jetblack-options 0.2.0

//! # Option valuations using a trinomial tree.
//!
//! The following arguments are common.
//!
//! * is_european (bool): True for European, false for American.
//! * is_call (bool): True for a call, false for a put.
//! * S (f64): The current asset price.
//! * K (f64): The option strike price
//! * T (f64): The time to maturity of the option in years.
//! * r (f64): The risk free rate.
//! * b (f64): The cost of carry of the asset.
//! * v (f64): The volatility of the asset.
//! * n (usize): The number of the steps in the tree.
//! * p (f64): The option price.
//! * max_iterations (usize): The maximum number of iterations before a price is returned.
//! * epsilon (f64): The largest acceptable error.

use libm::{exp, fmax, sqrt};
use std::cmp::max;

use crate::{fdm::FdmWithCarry, implied_volatility::solve_ivol};

fn sqr(x: f64) -> f64 {
    x * x
}

fn ipow(x: f64, p: i32) -> f64 {
    let mut sum = 1.0;
    for _ in 0..p {
        sum *= x;
    }
    sum
}

/// The greeks calculated by the model.
pub struct Greeks {
    pub price: f64,
    pub delta: f64,
    pub gamma: f64,
    pub theta: f64,
}

/// A trinomial tree options pricer returning the price and some greeks.
#[allow(non_snake_case)]
pub fn greeks(
    is_european: bool,
    is_call: bool,
    S: f64,
    K: f64,
    T: f64,
    r: f64,
    b: f64,
    v: f64,
    n: usize,
) -> Greeks {
    let z = if is_call { 1.0 } else { -1.0 };

    let dT = T / n as f64;
    let u = exp(v * sqrt(2.0 * dT));
    let d = exp(-v * sqrt(2.0 * dT));
    let pu = sqr((exp(b * dT / 2.0) - exp(-v * sqrt(dT / 2.0)))
        / (exp(v * sqrt(dT / 2.0)) - exp(-v * sqrt(dT / 2.0))));
    let pd = sqr((exp(v * sqrt(dT / 2.0)) - exp(b * dT / 2.0))
        / (exp(v * sqrt(dT / 2.0)) - exp(-v * sqrt(dT / 2.0))));
    let pm = 1.0 - pu - pd;
    let Df = exp(-r * dT);

    let mut option_value = vec![0.0; 1 + 2 * n];
    for i in 0..option_value.len() {
        let I = i as i32;
        let N = n as i32;
        option_value[i] = fmax(
            0.0,
            z * (S * ipow(u, max(I - N, 0)) * ipow(d, max(N - I, 0)) - K),
        );
    }

    let mut delta = f64::NAN;
    let mut gamma = f64::NAN;
    let mut theta = f64::NAN;

    for j in (0..n).rev() {
        for i in 0..=(j * 2) {
            option_value[i] =
                (pu * option_value[i + 2] + pm * option_value[i + 1] + pd * option_value[i]) * Df;

            if is_european {
                let I = i as i32;
                let J = j as i32;
                option_value[i] = fmax(
                    z * (S * ipow(u, max(I - J, 0)) * ipow(d, max(J - I, 0)) - K),
                    option_value[i],
                );
            }
        }

        if j == 1 {
            delta = (option_value[2] - option_value[0]) / (S * u - S * d);
            gamma = ((option_value[2] - option_value[1]) / (S * u - S)
                - (option_value[1] - option_value[0]) / (S - S * d))
                / (0.5 * (S * u - S * d));
            theta = option_value[1];
        }
    }

    theta = (theta - option_value[0]) / dT / 365.0;

    return Greeks {
        price: option_value[0],
        delta,
        gamma,
        theta,
    };
}

/// Calculate the price of an option using a trinomial tree.
#[allow(non_snake_case)]
pub fn price(
    is_european: bool,
    is_call: bool,
    S: f64,
    K: f64,
    T: f64,
    r: f64,
    b: f64,
    v: f64,
    n: usize,
) -> f64 {
    greeks(is_european, is_call, S, K, T, r, b, v, n).price
}

/// Calculate the volatility of an option that is implied by the price.
#[allow(non_snake_case)]
pub fn ivol(
    is_european: bool,
    is_call: bool,
    S: f64,
    K: f64,
    T: f64,
    r: f64,
    b: f64,
    p: f64,
    n: usize,
    max_iterations: usize,
    epsilon: f64,
) -> f64 {
    solve_ivol(
        p,
        |v| price(is_european, is_call, S, K, T, r, b, v, n),
        max_iterations,
        epsilon,
    )
}

/// Return a struct to calculate greeks numerically using finite difference methods.
pub fn fdm_greeks(is_european: bool, is_call: bool, n: usize) -> FdmWithCarry {
    #[allow(non_snake_case)]
    FdmWithCarry::new(move |S: f64, K: f64, T: f64, r: f64, b: f64, v: f64| {
        price(is_european, is_call, S, K, T, r, b, v, n)
    })
}

#[cfg(test)]
mod tests {
    use libm::fabs;

    use super::*;

    fn is_close_to(actual: f64, expected: f64, threshold: f64) -> bool {
        let diff = fabs(actual - expected);
        diff < threshold
    }

    #[test]
    fn it_should_calc_price() {
        #[allow(non_snake_case)]
        for (is_european, is_call, S, K, r, q, T, v, price, threshold) in [
            (
                true,
                true,
                110.0,
                100.0,
                0.1,
                0.08,
                6.0 / 12.0,
                0.125,
                11.073487046391277,
                1e-12,
            ),
            (
                true,
                false,
                110.0,
                100.0,
                0.1,
                0.08,
                6.0 / 12.0,
                0.125,
                0.5177978858542516,
                1e-12,
            ),
            (
                true,
                true,
                100.0,
                100.0,
                0.1,
                0.08,
                6.0 / 12.0,
                0.125,
                3.8675119431883433,
                1e-12,
            ),
            (
                true,
                false,
                100.0,
                100.0,
                0.1,
                0.08,
                6.0 / 12.0,
                0.125,
                3.0348490093800793,
                1e-12,
            ),
            (
                true,
                true,
                100.0,
                110.0,
                0.1,
                0.08,
                6.0 / 12.0,
                0.125,
                0.7880335120168521,
                1e-12,
            ),
            (
                true,
                false,
                100.0,
                110.0,
                0.1,
                0.08,
                6.0 / 12.0,
                0.125,
                10.099148293480416,
                1e-12,
            ),
            (
                false,
                true,
                110.0,
                100.0,
                0.1,
                0.08,
                6.0 / 12.0,
                0.125,
                11.069621888515023,
                1e-12,
            ),
            (
                false,
                false,
                110.0,
                100.0,
                0.1,
                0.08,
                6.0 / 12.0,
                0.125,
                0.5057260318285591,
                1e-12,
            ),
            (
                false,
                true,
                100.0,
                100.0,
                0.1,
                0.08,
                6.0 / 12.0,
                0.125,
                3.867381427247627,
                1e-12,
            ),
            (
                false,
                false,
                100.0,
                100.0,
                0.1,
                0.08,
                6.0 / 12.0,
                0.125,
                2.911379962084615,
                1e-12,
            ),
            (
                false,
                true,
                100.0,
                110.0,
                0.1,
                0.08,
                6.0 / 12.0,
                0.125,
                0.7880316003855702,
                1e-12,
            ),
            (
                false,
                false,
                100.0,
                110.0,
                0.1,
                0.08,
                6.0 / 12.0,
                0.125,
                9.344324380229715,
                1e-12,
            ),
        ] {
            let b = r - q;
            let actual = greeks(is_european, is_call, S, K, T, r, b, v, 200);
            assert!(
                is_close_to(actual.price, price, threshold),
                "price({}, {}, {}, {}, {}, {}, {}, {}, {})",
                is_european,
                is_call,
                S,
                K,
                T,
                r,
                b,
                v,
                200
            );
        }
    }
}