
Corresponds to the QuantLib AnalyticHestonEngine.
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:
Complex Log Formula
Integration
and also of the absolute/relative tolerance, maximum number of evaluations and the Andersen Piterbarg epsilon wrt the Fourier integration.
AnalyticPTDHeston
Subtype of Pricing Method
Corresponds to the QuantLib AnalyticPTDHestonEngine, which is the Piecewise Time Dependent version of the regular AnalyticHestonEngine.
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
Subtype of Pricing Method
Corresponds to the QuantLib BatesEngine.
4-factor model, with a closed-form solution, driven by stochastic underlying price, volatility and jumps with random occurrence and size.
The underlying price process is modelled according to Bates Model
Integral
Subtype of Pricing Method
Corresponds to the QuantLib IntegralEngine.
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
BaroneAdesiWhaleyApprox
Subtype of Pricing Method
Corresponds to the QuantLib BaroneAdesiWhaleyApproximationEngine.
It uses an approximating semi-analytical formula for pricing american options. Web reference available here
BjerksundStenslandApprox
Subtype of Pricing Method
Corresponds to the QuantLib BjerksundStenslandApproximationEngine.
It uses an approximating semi-analytical formula for pricing american options. Web reference available here
JuQuadraticApprox
Subtype of Pricing Method
Corresponds to the QuantLib JuQuadraticApproximationEngine.
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
Subtype of Pricing Method
Corresponds to the QuantLib AnalyticGJRGARCHEngine.
The underlying price is modelled according to GJRGARCH Model
JumpDiffusion
Subtype of Pricing Method
Corresponds to the QuantLib JumpDiffusionEngine.
3-factor model, with a semi closed-form solution, driven by stochastic underlying price and jumps with random occurrence and size.
The underlying price process is modelled according to Merton76 Model