## 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.