Added carr madan for generic distributions

Author: aleCombi

examples/carr_madan.jl

@@ -17,7 +17,7 @@ payoff = VanillaOption(strike, expiry, Hedgehog2.European(), Hedgehog2.Call(), H
 # Define carr madan method
 boundary = 32
 α = 1
-method = Hedgehog2.CarrMadanBS(α, boundary)
+method = Hedgehog2.CarrMadan(α, boundary; market_inputs=market_inputs)
 
 # Define pricer
 carr_madan_pricer = Pricer(payoff, market_inputs, method)

src/pricing_methods/carr_madan.jl

@@ -1,27 +1,30 @@
-using QuadGK, Dates
+using QuadGK, Dates, Distributions
 
 # at the moment, only for black scholes. The dynamics should be use to get more generality.
-struct CarrMadanBS <: AbstractPricingMethod 
+struct CarrMadan <: AbstractPricingMethod
     α 
     bound
+    log_distribution
 end
 
-function characteristic_function(t, market_inputs::BlackScholesInputs, method::CarrMadanBS)
-    r = market_inputs.rate
-    σ = market_inputs.sigma
-    return u -> exp(im * (r - σ^2 / 2) * u * t - u^2 * t * σ^2 / 2)
+function CarrMadan(α, bound; market_inputs::BlackScholesInputs)
+    distribution(t) = Normal((market_inputs.rate - market_inputs.sigma^2 / 2) * t, market_inputs.sigma * sqrt(t))
+    return CarrMadan(α, bound, distribution)
 end
 
-function compute_price(payoff::VanillaOption{European, Call, Spot}, market_inputs::BlackScholesInputs, method::CarrMadanBS)
+# in distribution.jl t is Real, hence we need to redefine it.
+cf(d::Normal, t) = exp(im * t * d.μ - d.σ^2 / 2 * t^2)
+
+function compute_price(payoff::VanillaOption{European, Call, Spot}, market_inputs::BlackScholesInputs, method::CarrMadan)
     damp = exp(- method.α * log(payoff.strike)) / 2π
     T = Dates.value(payoff.expiry - market_inputs.referenceDate) / 365
-    ϕ = characteristic_function(T, market_inputs, method)
+    ϕ(u) = cf(method.log_distribution(T), u)
     integrand(v) = call_transform(market_inputs.rate, T, ϕ, v, method) * exp(- im * v * log(payoff.strike))
     integral, error = quadgk(integrand, -method.bound, method.bound)
     return real(damp * integral)
 end
 
-function call_transform(rate, time, ϕ, v, method::CarrMadanBS)
+function call_transform(rate, time, ϕ, v, method::CarrMadan)
     numerator = exp(- rate * time) * ϕ(v - (method.α + 1)im)
     denominator = method.α^2 + method.α - v^2 + v * (2 * method.α + 1)im
     return numerator / denominator