1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171

//! Carr-Madan approach for an option using the underlying's 
//! characteristic function. Some useful characteristic functions
//! are provided in the [cf_functions](https://crates.io/crates/cf_functions) repository.
//! Carr and Madan's approach works well for computing a large number of equidistant 
//! strikes. [Link to Carr-Madan paper](https://wwwf.imperial.ac.uk/~ajacquie/IC_Num_Methods/IC_Num_Methods_Docs/Literature/CarrMadan.pdf).
//! 
extern crate rustfft;
extern crate num;
extern crate black_scholes;
extern crate rayon;
#[macro_use]
#[cfg(test)]
extern crate approx;

use num::traits::{Zero};
use rustfft::algorithm::Radix4;
use rustfft::FFT;
use rustfft::num_complex::Complex;
use std::f64::consts::PI;
use rayon::prelude::*;

/**
@ada the distance between nodes in the U domain
@returns the maximum value of the U domain
*/
fn get_max_k(ada: f64)->f64{
    PI/ada
}
/**
@numSteps the number of steps used to discretize the x and u domains
@returns the distance between nodes in the X domain
*/
fn get_lambda(num_steps:usize, b:f64)->f64{
    2.0*b/(num_steps as f64)
}

fn get_x(xmin:f64, dx:f64, index:usize)->f64{
    xmin+dx*(index as f64)
}

fn get_k_at_index(b:f64, lambda:f64, s0:f64, index:usize)->f64{
    s0*get_x(-b, lambda, index).exp()
}

/// Returns iterator which produces strikes
/// 
/// # Examples
/// ```
/// let eta = 0.04;
/// let s0 = 50.0;
/// let num_x = 256;
/// let strikes = carr_madan::get_strikes(eta, s0, num_x);
/// ```
pub fn get_strikes(
    eta:f64, s0:f64, num_x:usize
)->impl IndexedParallelIterator<Item = f64>{ 
    let b=get_max_k(eta);
    let lambda=get_lambda(num_x, b);
    (0..num_x).into_par_iter().map(move |index|get_k_at_index(b, lambda, s0, index))
}


fn call_aug<T>(v:&Complex<f64>, alpha:f64, cf:T)->Complex<f64> 
    where T: Fn(&Complex<f64>)->Complex<f64>
{ //used for Carr-Madan approach...v is typically complex
    let aug_u=v+(alpha+1.0);
    cf(&aug_u)/(alpha*alpha+alpha+v*v+(2.0*alpha+1.0)*v)
}

fn carr_madan_g<T, S>(num_steps:usize, eta:f64, alpha:f64, s0:f64, discount:f64, m_out:S, aug_cf:T)->Vec<(f64, f64)> 
    where T: Fn(&Complex<f64>, f64)->Complex<f64>+std::marker::Sync, S:Fn(f64, usize)->f64+std::marker::Sync
{
    let b=get_max_k(eta);
    let lambda=get_lambda(num_steps, b);
    let fft = Radix4::new(num_steps, false);
    let mut cmpl:  Vec<Complex<f64> > =(0..num_steps).into_par_iter().map(|index| {
        let pm=if index%2==0 {-1.0} else {1.0};
        let u=Complex::<f64>::new(0.0, (index as f64)*eta);
        let aug_u=Complex::<f64>::new(0.0, b*(index as f64)*eta);
        let f_answer=discount*aug_cf(&u, alpha)*aug_u.exp()*(3.0+pm);
        if index==0 {f_answer*0.5} else {f_answer}
    }).collect();
    let mut output:  Vec<Complex<f64> >=vec![Complex::zero(); num_steps];//why do I have to initialize this?
    fft.process(&mut cmpl, &mut output);
    get_strikes(eta, s0, num_steps)
        .zip(output.par_iter()
            .enumerate()
            .map(|(index, &x)| m_out(s0*x.re*(-alpha*get_x(-b, lambda, index)).exp()*eta/(PI*3.0), index))
        ).collect()
}
/// Returns call prices over a series of strikes
/// 
/// # Examples
/// ```
/// extern crate rustfft;
/// use rustfft::num_complex::Complex;
/// extern crate carr_madan;
/// # fn main() {
/// let alpha = 1.5;
/// let eta = 0.25;
/// let num_steps = 256;
/// let s0 = 50.0;
/// let r:f64 = 0.05;
/// let t:f64 = 1.5;
/// let sig = 0.4;
/// let discount = (-r*t).exp();
/// let bscf=|u:&Complex<f64>| ((r-sig*sig*0.5)*t*u+sig*sig*t*u*u*0.5).exp(); 
/// let call_prices = carr_madan::carr_madan_call(
///     num_steps, eta, alpha, s0,
///     discount, &bscf
/// ); //first element is strike, second is call price
/// # }
/// ```
pub fn carr_madan_call<T>(
    num_steps:usize, eta:f64, alpha:f64, 
    s0:f64, discount:f64, 
    cf:&T
)->Vec<(f64, f64)>
    where T:Fn(&Complex<f64>)->Complex<f64>+std::marker::Sync
{
    carr_madan_g(num_steps, eta, alpha, s0, discount, 
        |x, _| x,
        |&x, alpha| call_aug(&x, alpha, &cf) 
    )
}

#[cfg(test)]
mod tests {
    use super::*;
    #[test]
    fn get_x_returns_correctly() {
        let x=get_x(-5.0, 1.0, 3);
        assert_eq!(x, -2.0);
    }
    #[test]
    fn get_strikes_returns_correctly() {
        let strikes=get_strikes(1.0, 1.0, 5);
        assert_eq!(strikes.len(), 5);
    }
    #[test]
    fn carr_madan_call_returns_correctly() {
        let r=0.05;
        let sig=0.3;
        let t=1.0;
        let s0=50.0;
        let discount=(-r*t as f64).exp();
        let bscf=|u:&Complex<f64>| ((r-sig*sig*0.5)*t*u+sig*sig*t*u*u*0.5).exp();
        let num_x=(2 as usize).pow(10);
        let eta=0.25;
        let alpha=1.5;
        let my_options_price=carr_madan_call(
            num_x,  
            eta,
            alpha,
            s0, 
            discount,
            &bscf
        );
        let min_n=num_x/4;
        let max_n=num_x-num_x/4;
        for i in min_n..max_n{
            let (strike, price)=my_options_price[i];
            assert_abs_diff_eq!(
                black_scholes::call(s0, strike, discount, sig),
                price,
                epsilon=0.001

            );
        }
    }
}