SIMD multithreaded Monte Carlo Options Pricer

Prices European options and calculates Greeks using Monte Carlo Simulations and achieves up to ~40x speed-up with multithreading and SIMD operations compared to single threading and scalar operations. This library requires nightly.

Available modules:

• [mc_simd] - pricing options with SIMD operations
  • [mc_simd::call_price] - calculate the price of a call option given strike, spot, risk-free rate, dividend, and time to expiry
  • [mc_simd::put_price] - calculate the price of a put option
  • [mc_simd::call_price_av] - calculate the price of a call option with reduced variance
  • [mc_simd::put_price_av] - calculate the price of a put option with reduced variance
  • [mc_simd::call_delta] - calculate Delta for call options
  • [mc_simd::put_delta] - calculate Delta for put options
  • [mc_simd::gamma] - calculate Gamma
  • [mc_simd::vega] - calculate Vega
  • [mc_simd::call_rho] - calculate Rho for call options
  • [mc_simd::put_rho] - calculate Rho for put options
  • [mc_simd::call_theta] - calculate Theta for call options
  • [mc_simd::put_theta] - calculate Theta for put options
• [mc] - pricing options with scalar operations, used to compare performance
  • [mc::call_price]
  • [mc::put_price]

Background

Monte Carlo simulations are one of the most popular methods of pricing financial options, especially when a closed form equation is not available. However, these simulations require a large number of trials to calculate the underlying's price over option's lifetime, which motivates the use of SIMD to speed up these calculations.

SIMD allows us to perform operations on multiple data with one instruction, which is especially helpful in the inner loop of the Monte Carlo simulation where we're doing the same set of operations n number of times.

So instead of this,

for _ in 0..num_trials {
    let mut count = 0;
    ...
    for step in 0..num_steps {
        count += step.ln();
        ...
    }
}

we can do something like this, arriving at a ~8x speed-up.

for _ in 0..num_trials/8 {
    let mut count = f32x8::splat(0.0);
    for step in 0..num_steps {
        count += f32x8::splat(step).ln();
        ...
    }
}

We make use of wide for SIMD-compatible data types and simd-rand for vectorized random number generation to get 6-8x speed-up when generating random numbers inside the inner-most loop of the Monte Carlo simulation as opposed to individually generating 8 random, normally distributed numbers inside the inner-most loop. There may be concerns of bias when generating a small sample of random numbers, but xoroshiro256++ is safe from this bias.

We also reduce the number of math operations inside the inner-most loop of the Monte Carlo simulation and use Fused Multiply-Add. rayon is used to parallelize the Monte Carlo trials and get an extra performance boost.

To calculate the Greeks, we get option prices using a Monte Carlo simulation and apply the finite difference method (central difference) to determine Delta, Gamma, Theta, Vega, and Rho.

Usage

[dependencies]
monte_carlo_options_simd = { git = "https://github.com/ashayp22/monte-carlo-options-simd" }

use monte_carlo_options_simd::*;
use wide::*;

fn main() {
    let spot : f32 = 100.0;
    let strike : f32 = 110.0;
    let volatility : f32 = 0.25;
    let risk_free_rate : f32 = 0.05;
    let years_to_expiry : f32 = 0.5;
    let dividend_yield : f32 = 0.02;
    let steps: f32 = 100.0;
    let num_trials: f32 = 1000.0;

    // Get the option price
    let call_option_price: f32 = mc_simd::call_price(spot, strike, volatility, risk_free_rate, years_to_expiry, dividend_yield, steps, num_trials);
}

You may need to set RUSTFLAGS to get the “the best” code possible for the machine that you’re working on.

export RUSTFLAGS="-C target-cpu=native -C opt-level=3"
cargo run
...

Performance

Calculated on a Macbook M1:

For 112 spots, 80 trials and 100 steps (~11x improvement in speed):

• mc::call_price(): ~33.649ms
• mc_simd::call_price(): ~3.29ms

For 112 spots, 1000 trials and 100 steps (~27x improvement in speed):

• mc::call_price(): ~419ms
• mc_simd::call_price(): ~15ms

For 112 spots, 10000 trials and 100 steps (~40x improvement in speed):

• mc::call_price(): ~4.26s
• mc_simd::call_price(): ~104ms

For 112 spots, 2000 trials and 1000 steps (~40x improvement in speed):

• mc::call_price(): ~8.22s
• mc_simd::call_price(): ~205ms

For 112 spots, 20000 trials and 100 steps (~40x improvement in speed):

• mc::call_price(): ~8.27s
• mc_simd::call_price(): ~207ms

Notice that mc_simd performance increases compared to mc as the number of trials and steps get larger.

Resources

• Math behind Monte Carlo simulations for pricing options
• Another example of Monte Carlo simulations for pricing options
• Designing a SIMD Algorithm from Scratch

Credits

• Hassam Uddin for the suggestion to add multithreading and look at xoshiro256++ bias

License

GNU GPL v3